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y(mak)n(e)h(the)i(follo)n(wing)f(h)n(yp)r(otheses:)506 5010 y FA(\(H0\))41 b Fr(a)788 5022 y Fp(ij)870 5010 y Fq(2)23 b Fr(C)1017 4980 y Fs(\026)-37 b Fp(n)1059 5010 y Fu(\()p FA(R)1163 4980 y Fp(d)1201 5010 y Fu(\))28 b(for)f(1)c Fq(\024)f Fr(i;)14 b(j)28 b Fq(\024)23 b Fr(d)p Fu(;)506 5113 y FA(\(H1\))41 b Fr(\025)p Fq(j)p Fr(\020)6 b Fq(j)24 b(\024)f Fr(a)1036 5125 y Fp(ij)1094 5113 y Fu(\()p Fr(x)p Fu(\))14 b Fr(\020)1255 5125 y Fp(i)1284 5113 y Fr(\020)1320 5125 y Fp(j)1379 5113 y Fq(\024)22 b Fu(\003)p Fq(j)p Fr(\020)6 b Fq(j)p Fu(,)28 b(for)f(an)n(y)g Fr(\020)6 b(;)14 b(x)24 b Fq(2)f FA(R)2247 5083 y Fp(d)2286 5113 y Fu(;)506 5216 y FA(\(H2\))41 b Fr(F)35 b Fq(2)24 b Fr(C)980 5185 y Fs(\026)-37 b Fp(n)1021 5216 y Fu(\()p FA(R)1125 5185 y Fp(d)1182 5216 y Fq(\002)18 b FA(R)p Fu(\),)28 b Fr(F)12 b Fu(\()p Fr(x;)i Fq(\006)p Fu(1\))23 b(=)g(0,)k(0)22 b Fq(\024)h Fr(F)12 b Fu(\()p Fr(x;)i(\026)p Fu(\))24 b Fq(\024)f Fu(\003,)k(for)g(an)n(y)g Fr(x)c Fq(2)h FA(R)3083 5185 y Fp(d)3121 5216 y Fu(,)k Fr(\026)23 b Fq(2)g FA(R)p Fu(;)p eop %%Page: 3 3 3 2 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)541 b(3)506 450 y FA(\(H3\))41 b Fu(F)-7 b(or)40 b(an)n(y)f Fr(\022)47 b Fq(2)d Fu([0)p Fr(;)14 b Fu(1\),)43 b(inf)1603 465 y Fm(j)p Fp(\026)p Fm(j\024)p Fp(\022)1786 450 y Fr(F)12 b Fu(\()p Fr(x;)i(\026)p Fu(\))45 b Fq(\025)f Fr(\015)5 b Fu(\()p Fr(\022)r Fu(\),)44 b(where)39 b Fr(\015)5 b Fu(\()p Fr(\022)r Fu(\))41 b(is)f(a)g(decreasing,)744 550 y(strictly)28 b(p)r(ositiv)n(e)f(function)h(in)g(the)g(in)n(terv)-5 b(al)27 b([0)p Fr(;)14 b Fu(1\);)506 650 y FA(\(H4\))41 b Fu(\003)14 b(\(1)k Fq(\000)g(j)p Fr(\026)p Fq(j)p Fu(\))1119 620 y Fs(2)1180 650 y Fq(\025)k Fr(F)12 b Fu(\()p Fr(x;)i(\026)p Fu(\))24 b Fq(\025)f Fr(\025)14 b Fu(\(1)19 b Fq(\000)f(j)p Fr(\026)p Fq(j)p Fu(\))2008 620 y Fs(2)2045 650 y Fu(,)28 b(if)g Fq(j)p Fr(\026)p Fq(j)23 b(2)h Fu(\(1)18 b Fq(\000)g Fr(\025;)c Fu(1\);)506 749 y FA(\(H5\))41 b Fr(@)788 761 y Fp(\026)833 749 y Fr(F)12 b Fu(\()p Fr(x;)i Fq(\000)p Fu(1)k(+)g Fr(s)p Fu(\))23 b Fq(\025)g Fr(\025)14 b(s)28 b Fu(and)f Fr(@)1738 761 y Fp(\026)1783 749 y Fr(F)12 b Fu(\()p Fr(x;)i Fu(1)k Fq(\000)g Fr(s)p Fu(\))24 b Fq(\024)e(\000)p Fr(\025)14 b(s)p Fu(,)28 b(for)f(an)n(y)g Fr(s)c Fq(2)g Fu(\(0)p Fr(;)14 b(\025)p Fu(\);)506 852 y FA(\(H6\))41 b Fu(F)-7 b(or)39 b(a)g(\014xed)h Fr(x)j Fq(2)h FA(R)1460 822 y Fp(d)1498 852 y Fu(,)f Fr(@)1608 864 y Fp(\026)1652 852 y Fr(F)12 b Fu(\()p Fr(x;)i(\026)p Fu(\))41 b(is)e(increasing)f(for)h Fr(\026)k Fq(2)h Fu(\(1)26 b Fq(\000)g Fr(\025;)14 b Fu(1\))40 b(and)f(for)744 952 y Fr(\026)23 b Fq(2)h Fu(\()p Fq(\000)p Fu(1)p Fr(;)14 b Fq(\000)p Fu(1)j(+)h Fr(\025)p Fu(\);)506 1051 y FA(\(H7\))41 b Fr(a)788 1063 y Fp(ij)847 1051 y Fu(\()p Fr(x)23 b Fu(+)g Fr(k)s Fu(\))34 b(=)g Fr(a)1292 1063 y Fp(ij)1350 1051 y Fu(\()p Fr(x)p Fu(\))i(and)e Fr(F)12 b Fu(\()p Fr(x)24 b Fu(+)e Fr(k)s(;)14 b(\026)p Fu(\))34 b(=)g Fr(F)12 b Fu(\()p Fr(x;)i(\026)p Fu(\))35 b(for)f(an)n(y)g Fr(k)j Fq(2)e FA(Z)3042 1021 y Fp(d)3081 1051 y Fu(,)h Fr(x)e Fq(2)h FA(R)3383 1021 y Fp(d)3421 1051 y Fu(,)744 1151 y Fr(\026)23 b Fq(2)h FA(R)p Fu(.)456 1273 y(Here)30 b(ab)r(o)n(v)n(e)e(and)j(in)f(what)g(follo)n(ws,)g(w)n(e)g(assume)g(0)d Fr(<)g(\025)h Fq(\024)f Fu(\003)j(and)35 b(\026)-47 b Fr(n)28 b Fq(2)g FA(N)i Fu(suitably)g(large.)456 1372 y(Since)19 b(the)h(main)f(results)g(of)g(this)g(pap)r(er)g(are)g(of)g (geometric)f(t)n(yp)r(e,)j(w)n(e)e(did)h(not)f(try)g(to)g(minimize)456 1472 y(regularit)n(y)25 b(assumptions.)36 b(T)-7 b(aking)31 b(\026)-46 b Fr(n)23 b Fu(=)g(4)k(is)g(certainly)g(enough.)555 1571 y(Condition)d FA(\(H1\))f Fu(is)h(a)f(standard)g(uniform)h (ellipticit)n(y)g(assumption.)35 b(Conditions)23 b FA(\(H2\))p Fu({)456 1671 y FA(\(H6\))28 b Fu(state)h(what)g(w)n(e)g(mean)g(here)g (b)n(y)f(\\double-w)n(ell")g(p)r(oten)n(tial,)h(an)g(imp)r(ortan)n(t)g (example)456 1771 y(b)r(eing)f(giv)n(en)g(b)n(y)h Fr(F)36 b Fu(=)25 b Fr(Q)p Fu(\()p Fr(x)p Fu(\))14 b(\(1)19 b Fq(\000)g Fr(u)1610 1741 y Fs(2)1647 1771 y Fu(\))1679 1741 y Fs(2)1716 1771 y Fu(,)29 b(with)h Fr(Q)e Fu(p)r(ositiv)n(e,)g(b) r(ounded)i(and)e(in)n(v)-5 b(arian)n(t)28 b(under)456 1870 y(in)n(teger)k(translations.)53 b(Condition)33 b FA(\(H7\))h Fu(is)f(a)g(p)r(erio)r(dicit)n(y)g(assumption.)54 b(Constan)n(ts)33 b(de-)456 1970 y(p)r(ending)26 b(only)f(on)h Fr(d)g Fu(and)g(on)f(the)h(quan)n(tities)g(in)n(tro)r(duced)f(in)i FA(\(H1\))p Fu({)p FA(\(H7\))d Fu(will)i(b)r(e)h(referred)456 2070 y(to)g(as)g(\\univ)n(ersal)f(constan)n(ts".)555 2213 y(F)-7 b(unctionals)40 b(of)h(the)f(t)n(yp)r(e)h(considered)e (here)h(arise)f(in)h(the)h(Ginzburg-Landau-Allen-)456 2313 y(Cahn)34 b(theory)g(of)h(phase)f(transitions)g(\(see)g([Ro)n(w79) o(]\),)j(and)d(in)h(this)g(setting)g(the)g(constan)n(t)456 2413 y(solutions)30 b Fq(\006)p Fu(1)g(are)g(seen)h(as)g(\\pure)f (phases".)46 b(Similar)31 b(mo)r(dels)g(also)f(arise)g(in)i(the)f (study)h(of)456 2512 y(\\sup)r(er-\015uids")26 b(and)h(\\sup)r (er-conductors")e(\(see)j([GL50)o(])f(and)h([GP58)o(]\).)555 2612 y(In)k(the)f(mathematics)h(literature,)f(this)h(problem)f(app)r (ears)f(also)g(as)h(a)g(regularization)e(of)456 2712 y(the)37 b(study)h(of)f(minimal)h(h)n(yp)r(ersurfaces.)65 b(It)38 b(w)n(as)e(sho)n(wn)h(in)h([Mo)r(d87)o(])f(and)h([CC95)o(])f (that)456 2811 y(suitably)e(scaled)f(lev)n(el)h(sets)g(of)g(minimizers) g(of)g Fq(E)42 b Fu(con)n(v)n(erge)33 b(to)i(minimal)g(h)n(yp)r (er-surfaces.)456 2911 y(The)g(in)n(tuition)g(for)g(this)g(fact)g(is)g (that)g(minimizers)g(of)g(the)g(scaled)g(functional)g(will)g(try)g(to) 456 3010 y(sit)29 b(in)n(to)f(the)h(t)n(w)n(o)f(minima)h(of)g(the)g (double-w)n(ell)f(p)r(oten)n(tial)g(\(whic)n(h)h(corresp)r(ond)f(to)g (the)h(t)n(w)n(o)456 3110 y(ph)n(ysical)22 b(phases)g(of)h(the)h (medium\))g(as)e(m)n(uc)n(h)h(as)g(they)g(can,)h(pa)n(ying)e(the)h (least)g(p)r(ossible)g(price)456 3210 y(in)k(the)h(in)n(terface.)1145 3450 y(2.)41 b Fv(Not)-6 b(a)g(tion)32 b(and)f(st)-6 b(a)g(tement)33 b(of)e(resul)-6 b(ts)456 3600 y Fu(2.1.)40 b FA(Notation)c(and)h(some)d(standard)j(de\014nitions.)j Fu(Since)31 b(w)n(e)g(will)h(b)r(e)g(dealing)f(with)456 3699 y(p)r(erio)r(dic)f(functions)g(with)h(sev)n(eral)d(p)r(erio)r(ds,) j(it)f(will)h(b)r(e)f(useful)h(to)f(dev)n(elop)f(some)h(notation)456 3799 y(that)d(will)h(handle)g(this)g(comfortably)-7 b(.)555 3898 y(Giv)n(en)28 b Fr(v)e Fq(2)d FA(R)1012 3868 y Fp(d)1051 3898 y Fu(,)k(w)n(e)h(denote)1425 4042 y Fr(v)1468 4008 y Fm(?)1548 4042 y Fu(:=)22 b Fq(f)p Fr(w)j Fq(2)f FA(R)1935 4008 y Fp(d)2010 4042 y Fu(:)46 b Fr(v)22 b Fq(\001)c Fr(w)26 b Fu(=)d(0)p Fq(g)14 b Fr(:)555 4186 y Fu(Giv)n(en)36 b(a)g(direction)g Fr(!)k Fq(2)d FA(R)1493 4156 y Fp(d)1556 4186 y Fq(\000)24 b(f)p Fu(0)p Fq(g)p Fu(,)37 b(w)n(e)f(de\014ne)g(the) h(follo)n(wing)e(equiv)-5 b(alence)35 b(relation)456 4286 y Fq(\030)521 4298 y Fp(!)600 4286 y Fu(on)c FA(R)791 4256 y Fp(d)829 4286 y Fu(:)44 b(w)n(e)31 b(sa)n(y)f(that)i Fr(x)d Fq(\030)1494 4298 y Fp(!)1571 4286 y Fr(y)34 b Fu(if)e(and)f(only)g(if)h(there)f(exists)g Fr(k)h Fq(2)d FA(Z)2823 4256 y Fp(d)2883 4286 y Fq(\\)21 b Fr(!)3014 4256 y Fm(?)3101 4286 y Fu(suc)n(h)31 b(that)456 4385 y Fr(x)19 b Fq(\000)f Fr(y)25 b Fu(=)e Fr(k)s Fu(.)555 4485 y(W)-7 b(e)28 b(de\014ne)1648 4594 y FA(K)1723 4560 y Fp(d)1723 4614 y(!)1794 4594 y Fu(:=)23 b FA(R)1977 4560 y Fp(d)2015 4594 y Fr(=)f Fq(\030)2144 4606 y Fp(!)2229 4594 y Fr(:)555 4720 y Fu(Notice)28 b(that)g(if)h Fr(!)d Fq(2)e FA(Q)1302 4690 y Fp(d)1340 4720 y Fu(,)k(then)h FA(K)1656 4690 y Fp(d)1656 4741 y(!)1731 4720 y Fu(is)f(top)r (ologically)e(equiv)-5 b(alen)n(t)28 b(to)g(the)g Fr(d)p Fu(-dimensional)456 4820 y(cylinder)g FA(T)839 4790 y Fp(d)p Fm(\000)p Fs(1)983 4820 y Fq(\002)18 b FA(R)p Fu(.)41 b(In)29 b(the)g(general)f(case,)g FA(K)2014 4790 y Fp(d)2014 4841 y(!)2087 4820 y Fu(=)d FA(T)2243 4790 y Fp(d)p Fm(\000)p Fp(r)2386 4820 y Fq(\002)19 b FA(R)2542 4790 y Fp(r)2578 4820 y Fu(,)30 b(where)e Fr(r)k Fu(is)c(the)i(n)n(um)n (b)r(er)456 4920 y(of)d Fn(indep)l(endent)k(r)l(esonanc)l(es)p Fu(,)c(that)h(is,)g(the)g(dimension)f(of)h(the)g(mo)r(dule)1336 5063 y Fq(R)1406 5075 y Fp(!)1491 5063 y Fu(:=)36 b Fq(f)p Fr(k)26 b Fq(2)d Fr(!)1859 5029 y Fm(?)1933 5063 y Fq([)c FA(Z)2065 5029 y Fp(d)2132 4993 y Fo(\014)2132 5043 y(\014)2187 5063 y Fr(!)i Fq(\001)e Fr(k)26 b Fu(=)c(0)p Fq(g)p Fr(:)456 5216 y Fu(F)-7 b(or)27 b(almost)g(all)g Fr(!)f Fq(2)d FA(R)1216 5185 y Fp(d)1254 5216 y Fu(,)28 b(w)n(e)f(ha)n(v)n(e)g(that)h Fr(r)d Fu(=)e(0)k(and)h(therefore)e FA(K)2603 5185 y Fp(d)2603 5236 y(!)2674 5216 y Fu(=)d FA(R)2834 5185 y Fp(d)2872 5216 y Fu(.)p eop %%Page: 4 4 4 3 bop 456 251 a Fs(4)664 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)555 450 y Fu(W)g(e)26 b(will)f(b)r(e)g(lo)r(oking)f(for)g(functions)i(that)f(ha)n(v)n(e)f (the)h(p)r(erio)r(dicities)g(giv)n(en)f(b)n(y)h Fq(R)3095 462 y Fp(!)3143 450 y Fu(.)36 b(Hence,)456 550 y(it)28 b(is)f(natural)g(to)g(consider)g(the)h(functional)1749 689 y Fq(E)1793 701 y Fp(!)1878 689 y Fu(:=)36 b Fq(E)2046 705 y Fl(K)2104 689 y Fk(d)2104 722 y(!)456 840 y Fu(de\014ned)d(on)g (functions)g Fr(u)f Fu(:)g FA(K)1441 810 y Fp(d)1441 861 y(!)1521 840 y Fq(!)g FA(R)p Fu(.)52 b(\(Later)33 b(on,)h(w)n(e)f(will)g(sp)r(ecify)g(other)f(prop)r(erties)h(of)456 940 y(the)c(functions)h(on)f(whic)n(h)h(the)f(functional)h(is)f (de\014ned)h(suc)n(h)f(as)g(regularit)n(y)-7 b(,)28 b(deca)n(y)-7 b(,)29 b(in)n(tegra-)456 1039 y(bilit)n(y)-7 b(,)30 b(etc.\))45 b(W)-7 b(e)30 b(th)n(us)g(de\014ne)h(the)f(functional)g(b)n(y)g(the)g (form)n(ula)f(\(1.1\))h(but)g(w)n(e)g(extend)g(the)456 1139 y(in)n(tegration)c(only)h(to)h FA(K)1237 1109 y Fp(d)1237 1160 y(!)1284 1139 y Fu(.)555 1239 y(Notice)d(that,)i(as)d (it)i(is,)g Fq(E)1344 1251 y Fp(!)1417 1239 y Fu(is)f(not)g(a)g(b)r (ona-\014de)g(functional)g({)g(the)h(domain)f(of)g(in)n(tegration)456 1338 y(is)37 b(un)n(b)r(ounded)g({)g(but)h(rather)e(it)h(is)g(just)h(a) f(v)-5 b(ariational)36 b(principle.)65 b(Later,)39 b(w)n(e)e(will)g (\014nd)456 1438 y(sev)n(eral)c(regularizations)f(that)j(mak)n(e)f(it)h (in)n(to)f(a)h(w)n(ell)f(de\014ned)h(functional)g(in)g(appropriate)456 1537 y(function)28 b(spaces.)555 1637 y(The)36 b(easiest)e(case)h(is)g Fr(!)k Fq(2)d FA(Q)1534 1607 y Fp(d)1596 1637 y Fq(\000)23 b(f)p Fu(0)p Fq(g)p Fu(.)59 b(In)36 b(this)f(case,)i(there)e(is)g(only) g(one)g(un)n(b)r(ounded)456 1737 y(direction,)26 b(along)g(whic)n(h,)h (as)g(w)n(e)f(will)h(see,)g(w)n(e)g(ha)n(v)n(e)f(uniform)h(deca)n(y)f (estimates.)36 b(In)27 b(certain)456 1836 y(sense,)37 b(the)f(case)f(whic)n(h)h(is)g(hardest)f(is)h(when)g Fr(!)j Fu(do)r(es)c(not)h(satisfy)g(an)n(y)f(relations,)h(hence)456 1936 y FA(K)531 1906 y Fp(d)531 1957 y(!)602 1936 y Fu(=)22 b FA(R)761 1906 y Fp(d)800 1936 y Fu(.)555 2036 y(Ev)n(en)35 b(if)g(the)h(functional)f(\(1.1\))g(is)g(not,)i(in)f(general)d(mean)n (t)i(as)g(a)g(con)n(v)n(ergen)n(t)e(in)n(tegral,)456 2135 y(w)n(e)i(recall)f(that)i Fr(u)f Fu(is)g(said)g(to)h(b)r(e)g(a)72 b Fn(lo)l(c)l(al)38 b(minimizer)45 b Fu(\(also)35 b(called)g Fn(class-A)h Fu(minimizers)456 2235 y(in)27 b(calculus)h(of)f(v)-5 b(ariations)26 b(or)h Fn(gr)l(ound)j(states)k Fu(in)28 b(phase)f(transition)g(theory\))g(whenev)n(er,)g(for)456 2335 y(ev)n(ery)f(ball)h Fr(B)h Fq(\032)22 b FA(R)1086 2304 y Fp(d)1125 2335 y Fu(,)1606 2439 y Fq(E)1650 2451 y Fp(B)1707 2439 y Fu(\()p Fr(u)p Fu(\))h Fq(\024)g(E)1974 2451 y Fp(B)2031 2439 y Fu(\()p Fr(u)18 b Fu(+)g Fr(\036)p Fu(\))456 2560 y(for)27 b(all)g Fr(\036)h Fu(of)g(compact)f(supp)r(ort) g(con)n(tained)g(in)h Fr(B)t Fu(.)555 2660 y(Note)h(that)f(the)h (de\014nition)g(of)f(lo)r(cal)g(minimizer)g(do)r(es)g(not)g(require)g (that)g(the)h(expression)456 2759 y(\(1.1\))e(is)g(a)h(functional)f (when)h(extended)g(o)n(v)n(er)e(the)i(whole)f(space.)555 2859 y(In)37 b(the)h(cases)e(that)h(the)g(functional)g(is)g(indeed)h (con)n(v)n(ergen)n(t,)f(w)n(e)f(will)i(see)e(that)h Fr(u)g Fu(is)g(a)456 2958 y Fn(glob)l(al)h(minimizer)p Fu(,)h(i.e.)61 b Fq(E)7 b Fu(\()p Fr(u)p Fu(\))36 b(is)f(the)h(smallest)f(p)r(ossible) h(v)-5 b(alue)35 b(that)h(it)g(can)f(tak)n(e)g(in)h(the)456 3058 y(space)26 b(considered.)36 b(Global)27 b(minimizers)h(will)g(b)r (e)g(considered)e(in)i Fq(x)g Fu(4.1.)555 3158 y(W)-7 b(e)28 b(denote)1561 3284 y Fr(f)9 b Fu(\()p Fr(x;)14 b(\026)p Fu(\))23 b(=)1955 3228 y Fr(@)p 1930 3265 99 4 v 1930 3341 a(@)5 b(\026)2039 3284 y(F)12 b Fu(\()p Fr(x;)i(\026)p Fu(\))g Fr(:)456 3445 y Fu(W)-7 b(e)28 b(sa)n(y)e(that)i Fr(u)f Fu(is)h(a)f(critical)g(p)r(oin)n(t)h(when)f (it)h(satis\014es)456 3584 y(\(2.1\))794 b Fr(@)1465 3596 y Fp(i)1493 3584 y Fu(\()p Fr(a)1569 3596 y Fp(ij)1627 3584 y Fu(\()p Fr(x)p Fu(\))p Fr(@)1782 3596 y Fp(j)1818 3584 y Fr(u)p Fu(\()p Fr(x)p Fu(\)\))24 b(=)f Fr(f)9 b Fu(\()p Fr(x;)14 b(u)p Fu(\()p Fr(x)p Fu(\)\))456 3723 y(in)27 b(the)h(w)n(eak)f(sense)g(\(and,)h(in)g(fact,)g(classically)-7 b(,)26 b(thanks)h(to)g(elliptic)i(regularit)n(y)c(theory\).)555 3823 y(It)c(is)e(w)n(ell)h(kno)n(wn)f(that)i(as)e(so)r(on)g(as)g(the)i (space)e(of)h(functions)g(w)n(e)g(consider)f(is)h(large)e(enough,)456 3922 y(then)30 b(all)g(class-A)f(minimizers)h(are)f(critical)g(p)r(oin) n(ts,)i(but)f(the)h(con)n(v)n(erse)c(is,)k(in)f(general,)f(not)456 4022 y(true.)555 4121 y(The)g(solutions)g(of)35 b(\(2.1\))29 b(sometimes)g(are)f(called)h Fn(metastable)i(states)36 b Fu(in)29 b(the)h(ph)n(ysics)e(lit-)456 4221 y(erature.)456 4379 y(2.2.)40 b FA(The)32 b(Birkho\013)g(prop)s(ert)m(y.)456 4499 y(De\014nition)j(2.1)f Fu(\(Birkho\013)d(prop)r(ert)n(y\))p FA(.)42 b Fn(We)34 b(say)f(that)g(the)g(function)g Fr(u)c Fq(2)g Fr(C)6 b Fu(\()p FA(R)3113 4468 y Fp(d)3152 4499 y Fu(\))33 b Fn(enjoys)456 4598 y(the)28 b(Birkho\013)i(pr)l(op)l(erty) f(with)f(r)l(esp)l(e)l(ct)g(to)g Fr(!)j Fn(if,)f(for)f(any)f Fr(k)e Fq(2)e FA(Z)2463 4568 y Fp(d)2530 4598 y Fn(so)k(that)g Fr(k)18 b Fq(\001)d Fr(!)26 b Fq(\025)c Fu(0)p Fn(,)29 b(we)f(have)456 4698 y(that)h Fr(u)p Fu(\()p Fr(x)p Fu(\))24 b Fq(\024)f Fr(u)p Fu(\()p Fr(x)18 b Fu(+)g Fr(k)s Fu(\))30 b Fn(for)h(any)f Fr(x)23 b Fq(2)h FA(R)1745 4668 y Fp(d)1783 4698 y Fn(.)555 4817 y Fu(W)-7 b(e)28 b(remark)d(that,)j(ev)n(en)f(if)g (w)n(e)g(ha)n(v)n(e)f(form)n(ulated)h(the)g(Birkho\013)g(prop)r(ert)n (y)f(only)h(for)f(con-)456 4917 y(tin)n(uous)e(functions)h(\(and)g (this)g(is)g(what)f(w)n(e)h(use)f(in)h(this)g(pap)r(er\),)h(the)f(prop) r(ert)n(y)e(mak)n(es)h(sense)456 5016 y(for)32 b(measurable)f (functions)i(to)r(o,)g(with)g(ob)n(vious)e(mo)r(di\014cations.)51 b(Prop)r(erties)31 b(of)i(inclusion)456 5116 y(related)25 b(to)h(the)h(Birkho\013)f(prop)r(ert)n(y)f(pla)n(y)h(an)g(imp)r(ortan)n (t)g(r^)-42 b(ole)25 b(b)r(oth)i(in)f(dynamical)g(systems)456 5216 y(and)f(PDEs)g(\(see,)g(for)g(instance,)h([MF94],)g([KdlLR97)n(],) h([CdlL01)o(])e(and)h([V)-7 b(al04)o(]\).)36 b(Also,)26 b(suc)n(h)p eop %%Page: 5 5 5 4 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)541 b(5)456 450 y Fu(prop)r(ert)n(y)31 b(can)h(b)r(e)h(used)g(to)f(a)n(v)n(oid)f(some)h (problems)g(as)g(the)h(Hedlund-t)n(yp)r(e)g(coun)n(terexam-)456 550 y(ples)23 b(in)h(the)g(geo)r(desic)f(setting)h(\(see)f([Hed32]\).) 36 b(Indeed,)24 b(the)h(Birkho\013)e(prop)r(ert)n(y)f(implies)i(the)456 649 y Fn(doubling)30 b(pr)l(op)l(erty)e Fu(\(also)e(kno)n(wn)h(in)g (the)g(literature)g(as)f Fn(no-symmetry-br)l(e)l(aking-pr)l(op)l(erty)p Fu(\),)456 749 y(i.e.:)42 b(a)29 b(function)i(satisfying)e(the)h (Birkho\013)g(prop)r(ert)n(y)f(and)h(ha)n(ving)f(p)r(erio)r(d)h(m)n (ultiple)g(of)g(the)456 849 y(original)c(one,)h(is)g(alw)n(a)n(ys)f(p)r (erio)r(dic)i(with)g(the)g(original)e(p)r(erio)r(d.)36 b(More)27 b(precisely)-7 b(,)27 b(w)n(e)g(ha)n(v)n(e:)456 975 y FA(Prop)s(osition)45 b(2.2.)j Fn(L)l(et)41 b Fr(v)k Fn(satisfy)e(the)f(Birkho\013)h(pr)l(op)l(erty)g(with)f(r)l(esp)l(e)l (ct)g(to)f Fr(!)s Fn(.)75 b(Then)456 1074 y Fr(v)s Fu(\()p Fr(x)19 b Fu(+)f Fr(k)s Fu(\))23 b(=)g Fr(v)s Fu(\()p Fr(x)p Fu(\))p Fn(,)31 b(for)g(any)f Fr(k)c Fq(2)d FA(Z)1576 1044 y Fp(d)1634 1074 y Fq(\\)c Fr(!)1763 1044 y Fm(?)1818 1074 y Fn(.)456 1249 y(Pr)l(o)l(of.)43 b Fu(Since)25 b Fq(\006)p Fr(k)g Fq(2)e Fr(!)1196 1219 y Fm(?)1252 1249 y Fu(,)i(w)n(e)g(deduce)f(that)h Fr(v)s Fu(\()p Fr(x)12 b Fu(+)g Fr(k)s Fu(\))25 b Fq(\025)e Fr(v)s Fu(\()p Fr(x)p Fu(\))j(and)e Fr(v)s Fu(\()p Fr(x)12 b Fq(\000)g Fr(k)s Fu(\))25 b Fq(\025)d Fr(v)s Fu(\()p Fr(x)p Fu(\))k(for)e(an)n(y) 456 1349 y Fr(x)f Fq(2)h FA(R)677 1318 y Fp(d)715 1349 y Fu(.)37 b(Therefore,)26 b(for)h(an)n(y)g Fr(x)d Fq(2)f FA(R)1679 1318 y Fp(d)1717 1349 y Fu(,)1202 1501 y Fr(v)s Fu(\()p Fr(x)p Fu(\))h(=)f Fr(v)s Fu(\(\()p Fr(x)c Fu(+)f Fr(k)s Fu(\))h Fq(\000)f Fr(k)s Fu(\))23 b Fq(\025)g Fr(v)s Fu(\()p Fr(x)c Fu(+)f Fr(k)s Fu(\))23 b Fq(\025)g Fr(v)s Fu(\()p Fr(x)p Fu(\))14 b Fr(;)456 1653 y Fu(hence)27 b Fr(v)s Fu(\()p Fr(x)p Fu(\))e(=)d Fr(v)s Fu(\()p Fr(x)e Fu(+)e Fr(k)s Fu(\).)2102 b Fj(\003)456 1828 y Fu(2.3.)40 b FA(Results)29 b(on)h(p)s(erio)s(dic)g(minimizers)c(for)31 b(arbitrary)h(frequencies.)41 b Fu(The)27 b(follo)n(w-)456 1927 y(ing)g(result)h(is)g(a)g(sligh)n(t)f(strengthening)h(of)g(the)g (results)g(in)g([V)-7 b(al04)o(].)38 b(The)29 b(main)f(di\013erence)g (is)456 2027 y(that)23 b(w)n(e)f(pro)r(duce)h(uniform)g(deca)n(y)f (prop)r(erties)g(along)g(the)h(direction)g Fr(!)s Fu(.)35 b(This)23 b(will)g(b)r(e)g(useful)456 2127 y(later)k(in)g(con)n (trolling)f(sev)n(eral)g(limits)i(of)g(minimizers.)456 2253 y FA(Theorem)35 b(2.3.)42 b Fn(L)l(et)33 b Fr(!)g Fq(2)d FA(Q)1447 2223 y Fp(d)1507 2253 y Fq(\000)21 b(f)p Fu(0)p Fq(g)p Fn(.)48 b(Then,)36 b(ther)l(e)e(exists)f Fr(u)c Fq(2)h Fr(C)2708 2223 y Fs(2)2746 2253 y Fu(\()p FA(K)2853 2223 y Fp(d)2853 2273 y(!)2901 2253 y Fu(\))p Fn(,)35 b Fq(j)p Fr(u)p Fq(j)30 b(\024)f Fu(1)p Fn(,)35 b(and)456 2352 y(p)l(ositive)c(universal)f(c)l(onstants)f Fr(c)1511 2364 y Fs(1)1548 2352 y Fn(,)h Fr(c)1639 2364 y Fs(2)1706 2352 y Fn(and)h Fr(c)1904 2364 y Fs(3)1941 2352 y Fn(,)f(such)g(that)629 2478 y Fu(1\))41 b Fq(E)788 2490 y Fp(!)836 2478 y Fu(\()p Fr(u)p Fu(\))23 b Fr(<)g Fu(+)p Fq(1)629 2578 y Fu(2\))41 b Fq(E)788 2590 y Fp(!)836 2578 y Fu(\()p Fr(u)p Fu(\))23 b Fq(\024)g(E)1103 2590 y Fp(!)1151 2578 y Fu(\()p Fr(u)18 b Fu(+)g Fr(')p Fu(\))c Fr(;)184 b Fq(8)p Fr(')23 b Fq(2)g Fr(W)1931 2548 y Fs(1)p Fp(;)p Fs(2)2021 2578 y Fu(\()p FA(K)2128 2548 y Fp(d)2128 2598 y(!)2176 2578 y Fu(\))c Fq(\\)g Fr(L)2358 2548 y Fm(1)2428 2578 y Fu(\()p FA(K)2535 2548 y Fp(d)2535 2598 y(!)2583 2578 y Fu(\))p Fn(.)629 2677 y Fu(3\))41 b Fr(u)p Fu(\()p Fr(x)19 b Fu(+)f Fr(k)s Fu(\))23 b Fq(\025)g Fr(u)p Fu(\()p Fr(x)p Fu(\))30 b Fn(for)h(any)f Fr(k)c Fq(2)d FA(Z)1848 2647 y Fp(d)1917 2677 y Fn(with)30 b Fr(!)21 b Fq(\001)e Fr(k)26 b Fq(\025)c Fu(0)p Fn(.)629 2777 y Fu(4\))1177 2904 y Fq(j)p Fr(u)p Fu(\()p Fr(x)p Fu(\))e Fq(\000)e Fu(1)p Fq(j)k(\024)h Fr(c)1673 2916 y Fs(2)1710 2904 y Fr(e)1749 2864 y Fm(\000)p Fp(c)1831 2872 y Fi(3)1892 2842 y Fk(!)p 1873 2851 77 3 v 1873 2885 a Fh(j)p Fk(!)q Fh(j)1959 2864 y Fm(\001)p Fp(x)2034 2904 y Fr(;)184 b Fn(if)86 b Fr(!)21 b Fq(\001)d Fr(x)24 b Fq(\025)e Fr(c)2685 2916 y Fs(1)744 3039 y Fn(and)1160 3156 y Fq(j)p Fr(u)p Fu(\()p Fr(x)p Fu(\))d(+)f(1)p Fq(j)k(\024)h Fr(c)1655 3168 y Fs(2)1692 3156 y Fr(e)1731 3116 y Fp(c)1761 3124 y Fi(3)1822 3094 y Fk(!)p 1803 3103 V 1803 3137 a Fh(j)p Fk(!)q Fh(j)1889 3116 y Fm(\001)p Fp(x)1964 3156 y Fr(;)184 b Fn(if)86 b Fr(!)21 b Fq(\001)d Fr(x)24 b Fq(\024)f(\000)p Fr(c)2681 3168 y Fs(1)2717 3156 y Fr(:)555 3291 y Fn(In)31 b(p)l(articular,)i(\014xe)l(d)d(any)h Fr(\022)d Fq(2)d Fu(\(0)p Fr(;)14 b Fu(1\))p Fn(,)31 b(ther)l(e)h(exists)e Fr(M)k(>)25 b Fu(0)p Fn(,)31 b(dep)l(ending)h (only)g(on)f Fr(\022)i Fn(and)456 3391 y(on)38 b(universal)h (quantities,)j(such)c(that)h Fr(u)p Fu(\()p Fr(x)p Fu(\))h Fq(2)f Fu([)p Fr(\022)r(;)14 b Fu(1])38 b Fn(pr)l(ovide)l(d)j(that)e Fr(!)27 b Fq(\001)e Fr(x)39 b Fq(\025)g Fr(M)23 b Fq(j)p Fr(!)s Fq(j)38 b Fn(and)456 3490 y Fr(u)p Fu(\()p Fr(x)p Fu(\))23 b Fq(2)h Fu([)p Fq(\000)p Fu(1)p Fr(;)14 b Fq(\000)p Fr(\022)r Fu(])28 b Fn(pr)l(ovide)l(d)k(that)e Fr(!)21 b Fq(\001)d Fr(x)24 b Fq(\024)f(\000)p Fr(M)f Fq(j)p Fr(!)s Fq(j)p Fn(.)555 3616 y Fu(The)28 b(last)f(claim)h(in)g(Theorem)f (2.3)g(ma)n(y)g(b)r(e)h(summarized)f(b)n(y)h(sa)n(ying)e(that)i(the)g (in)n(terface)456 3716 y(\(that)34 b(is,)i(the)f(lev)n(el)e(sets)h (\\close)f(to)h(zero"\))f(of)h Fr(u)g Fu(\\lo)r(oks)e(lik)n(e)i(a)g(h)n (yp)r(erplane")f(or,)i(brie\015y)-7 b(,)456 3816 y(that)27 b(it)i(is)e(\\plane-lik)n(e".)555 3915 y(As)h(straigh)n(tforw)n(ard)d (consequence)i(of)g(Theorem)g(2.3)g(w)n(e)g(deduce)h(the)g(existence)g (of)f(het-)456 4015 y(ero)r(clinic)e(orbits)h(in)h(a)f(wide)g(class)g (of)g(ordinary)f(di\013eren)n(tial)h(equations.)35 b(Namely)-7 b(,)27 b(applying)456 4114 y(Theorem)f(2.3)h(for)g Fr(d)c Fu(=)g(1,)k(one)h(obtains)f(immediately:)456 4240 y FA(Corollary)39 b(2.4.)k Fn(L)l(et)34 b Fr(F)43 b Fq(2)32 b Fr(C)1481 4210 y Fs(\026)-37 b Fp(n)1523 4240 y Fu(\()p FA(T)22 b Fq(\002)g FA(R)p Fu(\))p Fn(,)36 b Fr(F)12 b Fu(\()p Fr(t;)i Fq(\006)p Fu(1\))31 b(=)g(0)p Fn(,)k Fr(F)12 b Fu(\()p Fr(t;)i(\026)p Fu(\))32 b Fq(\025)g Fu(0)i Fn(for)h(any)g Fr(\026)c Fq(2)h FA(R)p Fn(.)456 4340 y(Assume)39 b(c)l(onditions)j Fg(\(H3\))p Fn({)p Fg(\(H6\))h Fn(on)e Fr(F)12 b Fn(.)72 b(Then,)44 b(the)d(or)l(dinary)h(di\013er)l (ential)g(e)l(quation)462 4440 y Fu(\177)-48 b Fr(q)s Fu(\()p Fr(t)p Fu(\))23 b(=)g Fr(@)745 4452 y Fp(\026)789 4440 y Fr(F)12 b Fu(\()p Fr(t;)i(q)s Fu(\()p Fr(t)p Fu(\)\))p Fn(,)32 b(has)e(a)g(solution)g Fr(q)1747 4452 y Fs(1)1808 4440 y Fu(:)23 b FA(R)f Fq(\000)-14 b(!)23 b FA(R)30 b Fn(so)g(that)456 4592 y Fu(\(2.2\))1066 b(lim)1646 4642 y Fp(t)p Fm(!\0061)1869 4592 y Fr(q)1906 4604 y Fs(1)1943 4592 y Fu(\()p Fr(t)p Fu(\))24 b(=)e Fq(\006)p Fu(1)456 4775 y Fn(and)30 b(a)g(solution)g Fr(q)1040 4787 y Fs(2)1100 4775 y Fu(:)23 b FA(R)g Fq(\000)-14 b(!)23 b FA(R)29 b Fn(so)h(that)456 4928 y Fu(\(2.3\))1047 b(lim)1627 4978 y Fp(t)p Fm(!\0061)1850 4928 y Fr(q)1887 4940 y Fs(2)1925 4928 y Fu(\()p Fr(t)p Fu(\))23 b(=)g Fq(\007)p Fu(1)14 b Fr(:)456 5116 y Fn(F)-6 b(urthermor)l(e,)28 b Fq(j)p Fr(q)1019 5128 y Fp(i)1046 5116 y Fu(\()p Fr(t)p Fu(\))p Fq(j)c(\024)f Fu(1)p Fn(,)k Fr(q)1406 5128 y Fs(1)1444 5116 y Fu(\()p Fr(t)13 b Fu(+)g Fr(k)s Fu(\))22 b Fq(\025)h Fr(q)1822 5128 y Fs(1)1859 5116 y Fu(\()p Fr(t)p Fu(\))28 b Fn(and)g Fr(q)2177 5128 y Fs(2)2214 5116 y Fu(\()p Fr(t)13 b Fq(\000)g Fr(k)s Fu(\))23 b Fq(\025)f Fr(q)2592 5128 y Fs(2)2629 5116 y Fu(\()p Fr(t)p Fu(\))p Fn(,)29 b(for)f(any)g Fr(k)e Fq(2)d FA(N)k Fn(and)456 5216 y(any)j Fr(t)23 b Fq(2)g FA(R)p Fn(.)p eop %%Page: 6 6 6 5 bop 456 251 a Fs(6)664 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)555 450 y Fu(W)g(e)23 b(will)g(also)f(sho)n(w)g(that)h(all)f(p)r(erio)r(dic)h(minimizers)f (ha)n(v)n(e)g(lev)n(el)g(sets)g(con)n(tained)g(in)h(a)g(strip)456 550 y(of)k(univ)n(ersally)e(large)h(width.)37 b(Lo)r(osely)26 b(sp)r(eaking,)h(this)g(sa)n(ys)f(that)h(all)g(p)r(erio)r(dic)g (minimizers)456 649 y(\\are)f(plane-lik)n(e".)35 b(More)27 b(precisely)-7 b(,)27 b(w)n(e)g(will)h(pro)n(v)n(e)e(the)i(follo)n (wing)e(result:)456 780 y FA(Theorem)i(2.5.)38 b Fn(L)l(et)27 b Fr(!)f Fq(2)d FA(Q)1416 750 y Fp(d)1469 780 y Fq(\000)14 b(f)p Fu(0)p Fq(g)p Fn(.)36 b(Fix)28 b Fr(\022)1920 792 y Fs(0)1980 780 y Fq(2)c Fu(\(0)p Fr(;)14 b Fu(1\))p Fn(.)37 b(L)l(et)28 b Fr(u)22 b Fq(2)i Fr(W)2687 740 y Fs(1)p Fp(;)p Fs(2)2675 805 y(lo)r(c)2777 780 y Fu(\()p FA(K)2884 750 y Fp(d)2884 801 y(!)2932 780 y Fu(\))p Fn(,)29 b(b)l(e)e(such)h(that)629 909 y Fu(1\))41 b Fq(E)788 921 y Fp(!)836 909 y Fu(\()p Fr(u)p Fu(\))23 b Fr(<)g Fu(+)p Fq(1)p Fn(.)629 1008 y Fu(2\))41 b Fq(E)788 1020 y Fp(!)836 1008 y Fu(\()p Fr(u)p Fu(\))23 b Fq(\024)g(E)1103 1020 y Fp(!)1151 1008 y Fu(\()p Fr(u)18 b Fu(+)g Fr(')p Fu(\))30 b Fn(for)h(any)f Fr(')24 b Fq(2)f Fr(W)1986 978 y Fs(1)p Fp(;)p Fs(2)2076 1008 y Fu(\()p FA(K)2183 978 y Fp(d)2183 1029 y(!)2231 1008 y Fu(\))c Fq(\\)g Fr(L)2413 978 y Fm(1)2483 1008 y Fu(\()p FA(K)2590 978 y Fp(d)2590 1029 y(!)2638 1008 y Fu(\))p Fn(.)629 1108 y Fu(3\))41 b Fn(Mor)l(e)l(over,)33 b(ther)l(e)e(exists)f Fr(M)1653 1120 y Fs(0)1715 1108 y Fr(>)24 b Fu(0)p Fn(,)31 b(dep)l(ending)h(only)g(on)f(universal)g(c)l(onstants)f(and)744 1208 y(on)g Fr(\022)902 1220 y Fs(0)939 1208 y Fn(,)h(such)e(that)1268 1411 y Fq(fj)p Fr(u)p Fq(j)22 b Fr(<)g(\022)1552 1423 y Fs(0)1590 1411 y Fq(g)g(\022)1742 1294 y Fo(\032)1837 1354 y Fr(!)p 1814 1391 101 4 v 1814 1468 a Fq(j)p Fr(!)s Fq(j)1943 1411 y(\001)d Fr(x)k Fq(2)h Fu([)p Fr(p;)14 b(p)k Fu(+)g Fr(M)2460 1423 y Fs(0)2496 1411 y Fu(])2519 1294 y Fo(\033)2609 1411 y Fr(;)744 1613 y Fn(for)31 b(a)f(suitable)g Fr(p)23 b Fq(2)h FA(R)p Fn(.)555 1742 y Fu(The)38 b(ab)r(o)n(v)n(e)e(results)h(for)g(the)h(minima)f(of)h (Ginzburg-Landau-Allen-Cahn-t)n(yp)r(e)d(func-)456 1842 y(tionals)g(\014t)h(in)g(the)h(theory)e(of)h(plane-lik)n(e)f (structures)g(for)g(phase)g(transitions)g(in)i(p)r(erio)r(dic)456 1941 y(media,)27 b(whic)n(h)h(has)f(b)r(een)h(recen)n(tly)f(addressed)f (in)i([V)-7 b(al04)o(].)555 2041 y(Seee)28 b(also)e([Mos86)o(],)i ([Ban89)n(],)g([Ban90)o(],)f([CdlL01],)g([RS03],)h([RS04)o(],)g([T)-7 b(or04)n(],)28 b([PV05a)n(],)456 2141 y([PV05b)o(],)g([Bes05)o(])g(and) g([CdlL05)o(])g(for)g(related)f(results)h(in)g(di\013eren)n(t)h(con)n (texts.)37 b(W)-7 b(e)29 b(remark)456 2240 y(that)21 b(in)g(the)g(elliptic)h(in)n(tegrand)d(framew)n(ork)g(\(see,)j(e.g.,)g ([Mos86)o(],)g([Ban89)o(],)g([Ban90)o(],)g([RS03)o(])456 2340 y(and)31 b([RS04)o(]\))i(one)e(is)g(in)n(terested)h(in)g (\014nding)g(solutions)f(whose)g(graphs,)g(when)h(\\seen)f(from)456 2439 y(far",)g(b)r(eha)n(v)n(e)f(\\lik)n(e)h(h)n(yp)r(erplanes")f(and)h (the)h(solutions)e(are)g(th)n(us)i(exp)r(ected)f(to)h(gro)n(w)d(kind) 456 2539 y(of)c(linearly)f(at)h(in\014nit)n(y)-7 b(.)36 b(In)26 b(the)f(phase)g(transition)f(setting)h(w)n(e)g(deal)g(with,)h (all)f(the)g(solutions)456 2639 y(are)32 b(b)r(ounded,)j(th)n(us,)f (when)g(seen)f(from)f(far)h(in)g FA(R)2098 2609 y Fp(d)p Fs(+1)2221 2639 y Fu(,)i(their)e(graphs)e(are)i(ob)n(viously)e(close) 456 2738 y(to)h(horizon)n(tal)e Fr(d)p Fu(-dimensional)i(planes:)46 b(the)32 b(target)f(is)i(then)f(to)g(sho)n(w)g(that)g(also)f(the)i(lev) n(el)456 2838 y(sets)23 b(of)h(these)h(solutions)e(\(i.e.,)i(the)g (\\in)n(terfaces"\),)e(when)h(seen)g(from)g(far)f(in)h FA(R)2960 2808 y Fp(d)2999 2838 y Fu(,)h(b)r(eha)n(v)n(e)e(lik)n(e)456 2938 y(\()p Fr(d)9 b Fq(\000)g Fu(1\)-dimensional)21 b(planes.)35 b(F)-7 b(or)22 b(the)h(imp)r(ortan)n(t)f(r^)-42 b(ole)22 b(pla)n(y)n(ed)g(b)n(y)h(\015at)f(in)n(terfaces,)h(see)g(also) 456 3037 y([DG79)o(].)1236 3273 y(3.)41 b Fv(Minimizers)32 b(f)n(or)f(all)g(frequencies)555 3422 y Fu(It)g(is)g(w)n(ell)f(kno)n (wn)h(that)g(the)g(limit)g(\(understo)r(o)r(d)g(in)g(man)n(y)f(senses,) h(e.g.)46 b(lo)r(cally)30 b Fr(C)3277 3392 y Fs(0)3314 3422 y Fu(\))i(of)456 3522 y(lo)r(cal)20 b(minimizers)i(is)f(a)g(lo)r (cal)g(minimizer.)34 b(See)22 b(Lemma)f(3.1)f(b)r(elo)n(w)h(for)g(a)g (detailed)g(statemen)n(t)456 3622 y(and)27 b(pro)r(of)g(of)h(this)f (result.)555 3721 y(By)39 b(the)f(elliptic)i(regularit)n(y)c(theory)-7 b(,)41 b(w)n(e)d(see)g(that)h(the)g(set)g(of)f(minimizers)h(for)f(ev)n (ery)456 3821 y(frequency)27 b(has)g(uniformly)g(b)r(ounded)h(deriv)-5 b(ativ)n(es.)555 3920 y(Hence,)39 b(giv)n(en)d(a)g(sequence)g Fr(!)1545 3932 y Fp(n)1627 3920 y Fq(2)j FA(Q)1793 3890 y Fp(d)1868 3920 y Fu(so)c(that)i Fr(!)2219 3932 y Fp(n)2302 3920 y Fq(!)h Fr(!)i Fq(2)e FA(R)2680 3890 y Fp(d)2719 3920 y Fu(,)h(if)e(w)n(e)f(consider)f(the)456 4020 y(minimizers)26 b Fr(u)923 4032 y Fp(!)965 4040 y Fk(n)1036 4020 y Fu(pro)r(duced)h(in) g(Theorem)f(2.3)g(and)g(translated)g(so)h(that)g(the)g(in)n(terface)f (is)h(at)456 4120 y(a)g(uniformly)g(b)r(ounded)h(distance)f(from)g(the) g(origin,)g(w)n(e)g(see)g(that)g(w)n(e)g(can)g(pass)g(to)g(a)g(subse-) 456 4219 y(quence)f(and)h(obtain)f(something)h(that)g(con)n(v)n(erges)d (in)j(lo)r(cal)f Fr(C)2450 4189 y Fs(0)2488 4219 y Fu(.)37 b(Hence,)27 b(if)g(in)g(Theorem)f(2.3)456 4319 y(w)n(e)20 b(exc)n(hange)g(the)i(h)n(yp)r(othesis)f(that)g Fr(!)26 b Fq(2)d FA(Q)1861 4289 y Fp(d)1905 4319 y Fq(\000)6 b(f)p Fu(0)p Fq(g)20 b Fu(for)g Fr(!)26 b Fq(2)d FA(R)6 b Fq(\000)g(f)p Fu(0)p Fq(g)p Fu(,)20 b(w)n(e)h(obtain)g(a)g(function) 456 4419 y Fr(u)504 4431 y Fp(!)580 4419 y Fu(that)29 b(satis\014es)f(2\),)i(3\),)f(4\))g(of)f(Theorem)h(2.3.)40 b(The)28 b(conclusion)h(1\))f({)h(i.e.,)g(that)g(the)h(total)456 4518 y(energy)i(of)i(the)h(minimizer)f(w)n(as)f(\014nite)i({)e(ma)n(y)g (fail)i(b)r(ecause)e(the)i(domain)e(is)h(un)n(b)r(ounded.)456 4618 y(Similarly)-7 b(,)30 b(b)n(y)f(passing)g(to)h(limits)g(w)n(e)f (get)h(analogues)e(of)i(Theorem)f(2.5)g(for)g(an)n(y)g(frequency)456 4717 y(except)e(that)h(w)n(e)f(cannot)g(guaran)n(tee)f(that)i(the)g (functional)g(is)f(\014nite.)555 4817 y(The)k(fact)g(that)g(the)h(v)-5 b(ariational)29 b(principle)i Fq(E)2031 4829 y Fp(!)2109 4817 y Fu(is)g(not)g(a)f(b)r(ona-\014de)h(functional)g(is)g(what)456 4917 y(mak)n(es)e(it)i(imp)r(ossible)g(to)f(apply)h(straigh)n(tforw)n (ardly)c(the)k(direct)g(metho)r(ds)f(of)h(the)g(calculus)456 5016 y(of)c(v)-5 b(ariations.)555 5116 y(It)28 b(will)g(b)r(e)g(useful) h(to)e(remark)g(for)g(future)h(purp)r(oses)f(that)h(there)f(are)g(sev)n (eral)f(in)n(teresting)456 5216 y(geometric)g(features)h(of)h(lo)r(cal) f(minimizers)g(that)h(satisfy)f(the)h(Birkho\013)f(prop)r(ert)n(y)-7 b(.)p eop %%Page: 7 7 7 6 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)541 b(7)555 450 y Fu(First,)24 b(w)n(e)e(note)h(that)g(if)g Fr(u)f Fu(is)h(a)f(lo)r (cal)h(minimizer,)g(b)n(y)g(the)g(p)r(erio)r(dicit)n(y)f(assumption)h FA(\(H7\))p Fu(,)456 553 y(so)k(is)g Fr(u)689 523 y Fs(\()p Fp(k)q Fs(\))809 553 y Fu(de\014ned)h(for)f Fr(k)f Fq(2)d FA(Z)1427 523 y Fp(d)1494 553 y Fu(b)n(y:)1604 695 y Fr(u)1652 661 y Fs(\()p Fp(k)q Fs(\))1744 695 y Fu(\()p Fr(x)p Fu(\))h(=)f Fr(u)p Fu(\()p Fr(x)18 b Fu(+)g Fr(k)s Fu(\))p Fr(:)555 835 y Fu(If)k Fr(u)f Fu(is)h(Birkho\013,)g(w)n(e)f (kno)n(w)g(b)n(y)g(the)h(maxim)n(um)f(principle)h(\(see)f(Lemma)g(A.1)h (b)r(elo)n(w\))f(that,)456 937 y(giv)n(en)i Fr(u)717 907 y Fs(\()p Fp(k)q Fs(\))833 937 y Fu(and)h Fr(u)1039 907 y Fs(\()p Fp(`)p Fs(\))1122 937 y Fu(,)h(they)f(are)f(either)h (iden)n(tical)f(or)g(there)h(is)g(a)g(strict)f(comparison)g(b)r(et)n(w) n(een)456 1037 y(them)28 b(\(that)g(is,)f(either)h Fr(u)1270 1007 y Fs(\()p Fp(k)q Fs(\))1362 1037 y Fu(\()p Fr(x)p Fu(\))c Fr(<)f(u)1633 1007 y Fs(\()p Fp(`)p Fs(\))1716 1037 y Fu(\()p Fr(x)p Fu(\))29 b(for)e(an)n(y)g Fr(x)h Fu(or)f Fr(u)2365 1007 y Fs(\()p Fp(`)p Fs(\))2448 1037 y Fu(\()p Fr(x)p Fu(\))d Fr(<)f(u)2719 1007 y Fs(\()p Fp(k)q Fs(\))2811 1037 y Fu(\()p Fr(x)p Fu(\))29 b(for)e(an)n(y)g Fr(x)p Fu(\).)555 1137 y(Similarly)-7 b(,)31 b(if)g(w)n(e)f(consider)f (an)n(y)h(accum)n(ulation)f(p)r(oin)n(t)h(\(under)h(p)r(oin)n(t)n(wise) f(con)n(v)n(ergence,)456 1236 y(whic)n(h)25 b(is)h(equiv)-5 b(alen)n(t)26 b(to)g(accum)n(ulation)f(under)h(lo)r(cally)f(uniform)h Fr(C)2644 1206 y Fs(0)2707 1236 y Fu(con)n(v)n(ergence)e(b)r(ecause)456 1336 y(the)31 b Fr(u)650 1306 y Fs(\()p Fp(k)q Fs(\))743 1336 y Fu(,)h(b)r(eing)g(solutions)f(of)38 b(\(2.1\))31 b(ha)n(v)n(e)f(uniformly)h(b)r(ounded)h(second)f(deriv)-5 b(ativ)n(es\),)32 b(w)n(e)456 1435 y(obtain)e(also)f(a)h(lo)r(cal)f (minimizer)i(and,)f(hence)h(a)f(critical)f(p)r(oin)n(t.)45 b(Again)30 b(b)n(y)g(the)h(maxim)n(um)456 1535 y(principle,)j(an)n(y)f (t)n(w)n(o)g(of)g(these)h(functions)f(are)g(iden)n(tical)g(or)f(there)i (is)f(a)g(strict)g(comparison)456 1635 y(b)r(et)n(w)n(een)27 b(them.)555 1734 y(In)h(geometric)e(language,)g(w)n(e)i(ha)n(v)n(e)e (just)i(v)n(eri\014ed)f(that)h(the)g(closure)e(of)1513 1886 y Fq(L)1570 1898 y Fp(u)1650 1886 y Fu(:=)1810 1807 y Fo([)1775 1991 y Fp(k)q Fm(2)p Fl(Z)1902 1974 y Fk(d)1951 1886 y Fu(Graph)o(\()p Fr(u)2262 1852 y Fs(\()p Fp(k)q Fs(\))2355 1886 y Fu(\))456 2122 y(is)g(a)h Fn(lamination)h Fu(in)f FA(K)1193 2092 y Fp(d)1193 2143 y(!)1257 2122 y Fq(\002)16 b Fu([)p Fq(\000)p Fu(1)p Fr(;)e Fu(1].)36 b(W)-7 b(e)27 b(recall,)f(indeed,)h(that)g(laminations)f(are)g(just)h (closed)456 2222 y(sets)j(in)g(whic)n(h)g(w)n(e)g(can)g(de\014ne)g(lea) n(v)n(es)e(that)j(either)f(do)g(not)g(in)n(tersect)g(or)f(are)g(iden)n (tical.)44 b(In)456 2321 y(our)26 b(case,)h(the)h(lea)n(v)n(es)e(are)h (the)h(graphs)e(of)i(eac)n(h)e(of)i(the)g(minimizers.)555 2421 y(In)h(the)f(rational)f(case,)g(the)i(set)f(of)g(minimizers)g(is)g (discrete,)f(but)i(in)f(the)h(irrational)d(case,)456 2521 y(there)h(are)g(complicated)g(accum)n(ulations)f(of)i (translations)e(of)i(minimizers.)555 2620 y(It)i(can)e(happ)r(en)h (\(e.g.,)h(in)f(the)g(case)g(that)g(the)g Fr(a)2093 2632 y Fp(i;j)2197 2620 y Fu(=)c Fr(\016)2324 2632 y Fp(i;j)2402 2620 y Fu(,)k(that)g Fr(F)41 b Fu(do)r(es)29 b(not)g(dep)r(end)h(on)456 2723 y Fr(x)g Fu(and)g(that)g Fr(u)g Fu(is)f(a)h(one-dimensional)f (minimizers\))h(that)g Fq(L)2420 2735 y Fp(u)2491 2723 y Fu(=)c FA(K)2657 2693 y Fp(d)2657 2743 y(!)2725 2723 y Fq(\002)20 b Fu([)p Fq(\000)p Fu(1)p Fr(;)14 b Fu(1].)42 b(In)30 b(suc)n(h)g(a)456 2822 y(case,)e(follo)n(wing)g(a)h(standard)f (geometric)g(terminology)-7 b(,)28 b(w)n(e)g(sa)n(y)g(that)i Fq(L)2791 2834 y Fp(u)2863 2822 y Fu(is)f(a)g Fn(foliation)p Fu(.)43 b(It)456 2922 y(can)27 b(also)g(happ)r(en)i(ev)n(en)e(in)h(the) h(irrational)d(case)h(that)i Fq(L)2278 2934 y Fp(u)2345 2922 y Fq(6)p Fu(=)24 b FA(K)2509 2892 y Fp(d)2509 2943 y(!)2575 2922 y Fq(\002)19 b Fu([)p Fq(\000)p Fu(1)p Fr(;)14 b Fu(1])26 b(and)i(that)h(there)456 3022 y(are)e Fn(gaps)i Fu(in)f(the)h(lamination,)f(that)g(is,)g(connected)g(sets)g (in)h(the)f(complemen)n(t)g(of)g Fq(L)3155 3034 y Fp(u)3227 3022 y Fu(whose)456 3121 y(b)r(oundaries)d(are)g(just)h(t)n(w)n(o)g (lea)n(v)n(es)e(of)i Fq(L)1718 3133 y Fp(u)1762 3121 y Fu(.)36 b(W)-7 b(e)27 b(will)f(not)g(presen)n(t)f(here)h(explicit)g (examples)g(of)456 3221 y(this)h(phenomenon,)h(but)g(they)g(are)e (constructed)i(for)f(similar)f(mo)r(dels)i(in)g([Ban89)n(].)555 3321 y(In)j(the)g(irrational)d(case,)i(if)h(w)n(e)f(iden)n(tify)h(a)f (gap)g(in)g Fq(L)2274 3333 y Fp(u)2318 3321 y Fu(,)h(this)g(implies)f (that)h(there)f(is)h(also)456 3420 y(a)d(corresp)r(onding)e(gap)i(in)h Fq(L)1373 3439 y Fp(u)1412 3423 y Fi(\()p Fk(k)q Fi(\))1498 3420 y Fu(,)g Fr(k)e Fq(2)e FA(Z)1758 3390 y Fp(d)1797 3420 y Fu(.)40 b(Since)29 b Fq(L)2135 3439 y Fp(u)2174 3423 y Fi(\()p Fk(k)q Fi(\))2284 3420 y Fu(=)24 b Fq(L)2430 3432 y Fp(u)2474 3420 y Fu(,)29 b(w)n(e)f(see)g(that)h(there)f(ha)n(v)n (e)g(to)456 3520 y(b)r(e)j(an)g(in\014nite)h(n)n(um)n(b)r(er)e(of)i (gaps.)46 b(This)31 b(is)g(reminiscen)n(t)f(of)h(the)h(existence)f(of)g (gaps)f(in)h(the)456 3619 y(Can)n(tor)26 b(sets)h(in)h(the)g (Aubry-Mather)f(theory)g(of)g(dynamical)g(systems.)456 3739 y FA(Lemma)22 b(3.1.)35 b Fn(L)l(et)24 b Fr(u)1158 3751 y Fp(n)1226 3739 y Fn(b)l(e)h(a)f(se)l(quenc)l(e)g(of)h(c)l (ontinuous)e(lo)l(c)l(al)j(minimizers)f(of)g(the)f(variational)456 3839 y(pr)l(oblem)37 b Fu(\(1.1\))29 b Fn(c)l(onver)l(ging)i(p)l (ointwise)f(to)g(a)g(function)g Fr(u)p Fn(.)38 b(Then)31 b Fr(u)e Fn(is)h(a)g(lo)l(c)l(al)h(minimizer.)456 3995 y(Pr)l(o)l(of.)43 b Fu(Because)22 b(the)h(lo)r(cal)g(minimizers)g (satisfy)g(\(2.1\),)h(w)n(e)f(ha)n(v)n(e,)g(b)n(y)f(the)i(elliptic)f (regularit)n(y)456 4095 y(theory)32 b(that)i(the)g(second)f(deriv)-5 b(ativ)n(es)33 b(are)f(uniformly)i(b)r(ounded.)55 b(Therefore,)34 b(b)n(y)f(Ascoli-)456 4195 y(Arzel\022)-42 b(a)26 b(theorem)i(w)n(e)f (obtain)g(that)h Fr(u)1648 4207 y Fp(n)1720 4195 y Fu(con)n(v)n(erges)d (lo)r(cally)i Fr(C)2420 4164 y Fs(1)2485 4195 y Fu(to)h Fr(u)p Fu(.)555 4294 y(Assume,)45 b(b)n(y)40 b(con)n(tradiction,)k (that)d(there)g(exists)f(a)h(smo)r(oth)g(function)g Fr(\036)h Fu(of)f(supp)r(ort)456 4394 y(con)n(tained)27 b(in)g(a)h(ball)f Fr(B)32 b Fu(suc)n(h)27 b(that)1404 4533 y Fq(E)1448 4545 y Fp(B)1505 4533 y Fu(\()p Fr(u)18 b Fu(+)g Fr(\036)p Fu(\))h Fq(\000)f(E)1913 4545 y Fp(B)1970 4533 y Fu(\()p Fr(u)p Fu(\))24 b(=:)e Fq(\000)p Fr(\016)k(<)d Fu(0)p Fr(:)555 4673 y Fu(Then,)28 b(for)f(large)f(enough)h Fr(n)p Fu(,)h(w)n(e)f(ha)n(v)n(e)f(that)1352 4799 y Fq(jE)1419 4811 y Fp(B)1476 4799 y Fu(\()p Fr(u)1556 4811 y Fp(n)1619 4799 y Fu(+)18 b Fr(\036)p Fu(\))i Fq(\000)e(E)1930 4811 y Fp(B)1987 4799 y Fu(\()p Fr(u)g Fu(+)g Fr(\036)p Fu(\))p Fq(j)24 b(\024)e Fr(\016)s(=)p Fu(10)1352 4929 y Fq(jE)1419 4941 y Fp(B)1476 4929 y Fu(\()p Fr(u)1556 4941 y Fp(n)1601 4929 y Fu(\))d Fq(\000)f(E)1779 4941 y Fp(B)1836 4929 y Fu(\()p Fr(u)p Fu(\))p Fq(j)23 b(\024)g Fr(\016)s(=)p Fu(10)456 5069 y(and)k(therefore)1351 5192 y Fq(E)1395 5204 y Fp(B)1452 5192 y Fu(\()p Fr(u)1532 5204 y Fp(n)1595 5192 y Fu(+)19 b Fr(\036)p Fu(\))g Fq(\000)f(E)1906 5204 y Fp(B)1963 5192 y Fu(\()p Fr(u)2043 5204 y Fp(n)2088 5192 y Fu(\))23 b Fq(\024)g(\000)2306 5136 y Fu(4)p 2306 5173 42 4 v 2306 5249 a(5)2357 5192 y Fr(\016)j(<)c Fu(0)p eop %%Page: 8 8 8 7 bop 456 251 a Fs(8)664 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fu(whic)n(h)27 b(is)h(a)f(con)n(tradiction)f(with)i Fr(u)1588 462 y Fp(n)1661 450 y Fu(b)r(eing)f(a)g(minimizer.)1044 b Fj(\003)456 600 y Fu(3.1.)40 b FA(Results)25 b(on)h(the)g(existence)f(critical)i(p) s(oin)m(ts)e(other)h(than)h(minimizers.)37 b Fu(Giv)n(en)456 699 y(the)24 b(ab)r(o)n(v)n(e)e(discussion,)i(the)g(b)r(est)g(that)g (one)f(can)h(hop)r(e)f(along)g(the)h(lines)f(of)h(pro)r(ducing)f(a)g (new)456 799 y(solution)18 b(is)g(to)h(sho)n(w)e(that,)k(if)e(there)g (are)e(gaps)h(in)h Fq(L)p Fu(,)h(a)f(lamination)f(whose)g(lea)n(v)n(es) f(are)g(Birkho\013)456 898 y(minimizers,)29 b(then)g(there)g(is)g(a)g (solution)f(inside)h(the)g(gaps.)40 b(Clearly)-7 b(,)29 b(for)f(the)i(rational)d(case,)456 998 y(since)f(the)i(set)f(of)g (translations)f(is)h(discrete,)f(there)h(will)h(alw)n(a)n(ys)d(b)r(e)i (gaps,)f(and)h(hence,)h(there)456 1098 y(will)f(alw)n(a)n(ys)f(b)r(e)i (another)f(solution)g(b)r(esides)g(the)h(translations)f(of)g(the)h (Birkho\013)f(minimizer.)555 1197 y(In)36 b(general,)h(w)n(e)f(will)g (consider)f(a)g(lamination)h(of)f(minimizers.)62 b(W)-7 b(e)36 b(will)h(not)f(need)g(to)456 1297 y(assume)26 b(that)h(the)g(lamination)g(is)g(the)g(closure)f(of)h(the)g (translations)e(of)i(a)g(single)f(minimizer,)456 1397 y(ev)n(en)d(if)i(this)f(is)g(what)h(w)n(e)e(had)h(discussed)g(b)r (efore.)35 b(As)25 b(w)n(e)e(will)i(see,)f(it)h(is)f(quite)g(p)r (ossible)g(that)456 1496 y(there)j(are)g(laminations)f(generated)h(b)n (y)g(the)h(translation)f(of)g(sev)n(eral)f(minimizers.)555 1596 y(Hence,)i(our)f(result)g(is:)456 1713 y FA(Theorem)g(3.2.)37 b Fn(L)l(et)26 b Fr(u)1233 1683 y Fs(\(0\))1349 1713 y Fn(and)i Fr(u)1556 1683 y Fs(\(1\))1671 1713 y Fn(b)l(e)f(two)g(lo)l (c)l(al)h(minimizers)g(satisfying)h(the)e(c)l(onclusions)456 1813 y(in)i(The)l(or)l(em)i(2.3)g(exc)l(ept)f(1\),)g(so)g(that)1646 1954 y Fr(u)1694 1920 y Fs(\(1\))1783 1954 y Fu(\()p Fr(x)p Fu(\))24 b Fr(>)e(u)2053 1920 y Fs(\(0\))2142 1954 y Fu(\()p Fr(x)p Fu(\))456 2089 y Fn(for)30 b(any)g Fr(x)24 b Fq(2)f FA(K)971 2059 y Fp(d)971 2109 y(!)1019 2089 y Fn(.)39 b(Then,)31 b(ther)l(e)f(exists)f Fr(u)22 b Fq(2)i Fr(C)1973 2059 y Fs(2)2010 2089 y Fu(\()p FA(K)2117 2059 y Fp(d)2117 2109 y(!)2166 2089 y Fu(\))30 b Fn(satisfying:)661 2206 y Fq(\017)41 b Fr(u)30 b Fn(is)g(a)g(critic)l(al)h(p)l(oint.)661 2306 y Fq(\017)41 b Fn(F)-6 b(or)30 b(any)g Fr(k)c Fq(2)e FA(Z)1264 2276 y Fp(d)1333 2306 y Fn(so)30 b(that)f Fr(k)22 b Fq(\001)c Fr(!)26 b Fq(\025)c Fu(0)p Fn(,)30 b Fr(u)p Fu(\()p Fr(x)p Fu(\))24 b Fq(\024)e Fr(u)p Fu(\()p Fr(x)d Fu(+)f Fr(k)s Fu(\))p Fn(,)30 b(for)h(any)f Fr(x)24 b Fq(2)f FA(K)3124 2276 y Fp(d)3124 2326 y(!)3172 2306 y Fn(;)661 2408 y Fq(\017)41 b Fr(u)792 2378 y Fs(\(0\))881 2408 y Fu(\()p Fr(x)p Fu(\))24 b Fr(<)f(u)p Fu(\()p Fr(x)p Fu(\))g Fr(<)g(u)1422 2378 y Fs(\(1\))1511 2408 y Fu(\()p Fr(x)p Fu(\))p Fn(.)555 2525 y Fu(Later,)j(when)h(w)n(e)g(ha)n(v)n(e)f (in)n(tro)r(duced)g(more)g(notation,)h(w)n(e)f(will)h(state)g(some)f (more)g(precise)456 2625 y(results.)36 b(As)28 b(it)g(turns)f(out,)h (the)g(solutions)f(th)n(us)h(pro)r(duced)f(could)g(ha)n(v)n(e)g(an)g (in\014nite)h(energy)-7 b(.)456 2725 y(Nev)n(ertheless,)43 b(w)n(e)d(will)h(sho)n(w)f(that)h(they)g(ha)n(v)n(e)f(a)g(\014nite)h Fn(r)l(enormalize)l(d)j(ener)l(gy)k Fu(de\014ned)456 2824 y(in)35 b(\(5.8\).)60 b(This)35 b(renormalized)f(energy)g(will)h (allo)n(w)f(us)i(to)f(form)n(ulate)f(a)h(criterion)f(for)h(the)456 2924 y(existence)27 b(of)g(a)h(foliation)f(b)n(y)g(minimizers)h(or)e (not.)555 3024 y(The)e(pro)r(of)e(of)h(Theorem)g(3.2)f(do)r(es)h(not)g (dep)r(end)h(on)f(the)h(exp)r(onen)n(tial)e(b)r(ounds)i(concluded)456 3123 y(in)f(Theorem)g(2.3.)34 b(Of)23 b(course,)g(when)h(the)f Fr(u)1845 3093 y Fs(\(0\))1957 3123 y Fu(and)h Fr(u)2163 3093 y Fs(\(1\))2275 3123 y Fu(w)n(e)e(consider)h(are)f(limits)i(of)f (rational)456 3223 y(minimizers,)f(they)f(satisfy)g(the)g(exp)r(onen)n (tial)g(b)r(ounds)g(and)g(therefore,)g(since)g Fr(u)2950 3193 y Fs(\(0\))3062 3223 y Fr(<)i(u)f(<)h(u)3356 3193 y Fs(\(1\))456 3323 y Fu(so)18 b(do)r(es)g Fr(u)p Fu(.)34 b(In)19 b(the)g(case)g(that)g Fr(u)1448 3292 y Fs(\(0\))1555 3323 y Fu(and)g Fr(u)1756 3292 y Fs(\(1\))1864 3323 y Fu(are)f(obtained)g(as)h(limits)g(of)g(rational)f(minimizers,)456 3422 y(they)30 b(inherit)g(the)h(prop)r(ert)n(y)e(that)h(the)h(in)n (terface)e(is)i(con)n(tained)e(in)i(a)e(strip)h(whose)g(width)h(is)456 3522 y(b)r(ounded)c(b)n(y)g(a)g(univ)n(ersal)f(n)n(um)n(b)r(er.)37 b(Therefore,)26 b(the)i(solutions)f Fr(u)g Fu(w)n(e)g(construct)f(also) h(ha)n(v)n(e)456 3621 y(a)g(width)h(b)r(ounded)g(b)n(y)f(a)h(univ)n (ersal)e(constan)n(t.)555 3721 y(One)h(question)f(w)n(e)g(ha)n(v)n(e)g (not)g(settled)i(in)e(this)h(pap)r(er)g(is)f(whether)h(all)f (minimizers)h(can)f(b)r(e)456 3821 y(appro)n(ximated)17 b(b)n(y)h(p)r(erio)r(dic)h(ones.)33 b(Closely)18 b(related)g(questions) h(are)e(whether)i(all)g(minimizers)456 3920 y(are)30 b(Birkho\013,)j(whether)f(all)f(minimizers)h(satisfy)f(con)n(v)n(erge)f (to)h(free)h(phases)f(exp)r(onen)n(tially)456 4020 y(fast.)36 b(The)24 b(later)h(question)f(is)h(also)f(connected)g(to)h(whether)g (the)g(structure)f(of)h(the)g(in)n(terfaces)456 4120 y(is)30 b(simple)g(enough)f(at)h(in\014nit)n(y)h(so)e(that)i(one)e(can) h(construct)g(barrier)e(functions)i(as)g(is)g(done)456 4219 y(in)d(the)h(pro)r(of)f(of)h(Prop)r(osition)e(4.3.)456 4369 y(3.2.)40 b FA(Remarks)31 b(on)i(the)f(literature,)g(sk)m(etc)m(h) i(of)e(the)g(pro)s(ofs)g(and)h(organization)f(of)456 4468 y(the)f(pap)s(er.)456 4618 y Fu(3.2.1.)40 b Fn(R)l(emarks)23 b(on)g(the)g(liter)l(atur)l(e.)42 b Fu(In)21 b(the)g(elliptic)f(in)n (tegrand)g(setting,)i(a)e(result)g(related)g(to)456 4717 y(Theorem)27 b(3.2)h(for)h(the)g(rational)e(case)h(has)g(b)r(een)h (announced)f(b)n(y)h(P)n(aul)f(Rabino)n(witz)g(on)g(the)456 4817 y(o)r(ccasion)34 b(of)i(the)g(In)n(ternational)f(Symp)r(osium)h (on)f(V)-7 b(ariational)35 b(Metho)r(ds)h(and)f(Nonlinear)456 4917 y(Di\013eren)n(tial)k(Equations,)j(Rome,)g(2005.)71 b(The)40 b(study)g(of)f(the)h(irrational)e(case)h(has)g(b)r(een)456 5016 y(recen)n(tly)32 b(started)h(in)h([Bes05)n(])g(for)f(elliptic)h (in)n(tegrands,)g(under)f(an)g(additional)g(h)n(yp)r(othesis)456 5116 y(in)n(tro)r(duced)40 b(b)n(y)h([Ban87)n(].)77 b(T)-7 b(o)41 b(the)g(b)r(est)h(of)e(our)h(kno)n(wledge,)h(a)f(full)h (treatmen)n(t)e(of)h(the)456 5216 y(irrational)25 b(case)i(w)n(as)g (not)g(a)n(v)-5 b(ailable)27 b(y)n(et.)p eop %%Page: 9 9 9 8 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)541 b(9)555 450 y Fu(W)-7 b(e)35 b(also)e(p)r(oin)n(t)h(out)h(that,)h(for)e Fr(d)g Fu(=)g(1,)h(w)n(e)f(obtain)g(from)g(Theorem)f(3.2)h(the)h(follo) n(wing)456 550 y(m)n(ultiplicit)n(y)28 b(result)f(for)g(hetero)r (clinics:)456 683 y FA(Corollary)36 b(3.3.)42 b Fn(In)33 b(the)g(setting)f(of)i(Cor)l(ol)t(lary)h(2.4,)g(the)e(or)l(dinary)h (di\013er)l(ential)g(e)l(quation)462 783 y Fu(\177)-48 b Fr(q)s Fu(\()p Fr(t)p Fu(\))45 b(=)f Fr(@)788 795 y Fp(\026)833 783 y Fr(F)12 b Fu(\()p Fr(t;)i(q)s Fu(\()p Fr(t)p Fu(\)\))p Fn(,)46 b(has)c(at)g(le)l(ast)f(two)h(solutions)g (satisfying)50 b Fu(\(2.2\))42 b Fn(and)g(at)f(le)l(ast)h(two)456 882 y(solutions)23 b(satisfying)32 b Fu(\(2.3\))p Fn(.)k(A)n(lso,)26 b(if)e Fr(q)i Fn(is)e(any)g(of)g(such)f(solutions,)j(we)d(have)i(that)e Fq(j)p Fr(q)s Fu(\()p Fr(t)p Fu(\))p Fq(j)h(\024)e Fu(1)p Fn(,)456 982 y(and)30 b(either)g Fr(q)s Fu(\()p Fr(t)19 b Fu(+)f Fr(k)s Fu(\))23 b Fq(\025)g Fr(q)s Fu(\()p Fr(t)p Fu(\))30 b Fn(or)g Fr(q)s Fu(\()p Fr(t)19 b Fq(\000)f Fr(k)s Fu(\))23 b Fq(\025)g Fr(q)s Fu(\()p Fr(t)p Fu(\))p Fn(,)31 b(for)f(any)h Fr(k)25 b Fq(2)f FA(N)29 b Fn(and)i(any)f Fr(t)23 b Fq(2)g FA(R)p Fn(.)555 1115 y Fu(There)i(are)g(a)g(v)-5 b(ariet)n(y)24 b(of)i(results)f(on)g(asymptotic)g(tra)5 b(jectories)23 b(of)j(Hamiltonian)f(systems)456 1215 y(whic)n(h)e(are)g(related)g(to)g(the)h(ones)f(in)h(Corollary)d(2.4)h (and)i(Corollary)d(3.3)h(here)h(ab)r(o)n(v)n(e:)34 b(see,)24 b(for)456 1315 y(instance,)31 b([CZES90)o(],)h([Bol95)n(])f(and)g (references)f(therein.)47 b(In)31 b(this)g(sense,)h(our)e(results)g(ma) n(y)456 1414 y(also)c(b)r(een)i(considered)f(as)g(PDE)g(v)n(ersions)f (of)h(some)g(results)g(in)h(dynamical)f(systems.)555 1514 y(There)35 b(seems)f(also)g(to)h(b)r(e)g(a)g(close)f(connection)g (with)i(the)f(Aubry-Mather)f(theory)-7 b(.)58 b(In)456 1614 y(particular,)30 b(in)h(the)g(case)f(of)h(irrational)e(frequency) -7 b(,)32 b(one)e(can)h(considered)f(the)h(existence)g(of)456 1713 y(gaps)f(as)i(b)r(eing)g(v)n(ery)e(similar)i(to)f(the)i(existence) e(of)h(the)g(Aubry-Mather)f(Can)n(tor)g(sets)h(and)456 1813 y(the)j(case)g(that)g(the)h(minimizers)f(form)g(a)g(foliation)g (corresp)r(onds)f(to)h(the)g(existence)g(of)h(an)456 1912 y(in)n(v)-5 b(arian)n(t)38 b(circle.)72 b(F)-7 b(or)39 b(t)n(wist)h(mappings,)i(the)e(fact)g(that)g(when)g(there)f(are)g(gaps) f(in)i(the)456 2012 y(Aubry-Mather)25 b(sets)i(there)f(are)g(other)g (Can)n(tor)f(sets)i(homo)r(clinic)f(to)h(the)g(previous)e(one)h(w)n(as) 456 2112 y(pro)n(v)n(ed)j(in)h([Mat86)o(].)46 b(In)31 b Fq(x)f Fu(5.9,)h(w)n(e)f(discuss)g(some)g(relations)f(of)i(the)g (reduced)f(energy)f(that)456 2211 y(w)n(e)23 b(in)n(tro)r(duce)g(in)h (\(5.8\))f(with)h(the)g(P)n(eierls)d(barrier)h(in)n(tro)r(duced)h(in)h ([Mat86)o(])g(extending)f(ideas)456 2311 y(of)28 b([ALD83].)39 b(A)29 b(more)f(detailed)g(comparison)f(b)r(et)n(w)n(een)h(the)h (results)f(for)g(PDE's)g(and)g(those)456 2411 y(for)f(Aubry-Mather)f (theory)h(can)g(b)r(e)h(found)g(in)g([Mos86)o(].)456 2625 y(3.2.2.)40 b Fn(Sketch)e(of)g(the)g(pr)l(o)l(of.)43 b Fu(The)37 b(main)f(ideas)f(for)h(pro)n(ving)f(the)h(ab)r(o)n(v)n(e)f (results)h(are)f(the)456 2725 y(follo)n(wing.)72 b(Theorems)39 b(2.3)g(and)h(2.5)f(follo)n(w)g(from)h(some)f(results)g(of)h([V)-7 b(al04)o(])40 b(and)g(some)456 2825 y(estimates)25 b(on)h(the)h(exp)r (onen)n(tial)f(approac)n(h)e(to)i(pure)g(phases)f(of)h(the)h(solutions) e(found)i(there.)555 2924 y(The)f(pro)r(of)g(of)g(Theorem)f(3.2)h(uses) f(some)h(prop)r(erties)f(of)h(the)h(asso)r(ciated)e(heat)h(\015o)n(w.) 35 b(The)456 3024 y(heat)27 b(\015o)n(w)f(is)h(our)f(to)r(ol)h(to)g(o)n (v)n(ercome)e(the)i(in)n(trinsic)g(lac)n(k)f(of)h(compactness)f(of)h (the)g(problems)456 3123 y(\(namely)-7 b(,)31 b(its)g(translation)f(in) n(v)-5 b(ariance)29 b(and)h(the)h(fact)g(that)g(the)g(domain)g(of)f (the)h(functional)456 3223 y(is)d(un)n(b)r(ounded\),)i(whic)n(h)e(mak)n (es)g(the)h(standard)e(nonlinear)h(analysis)f(metho)r(ds)i(not)g (directly)456 3323 y(applicable.)k(Namely)-7 b(,)22 b(the)e(pro)r(of)f (of)h(Theorem)f(3.2)h(is)f(b)n(y)h(con)n(tradiction:)32 b(if)20 b(no)g(other)f(critical)456 3422 y(p)r(oin)n(ts)28 b(existed,)g(the)g(heat)g(\015o)n(w)g(w)n(ould)g(gain)f(compactness)g (and)h(pro)r(duce)g(a)g(new)g(solution.)456 3522 y(Argumen)n(ts)f(of)h (this)g(t)n(yp)r(e)g(ha)n(v)n(e)e(b)r(een)j(used)f(in)g([KdlLR97)o(].) 37 b(In)28 b(our)f(case,)g(the)i(heat)e(\015o)n(w)h(is)456 3622 y(the)22 b(gradien)n(t)g(of)g(a)g(functional,)i(whic)n(h)e(w)n(e)g (call)g(the)h(renormalized)e(energy)i(\(5.8\))o(.)36 b(The)22 b(pro)r(of)456 3721 y(is)h(somewhat)g(reminiscen)n(t)h(of)g (the)g(the)g(Ljusternik-Sc)n(hnirelmann)f(theory)g(\(see)h([LS34)o(])g (and)456 3821 y([Szu89)o(]\))39 b(but)h(w)n(e)f(ha)n(v)n(e)f(to)h(o)n (v)n(ercome)e(some)h(di\016culties)i(arising)e(from)h(the)g(un)n(b)r (ounded)456 3920 y(nature)26 b(of)h(the)g(problem)g(so)f(that)h(the)h (c)n(hoice)e(of)h(top)r(ologies)e(is)i(not)g(so)f(ob)n(vious,)g(so)g(w) n(e)h(\014nd)456 4020 y(it)j(adv)-5 b(an)n(tageous)28 b(to)h(consider)g(the)i(gradien)n(t)d(\015o)n(w)i(\(whic)n(h)g(in)g (our)f(con)n(text)g(is)h(a)g(parab)r(olic)456 4120 y(equation)20 b(that)i(w)n(e)f(call)g(the)h(heat)g(\015o)n(w\))f(directly)g(rather)f (than)i(to)f(use)g(some)g(of)h(the)f(standard)456 4219 y(v)n(ersions)30 b(of)j(the)f(theory)g(as)g(in)h([Sc)n(h64)o(])f(and)g ([Bro65)n(].)52 b(In)33 b(our)e(con)n(text,)j(the)e(heat)h(\015o)n(w)f (is)456 4319 y(particularly)e(useful)h(since)h(it)g(preserv)n(es)d(the) j(order,)f(whic)n(h)g(is)h(an)f(imp)r(ortan)n(t)g(part)g(of)g(our)456 4419 y(conclusions.)36 b(The)28 b(order)e(prop)r(erties)h(will)h(also)f (giv)n(e)g(some)g(in)n(tegral)f(a-priori)g(b)r(ounds)i(that)456 4518 y(supplemen)n(t)g(the)g(usual)f(lo)r(cal)g(regularit)n(y)e(gains)i (to)h(giv)n(e)e(strong)h(compactness)f(prop)r(erties.)555 4618 y(In)g(order)e(to)i(apply)f(the)h(heat)g(\015o)n(w)f(e\013ectiv)n (ely)-7 b(,)26 b(w)n(e)f(ha)n(v)n(e)g(to)g(use)h(that)g(it)g(decreases) e(some)456 4717 y(w)n(ell)31 b(de\014ned)i(renormalized)d(energy)-7 b(,)32 b(de\014ned)g(in)h(\(5.8\))o(,)g(and)f(w)n(e)g(ha)n(v)n(e)f(to)g (pro)n(vide)g(a)h(v)n(ery)456 4817 y(simple)23 b(a-priori)e(estimate)h (that)i(is)e(obtained)h(directly)g(from)f(the)h(fact)g(that)h(the)f (translations)456 4917 y(form)k(a)g(lamination)g(\(see)h(Lemma)f(5.1)g (here)g(b)r(elo)n(w\).)555 5016 y(W)-7 b(e)32 b(men)n(tion)g(that)g (the)g(use)g(of)f(the)h(renormalized)e(energy)h(and)g(the)i(conclusion) d(of)i(ex-)456 5116 y(istence)e(of)g(homo)r(clinic)h(orbits)f(in)g(the) h(gaps)e(is)i(somewhat)f(reminiscen)n(t)g(of)g(the)h(results)f(on)456 5216 y(existence)f(of)h(homo)r(clinic)f(Can)n(tor)g(sets)g(in)h([Mat86) o(],)h(whic)n(h)e(uses)h(v)n(ery)e(di\013eren)n(t)i(metho)r(ds)p eop %%Page: 10 10 10 9 bop 456 251 a Fs(10)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fu(than)36 b(those)h(in)g(this)g(pap)r(er.)63 b(W)-7 b(e)37 b(think)h(that)f(it)g (w)n(ould)f(b)r(e)h(v)n(ery)e(in)n(teresting)h(to)h(under-)456 550 y(tak)n(e)30 b(a)h(more)g(systematic)g(comparison)e(b)r(et)n(w)n (een)i(the)h(metho)r(ds)g(of)f(this)h(pap)r(er)f(and)g(those)456 649 y(in)g([Mat86)o(].)47 b(The)31 b(relation)f(of)h(gradien)n(t)f (\015o)n(w)g(metho)r(ds)h(and)g(Aubry-Mather)f(theory)g(w)n(as)456 749 y(p)r(oin)n(ted)d(out)h(in)g([Ang90a)n(].)456 934 y(3.2.3.)40 b Fn(Or)l(ganization)f(of)h(the)f(p)l(ap)l(er.)k Fu(The)38 b(pap)r(er)f(is)h(organized)e(as)h(follo)n(ws.)66 b(In)37 b Fq(x)h Fu(4,)i(w)n(e)456 1034 y(pro)n(v)n(e)26 b(Theorems)g(2.3)h(and)h(2.5.)36 b(The)28 b(pro)r(ofs)f(will)h(mak)n(e) f(use)h(of)f(the)h(exp)r(onen)n(tial)g(deca)n(y)f(of)456 1134 y(the)g(solutions)g(and)g(of)g(some)f(results)h(in)g([V)-7 b(al04)o(].)37 b(In)28 b Fq(x)f Fu(5,)g(w)n(e)g(in)n(tro)r(duce)f(a)h (\\renormalized")456 1233 y(energy)-7 b(.)66 b(This)38 b(is)f(needed)h(b)r(ecause)g(the)g(\\regular")d(energy)h Fq(E)46 b Fu(ma)n(y)37 b(b)r(ecome)g(in\014nite)i(on)456 1333 y(un)n(b)r(ounded)20 b(domains.)33 b(W)-7 b(e)20 b(then)h(consider)d(the)i(heat)g(\015o)n(w)f(asso)r(ciated)f(to)i(this) g(renormalized)456 1432 y(energy)-7 b(.)57 b(The)35 b(basin)g(of)g (attraction)e(of)i(suc)n(h)g(a)f(heat)h(\015o)n(w)f(is)h(th)n(us)g (discussed,)h(under)f(the)456 1532 y(assumption)20 b(that)h(the)g (claim)g(of)f(Theorem)g(3.2)g(is)h(false.)34 b(This)21 b(will)g(lead)f(to)h(the)g(construction)456 1632 y(of)d(a)h(further)f (critical)h(p)r(oin)n(t,)h(b)n(y)f(heat)g(\015o)n(wing)e(a)i(suitable)f (path)h(of)g(initial)g(data,)h(th)n(us)f(pro)n(ving)456 1731 y(Theorem)31 b(3.2.)50 b(Some)32 b(features)g(of)g(the)h(energy)e (barriers)f(are)i(also)f(discussed,)i(in)f(relation)456 1831 y(with)c(analogous)d(phenomena)i(arising)f(in)i(Aubry-Mather)f (theory)-7 b(.)555 1931 y(The)30 b(pap)r(er)f(ends)h(with)g(an)g(app)r (endix.)43 b(The)30 b(aim)f(of)h Fq(x)g Fu(A.1)f(and)h Fq(x)g Fu(A.2)f(\(whic)n(h)h(ma)n(y)f(b)r(e)456 2030 y(skipp)r(ed)g(b)n(y)f(exp)r(ert)h(readers\))f(is)g(to)h(state)g(some)f (standard)g(elliptic)i(and)e(parab)r(olic)g(results)456 2130 y(in)h(a)g(w)n(a)n(y)f(that)i(\014ts)f(our)g(purp)r(oses.)41 b(In)29 b Fq(x)h Fu(A.3,)f(w)n(e)g(collect)g(some)g(results)g(of)g([V) -7 b(al04)o(])30 b(whic)n(h)456 2229 y(are)c(used)i(in)g(the)g(pro)r (ofs)e(of)i(the)g(main)f(results)h(of)f(this)h(pap)r(er.)459 2450 y(4.)42 b Fv(Exponential)30 b(deca)-6 b(y)32 b(f)n(or)g(the)g (solutions)e(and)h(the)h(plane-like)f(minimizers)555 2599 y Fu(W)-7 b(e)28 b(deduce)f(from)g(the)g(comparison)f(principle)h (Lemma)g(A.1)g(an)g(exp)r(onen)n(tial)f(b)r(ound)i(on)456 2699 y(the)g(solutions)e(of)i(our)f(equation)g(in)h(the)g(rational)e (frequency)h(case.)456 2827 y FA(Corollary)33 b(4.1.)40 b Fn(L)l(et)29 b Fr(u)23 b Fq(2)g Fr(W)1454 2787 y Fs(1)p Fp(;)p Fs(2)1442 2852 y(lo)r(c)1544 2827 y Fu(\()p FA(K)1651 2797 y Fp(d)1651 2848 y(!)1699 2827 y Fu(\))30 b Fn(satisfy)h(we)l (akly)1509 2980 y Fr(@)1553 2992 y Fp(i)1580 2980 y Fu(\()p Fr(a)1656 2992 y Fp(i;j)1735 2980 y Fr(@)1779 2992 y Fp(j)1814 2980 y Fr(u)p Fu(\))23 b(=)f Fr(@)2048 2992 y Fp(\026)2093 2980 y Fr(F)12 b Fu(\()p Fr(x;)i(u)p Fu(\))g Fr(:)456 3132 y Fn(L)l(et)35 b(the)h(c)l(o)l(e\016cients)h Fr(a)1223 3144 y Fp(ij)1317 3132 y Fn(b)l(e)f(Lipschitz)i(c)l (ontinuous)d(and)h(satisfy)h(the)g(uniform)f(el)t(lipticity)456 3232 y(c)l(ondition)d(given)f(in)g Fg(\(H1\))p Fn(.)46 b(L)l(et)32 b Fr(F)43 b Fn(satisfy)33 b(the)f(assumptions)g(in)g Fg(\(H0\))p Fn({)p Fg(\(H7\))j Fn(and)d(let)g Fr(\025)456 3331 y Fn(b)l(e)c(the)h(quantity)f(intr)l(o)l(duc)l(e)l(d)h(ther)l(e.) 38 b(Assume)28 b(that)h Fu(1)22 b Fq(\025)h Fr(u)p Fu(\()p Fr(x)p Fu(\))h Fq(\025)e Fu(1)16 b Fq(\000)f Fr(\025)29 b Fn(for)h(any)f Fr(x)23 b Fq(2)h FA(R)3275 3301 y Fp(d)3341 3331 y Fn(for)456 3431 y(which)719 3398 y Fp(!)p 699 3412 84 4 v 699 3460 a Fm(j)p Fp(!)r Fm(j)811 3431 y Fq(\001)18 b Fr(x)24 b Fq(\025)e Fr(M)39 b Fn(and)30 b(that)456 3631 y Fu(\(4.1\))936 b(lim)1493 3681 y Fp(s)p Fm(\000)-11 b(!)p Fs(+)p Fm(1)1762 3631 y Fr(u)1810 3539 y Fo(\020)1859 3631 y Fr(x)19 b Fu(+)f Fr(s)2080 3575 y(!)p 2057 3612 101 4 v 2057 3688 a Fq(j)p Fr(!)s Fq(j)2168 3539 y Fo(\021)2241 3631 y Fu(=)k(1)14 b Fr(:)456 3827 y Fn(Then,)30 b(ther)l(e)g(exists)g(a)g(universal)g(c)l(onstant)f Fr(a)23 b(>)g Fu(0)29 b Fn(so)h(that)1498 3988 y Fr(u)p Fu(\()p Fr(x)p Fu(\))23 b Fq(\025)g Fu(1)18 b Fq(\000)g Fr(\025e)1998 3948 y Fp(a)p Fs(\()p Fp(M)6 b Fm(\000)2211 3926 y Fk(!)p 2192 3935 77 3 v 2192 3969 a Fh(j)p Fk(!)q Fh(j)2278 3948 y Fm(\001)p Fp(x)p Fs(\))2379 3988 y Fr(:)456 4176 y Fn(Pr)l(o)l(of.)43 b Fu(Let)30 b Fr(\014)t Fu(\()p Fr(x)p Fu(\))e(=)e(1)19 b Fq(\000)h Fr(\025e)1381 4136 y Fp(a)p Fs(\()p Fp(M)6 b Fm(\000)1593 4114 y Fk(!)p 1574 4123 V 1574 4157 a Fh(j)p Fk(!)q Fh(j)1660 4136 y Fm(\001)p Fp(x)p Fs(\))1778 4176 y Fu(if)1886 4144 y Fp(!)p 1866 4158 84 4 v 1866 4205 a Fm(j)p Fp(!)r Fm(j)1979 4176 y Fq(\001)20 b Fr(x)27 b Fq(\025)f Fr(M)9 b Fu(,)30 b(with)h Fr(a)26 b Fq(2)h Fu(\(0)p Fr(;)14 b Fu(1\))30 b(to)f(b)r(e)h(c)n(hosen)f(in)456 4285 y(the)f(sequel.)36 b(Then,)1446 4403 y Fr(@)1490 4415 y Fp(i)1518 4403 y Fu(\()p Fr(a)1594 4415 y Fp(i;j)1673 4403 y Fr(@)1717 4415 y Fp(j)1752 4403 y Fr(\014)t Fu(\))23 b Fq(\025)g(\000)p Fr(C)2076 4369 y Fp(?)2114 4403 y Fr(a)p Fu(\(1)18 b Fq(\000)g Fr(\014)t Fu(\))c Fr(;)456 4538 y Fu(for)27 b(a)g(suitable)g Fr(C)1026 4508 y Fp(?)1088 4538 y Fr(>)22 b Fu(0.)37 b(De\014ne)28 b Fr( )e Fu(=)d Fr(\014)g Fq(\000)18 b Fr(u)p Fu(.)36 b(Th)n(us,)28 b(from)f FA(\(H5\))p Fu(,)670 4690 y Fr(@)714 4702 y Fp(i)742 4690 y Fu(\()p Fr(a)818 4702 y Fp(i;j)896 4690 y Fr(@)940 4702 y Fp(j)975 4690 y Fr( )s Fu(\))19 b Fq(\000)f Fr(C)1231 4656 y Fp(?)1269 4690 y Fr(a )86 b Fq(\025)d(\000)p Fr(C)1731 4656 y Fp(?)1769 4690 y Fr(a)p Fu(\(1)18 b Fq(\000)g Fr(\014)t Fu(\))h Fq(\000)f Fr(@)2217 4702 y Fp(\026)2262 4690 y Fr(F)12 b Fu(\()p Fr(x;)i(u)p Fu(\))k Fq(\000)g Fr(C)2689 4656 y Fp(?)2728 4690 y Fr(a\014)23 b Fu(+)18 b Fr(C)2990 4656 y Fp(?)3028 4690 y Fr(au)23 b Fq(\025)1453 4815 y(\025)83 b(\000)p Fr(C)1731 4781 y Fp(?)1769 4815 y Fr(a)18 b Fu(+)g Fr(\025)p Fu(\(1)h Fq(\000)f Fr(u)p Fu(\))g(+)g Fr(C)2384 4781 y Fp(?)2423 4815 y Fr(au)k Fu(=)1453 4939 y(=)83 b(\()p Fr(\025)19 b Fq(\000)f Fr(C)1848 4905 y Fp(?)1886 4939 y Fr(a)p Fu(\)\(1)h Fq(\000)f Fr(u)p Fu(\))23 b Fq(\025)f Fu(0)14 b Fr(;)456 5092 y Fu(if)558 5059 y Fp(!)p 538 5073 V 538 5121 a Fm(j)p Fp(!)r Fm(j)642 5092 y Fq(\001)d Fr(x)24 b Fq(\025)f Fr(M)9 b Fu(,)24 b(pro)n(vided)f(that)h Fr(a)f(<)g(C)1706 5062 y Fp(?)1744 5092 y Fr(=\025)p Fu(.)36 b(Assume,)25 b(b)n(y)e(con)n(tradiction,)h (that)g(there)g(exists)456 5206 y Fr(x)503 5175 y Fp(?)577 5206 y Fu(so)35 b(that)904 5173 y Fp(!)p 884 5187 V 884 5234 a Fm(j)p Fp(!)r Fm(j)1001 5206 y Fq(\001)24 b Fr(x)1095 5175 y Fp(?)1170 5206 y Fq(\025)36 b Fr(M)44 b Fu(and)35 b Fr(u)p Fu(\()p Fr(x)1692 5175 y Fp(?)1731 5206 y Fu(\))h Fr(<)g(\014)t Fu(\()p Fr(x)2030 5175 y Fp(?)2069 5206 y Fu(\).)61 b(Then,)38 b(there)d(exists)g Fr(")2931 5218 y Fs(0)3004 5206 y Fr(>)h Fu(0)f(so)g(that)p eop %%Page: 11 11 11 10 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(11)456 450 y Fr( )s Fu(\()p Fr(x)592 420 y Fp(?)631 450 y Fu(\))23 b Fr(>)g(")813 462 y Fs(0)850 450 y Fu(.)36 b(Notice)28 b(that)g(the)g(set)1374 636 y(\012)23 b(:=)g Fq(f)p Fr( )i(>)e(")1816 648 y Fs(0)1853 636 y Fq(g)18 b(\\)1987 544 y Fo(n)2075 580 y Fr(!)p 2052 617 101 4 v 2052 693 a Fq(j)p Fr(!)s Fq(j)2181 636 y(\001)h Fr(x)k(>)g(M)2471 544 y Fo(o)456 841 y Fu(m)n(ust)k(b)r(e)h(b)r(ounded)g(in)g(the)g(direction)f(of)h Fr(!)s Fu(,)f(since)1558 1022 y(lim)1489 1072 y Fp(s)p Fm(\000)-12 b(!)p Fs(+)p Fm(1)1758 1022 y Fr( )1815 930 y Fo(\020)1864 1022 y Fr(x)19 b Fu(+)f Fr(s)2085 966 y(!)p 2062 1003 V 2062 1079 a Fq(j)p Fr(!)s Fq(j)2173 930 y Fo(\021)2245 1022 y Fu(=)23 b(0)14 b Fr(:)456 1227 y Fu(Then,)27 b(a)h(con)n(tradiction)e(easily)h(follo)n(ws)f(from)i (Lemma)f(A.1.)966 b Fj(\003)456 1414 y Fu(4.1.)40 b FA(Existence)26 b(and)g(qualitativ)m(e)h(prop)s(erties)e(of)h(global)f(minima)d(when)k Fr(!)g Fq(2)d FA(Q)3333 1384 y Fp(d)3380 1414 y Fq(\000)456 1514 y(f)p Fu(0)p Fq(g)p FA(.)39 b Fu(In)22 b(this)f(section,)i(w)n(e)e (sho)n(w)f(the)i(existence)f(of)g(a)g(global)f(minim)n(um)i(for)f Fq(E)2915 1526 y Fp(!)2984 1514 y Fu(and)g(w)n(e)g(p)r(oin)n(t)456 1613 y(out)h(that)h(it)h(will)f(approac)n(h)d(the)k(b)r(oundary)e(v)-5 b(alues)22 b(exp)r(onen)n(tially)g(fast.)35 b(W)-7 b(e)24 b(will)f(also)e(p)r(oin)n(t)456 1713 y(out)27 b(that)h(all)f(global)g (minima)h(satisfy)f(the)h(Birkho\013)f(prop)r(ert)n(y)-7 b(.)555 1812 y(First,)22 b(w)n(e)e(pro)n(v)n(e)e(the)i(exp)r(onen)n (tial)g(con)n(v)n(ergence)e(at)i(in\014nit)n(y)g(for)g(the)g(lo)r(cal)g (minima)g(found)456 1912 y(in)27 b([V)-7 b(al04)o(]:)456 2042 y FA(Corollary)37 b(4.2.)42 b Fn(L)l(et)32 b Fr(u)h Fn(b)l(e)g(the)g(lo)l(c)l(al)h(minimizer)g(given)f(by)h(The)l(or)l(em)g (A.8.)48 b(Then,)35 b(ther)l(e)456 2142 y(exist)29 b(p)l(ositive)i (universal)g(c)l(onstants)d Fr(c)1704 2154 y Fs(1)1742 2142 y Fn(,)i Fr(c)1833 2154 y Fs(2)1900 2142 y Fn(and)g Fr(c)2097 2154 y Fs(3)2134 2142 y Fn(,)g(so)g(that)1277 2308 y Fq(j)p Fr(u)p Fu(\()p Fr(x)p Fu(\))19 b Fq(\000)f Fu(1)p Fq(j)23 b(\024)g Fr(c)1773 2320 y Fs(2)1810 2308 y Fr(e)1849 2268 y Fm(\000)p Fp(c)1931 2276 y Fi(3)1992 2245 y Fk(!)p 1972 2254 77 3 v 1972 2289 a Fh(j)p Fk(!)q Fh(j)2059 2268 y Fm(\001)p Fp(x)2134 2308 y Fr(;)184 b(if)93 b(!)21 b Fq(\001)d Fr(x)24 b Fq(\025)f Fr(c)2813 2320 y Fs(1)1061 2442 y Fu(and)105 b Fq(j)p Fr(u)p Fu(\()p Fr(x)p Fu(\))20 b(+)e(1)p Fq(j)k(\024)h Fr(c)1796 2454 y Fs(2)1833 2442 y Fr(e)1872 2402 y Fp(c)1902 2410 y Fi(3)1963 2380 y Fk(!)p 1944 2389 V 1944 2423 a Fh(j)p Fk(!)q Fh(j)2030 2402 y Fm(\001)p Fp(x)2105 2442 y Fr(;)184 b(if)93 b(!)21 b Fq(\001)e Fr(x)k Fq(\024)g(\000)p Fr(c)2849 2454 y Fs(1)2899 2442 y Fr(:)456 2651 y Fn(Pr)l(o)l(of.)43 b Fu(Since)23 b Fq(E)972 2663 y Fp(!)1020 2651 y Fu(\()p Fr(u)p Fu(\))g Fr(<)g Fu(+)p Fq(1)p Fu(,)h(it)g(follo)n(ws)e(that)94 b(lim)1960 2701 y Fp(s)p Fm(\000)-11 b(!\0061)2230 2651 y Fr(u)p Fu(\()p Fr(x)10 b Fu(+)g Fr(s)2514 2595 y(!)p 2491 2632 101 4 v 2491 2708 a Fq(j)p Fr(!)s Fq(j)2602 2651 y Fu(\))23 b(=)g Fq(\006)p Fu(1)f(for)h(an)n(y)g Fr(x)g Fq(2)g FA(K)3373 2621 y Fp(d)3373 2671 y(!)3421 2651 y Fu(:)456 2799 y(if)33 b(not,)i(using)d(that)i Fr(u)e Fu(is)h(uniformly)g(H\177)-42 b(older)32 b(con)n(tin)n(uous)g (\(see)h([GG82]\),)i(there)d(w)n(ould)h(b)r(e)456 2899 y(a)e(family)g(of)h(disjoin)n(t)g(balls)f(of)g(uniform)h(radius)e(on)i (whic)n(h)f Fq(j)p Fr(u)p Fq(j)f(\024)f Fu(1)20 b Fq(\000)h Fr(\016)s Fu(,)33 b(for)e(some)g Fr(\016)i(>)c Fu(0,)456 2998 y(and)e(this)i(w)n(ould)e(pro)n(vide)g(an)g(in\014nite)i(amoun)n (t)e(of)h(energy)-7 b(.)37 b(Th)n(us,)28 b(the)g(claim)g(follo)n(ws)f (from)456 3098 y(Corollary)e(4.1.)2427 b Fj(\003)555 3285 y Fu(Analogous)38 b(deca)n(ys)g(for)h(higher)f(deriv)-5 b(ativ)n(es)38 b(are)h(obtained)g(from)f(Corollary)f(4.1)i(and)456 3385 y(elliptic)28 b(b)r(o)r(otstrap.)36 b(Namely)-7 b(,)27 b(if)i Fr(v)d Fu(:=)d Fr(u)18 b Fq(\006)g Fu(1,)27 b(then)1303 3545 y Fr(@)1347 3557 y Fp(i)1375 3545 y Fu(\()p Fr(a)1451 3557 y Fp(ij)1510 3545 y Fr(@)1554 3557 y Fp(j)1589 3545 y Fr(v)s Fu(\))c(=)g Fr(f)9 b Fu(\()p Fr(x;)14 b(v)s Fu(\()p Fr(x)p Fu(\))20 b Fq(\007)e Fu(1\))23 b(=:)f Fr(g)s Fu(\()p Fr(x)p Fu(\))14 b Fr(;)456 3706 y Fu(th)n(us,)41 b(from)d(\(8.86\))g(of)g([GT83)o(])h(\(recalling)e (also)h(\(4.3\){\(4.17\))e(and)j(\(6.10\))e(for)h(notations)456 3806 y(there\),)27 b(one)h(has)f(that)670 3992 y Fq(jr)p Fr(u)p Fu(\()p Fr(x)p Fu(\))p Fq(j)84 b(\024)e(k)p Fr(v)s Fq(k)1302 4007 y Fp(C)1354 3990 y Fi(1)p Fk(;\013)1442 4007 y Fs(\()p Fp(B)1518 4015 y Fi(1)1551 4007 y Fs(\()p Fp(x)p Fs(\)\))1694 3992 y Fq(\024)36 b Fu(const)1998 3900 y Fo(\020)2048 3992 y Fq(k)p Fr(v)s Fq(k)2175 4007 y Fp(L)2221 3990 y Fh(1)2281 4007 y Fs(\()p Fp(B)2357 4015 y Fi(2)2389 4007 y Fs(\()p Fp(x)p Fs(\)\))2527 3992 y Fu(+)18 b Fq(k)p Fr(g)s Fq(k)2737 4007 y Fp(L)2783 3990 y Fh(1)2842 4007 y Fs(\()p Fp(B)2918 4015 y Fi(2)2950 4007 y Fs(\()p Fp(x)p Fs(\)\))3070 3900 y Fo(\021)3143 3992 y Fq(\024)1028 4141 y(\024)96 b Fu(const)13 b Fq(k)p Fr(v)s Fq(k)1519 4156 y Fp(L)1565 4140 y Fh(1)1625 4156 y Fs(\()p Fp(B)1701 4164 y Fi(2)1733 4156 y Fs(\()p Fp(x)p Fs(\)\))1876 4141 y Fq(\024)-1485 b Fu(\(4.2\))1028 4300 y Fq(\024)96 b Fu(const)13 b Fr(e)1431 4260 y Fm(\000)e Fs(const)1657 4266 y Fq(j)1709 4238 y Fk(!)p 1690 4247 77 3 v 1690 4281 a Fh(j)p Fk(!)q Fh(j)1776 4260 y Fm(\001)p Fp(x)1834 4266 y Fq(j)1875 4300 y Fr(:)456 4461 y Fu(And)28 b(then,)g(b)n(y)f(Sc)n(hauder)g(Estimates)g(\(see,)g(e.g.,)h(\(6.23\))f (in)g([GT83]\),)627 4647 y Fq(j)p Fr(D)721 4612 y Fs(2)758 4647 y Fr(u)p Fu(\()p Fr(x)p Fu(\))p Fq(j)84 b(\024)f(k)p Fr(v)s Fq(k)1299 4662 y Fp(C)1351 4645 y Fi(2)p Fk(;\013)1439 4662 y Fs(\()p Fp(B)1515 4673 y Fi(1)p Fk(=)p Fi(2)1606 4662 y Fs(\()p Fp(x)p Fs(\)\))1749 4647 y Fq(\024)36 b Fu(const)2054 4555 y Fo(\020)2103 4647 y Fq(k)p Fr(v)s Fq(k)2230 4662 y Fp(L)2276 4645 y Fh(1)2336 4662 y Fs(\()p Fp(B)2412 4670 y Fi(1)2444 4662 y Fs(\()p Fp(x)p Fs(\)\))2582 4647 y Fu(+)18 b Fq(k)p Fr(g)s Fq(k)2792 4662 y Fp(C)2844 4645 y Fk(\013)2885 4662 y Fs(\()p Fp(B)2961 4670 y Fi(1)2993 4662 y Fs(\()p Fp(x)p Fs(\)\))3113 4555 y Fo(\021)3185 4647 y Fq(\024)1024 4796 y(\024)96 b Fu(const)14 b Fq(k)p Fr(v)s Fq(k)1516 4811 y Fp(C)1568 4795 y Fk(\013)1609 4811 y Fs(\()p Fp(B)1685 4819 y Fi(1)1717 4811 y Fs(\()p Fp(x)p Fs(\)\))1860 4796 y Fq(\024)-1469 b Fu(\(4.3\))1024 4955 y Fq(\024)96 b Fu(const)14 b Fr(e)1428 4915 y Fm(\000)d Fs(const)1653 4921 y Fq(j)1706 4893 y Fk(!)p 1687 4902 V 1687 4936 a Fh(j)p Fk(!)q Fh(j)1773 4915 y Fm(\001)p Fp(x)1831 4921 y Fq(j)1871 4955 y Fr(:)555 5116 y Fu(W)-7 b(e)26 b(no)n(w)f(sho)n(w)g(that)h(all)f(p)r(erio)r(dic)h(minimizers)f (satisfy)h(the)g(Birkho\013)f(prop)r(ert)n(y)f(\(up)i(to)g(a)456 5216 y(sign)h(c)n(hange\):)p eop %%Page: 12 12 12 11 bop 456 251 a Fs(12)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y FA(Prop)s(osition)30 b(4.3.)40 b Fn(L)l(et)29 b Fr(u)23 b Fq(2)g Fr(W)1544 410 y Fs(1)p Fp(;)p Fs(2)1532 475 y(lo)r(c)1634 450 y Fu(\()p FA(K)1741 420 y Fp(d)1741 471 y(!)1789 450 y Fu(\))c Fq(\\)g Fr(L)1971 420 y Fm(1)2041 450 y Fu(\()p FA(K)2148 420 y Fp(d)2148 471 y(!)2196 450 y Fu(\))30 b Fn(b)l(e)g(such)g(that)f Fq(E)2762 462 y Fp(!)2810 450 y Fu(\()p Fr(u)p Fu(\))23 b Fr(<)g Fu(+)p Fq(1)29 b Fn(and)456 594 y Fu(\(4.4\))986 b Fq(E)1657 606 y Fp(!)1705 594 y Fu(\()p Fr(u)p Fu(\))23 b Fq(\024)f(E)1971 606 y Fp(!)2019 594 y Fu(\()p Fr(u)d Fu(+)f Fr(')p Fu(\))456 738 y Fn(for)28 b(any)g Fr(')23 b Fq(2)h Fr(W)989 708 y Fs(1)p Fp(;)p Fs(2)1079 738 y Fu(\()p FA(K)1186 708 y Fp(d)1186 758 y(!)1234 738 y Fu(\))14 b Fq(\\)g Fr(L)1406 708 y Fm(1)1476 738 y Fu(\()p FA(K)1583 708 y Fp(d)1583 758 y(!)1631 738 y Fu(\))p Fn(.)38 b(Then,)29 b(either)g Fr(u)e Fn(or)h Fq(\000)p Fr(u)f Fn(is)g(Birkho\013)j(with)e(r)l(esp)l (e)l(ct)f(to)456 837 y Fr(!)32 b Fn(\(se)l(e)e(De\014nition)f(2.1\).) 456 1000 y(Pr)l(o)l(of.)43 b Fu(W)-7 b(e)28 b(\014rst)f(pro)n(v)n(e)f (that,)i(for)f(an)n(y)g Fr(k)f Fq(2)d FA(Z)1946 970 y Fp(d)1985 1000 y Fu(,)456 1144 y(\(4.5\))54 b(either)27 b Fr(u)p Fu(\()p Fr(x)19 b Fu(+)f Fr(k)s Fu(\))23 b Fq(\025)g Fr(u)p Fu(\()p Fr(x)p Fu(\))28 b(for)f(an)n(y)g Fr(x)d Fq(2)f FA(K)2029 1114 y Fp(d)2029 1164 y(!)2105 1144 y Fu(or)j Fr(u)p Fu(\()p Fr(x)19 b Fu(+)f Fr(k)s Fu(\))23 b Fq(\024)g Fr(u)p Fu(\()p Fr(x)p Fu(\))28 b(for)f(an)n(y)g Fr(x)c Fq(2)h FA(K)3319 1114 y Fp(d)3319 1164 y(!)3367 1144 y Fu(.)456 1288 y(The)32 b(pro)r(of)g(of)39 b(\(4.5\))32 b(is)g(b)n(y)g(con)n(tradiction.)51 b(Let)32 b Fr(k)i Fq(2)e FA(Z)2298 1258 y Fp(d)2337 1288 y Fu(.)51 b(Assume)33 b(that)f(there)h(exists)f Fr(x)3407 1300 y Fs(1)456 1387 y Fu(and)27 b Fr(x)664 1399 y Fs(2)729 1387 y Fu(so)g(that)456 1531 y(\(4.6\))420 b Fr(u)p Fu(\()p Fr(x)1174 1543 y Fs(1)1230 1531 y Fu(+)18 b Fr(k)s Fu(\))23 b Fr(>)g(u)p Fu(\()p Fr(x)1629 1543 y Fs(1)1666 1531 y Fu(\))167 b(and)e Fr(u)p Fu(\()p Fr(x)2291 1543 y Fs(2)2347 1531 y Fu(+)18 b Fr(k)s Fu(\))24 b Fr(<)e(u)p Fu(\()p Fr(x)2746 1543 y Fs(2)2784 1531 y Fu(\))14 b Fr(:)456 1675 y Fu(Then,)27 b(there)h(exists)f Fr(x)1184 1645 y Fm(\003)1250 1675 y Fu(so)g(that)456 1819 y(\(4.7\))978 b Fr(u)p Fu(\()p Fr(x)1732 1784 y Fm(\003)1789 1819 y Fu(+)18 b Fr(k)s Fu(\))23 b(=)g Fr(u)p Fu(\()p Fr(x)2188 1784 y Fm(\003)2226 1819 y Fu(\))14 b Fr(:)456 1963 y Fu(Let)745 2071 y Fr(u)793 2037 y Fm(\003)831 2071 y Fu(\()p Fr(x)p Fu(\))24 b(:=)f(max)o Fq(f)p Fr(u)p Fu(\()p Fr(x)p Fu(\))p Fr(;)28 b(u)p Fu(\()p Fr(x)19 b Fu(+)f Fr(k)s Fu(\))p Fq(g)c Fr(;)179 b(u)2096 2083 y Fm(\003)2134 2071 y Fu(\()p Fr(x)p Fu(\))24 b(:=)e(min)q Fq(f)p Fr(u)p Fu(\()p Fr(x)p Fu(\))p Fr(;)28 b(u)p Fu(\()p Fr(x)18 b Fu(+)g Fr(k)s Fu(\))p Fq(g)c Fr(:)456 2198 y Fu(Since)27 b Fq(E)716 2210 y Fp(!)764 2198 y Fu(\()p Fr(u)p Fu(\))c Fr(<)g Fu(+)p Fq(1)p Fu(,)k(w)n(e)h(ha)n(v)n(e)e(that)i Fr(u)22 b Fq(2)i Fr(W)1918 2168 y Fs(1)p Fp(;)p Fs(2)2008 2198 y Fu(\()p FA(K)2115 2168 y Fp(d)2115 2218 y(!)2163 2198 y Fu(\),)k(therefore)456 2346 y(\(4.8\))607 b Fr(u)1282 2311 y Fm(\003)1338 2346 y Fq(\000)18 b Fr(u;)27 b(u)1567 2358 y Fm(\003)1623 2346 y Fq(\000)18 b Fr(u)23 b Fq(2)g Fr(W)1945 2311 y Fs(1)p Fp(;)p Fs(2)2035 2346 y Fu(\()p FA(K)2142 2311 y Fp(d)2142 2366 y(!)2191 2346 y Fu(\))18 b Fq(\\)h Fr(L)2372 2311 y Fm(1)2442 2346 y Fu(\()p FA(K)2549 2311 y Fp(d)2549 2366 y(!)2597 2346 y Fu(\))14 b Fr(;)456 2490 y Fu(and)27 b(so,)g(from)g(\(4.4\),)456 2633 y(\(4.9\))630 b Fq(E)1301 2645 y Fp(!)1349 2633 y Fu(\()p Fr(u)1429 2599 y Fm(\003)1467 2633 y Fu(\))23 b Fq(\025)g(E)1654 2645 y Fp(!)1702 2633 y Fu(\()p Fr(u)p Fu(\))51 b(and)f Fq(E)2093 2645 y Fp(!)2141 2633 y Fu(\()p Fr(u)2221 2645 y Fm(\003)2259 2633 y Fu(\))23 b Fq(\025)g(E)2446 2645 y Fp(!)2494 2633 y Fu(\()p Fr(u)p Fu(\))14 b Fr(:)456 2777 y Fu(Also,)32 b(b)n(y)g(splitting)g(the)g(domains)f(of)g(in)n (tegration)g(\(using)g(the)i(so)e(called)g(Rellic)n(h)h(Lemma,)456 2877 y(see,)27 b(e.g.,)g(page)g(50)g(on)g([KS00)o(]\),)h(one)f(gets)g (that)1461 3021 y Fq(E)1505 3033 y Fp(!)1552 3021 y Fu(\()p Fr(u)1632 2986 y Fm(\003)1670 3021 y Fu(\))19 b(+)f Fq(E)1848 3033 y Fp(!)1896 3021 y Fu(\()p Fr(u)1976 3033 y Fm(\003)2014 3021 y Fu(\))23 b(=)g(2)p Fq(E)2243 3033 y Fp(!)2290 3021 y Fu(\()p Fr(u)p Fu(\))14 b Fr(;)456 3164 y Fu(hence,)27 b(b)n(y)i(\(4.9\))o(,)456 3308 y(\(4.10\))808 b Fq(E)1521 3320 y Fp(!)1569 3308 y Fu(\()p Fr(u)1649 3274 y Fm(\003)1687 3308 y Fu(\))23 b(=)g Fq(E)1874 3320 y Fp(!)1921 3308 y Fu(\()p Fr(u)2001 3320 y Fm(\003)2039 3308 y Fu(\))h(=)e Fq(E)2226 3320 y Fp(!)2274 3308 y Fu(\()p Fr(u)p Fu(\))14 b Fr(:)456 3452 y Fu(In)27 b(particular,)g(from)g(\(4.8\),)g(if)h Fr(\036)c Fq(2)f Fr(C)1675 3422 y Fm(1)1669 3473 y Fs(0)1746 3452 y Fu(\()p FA(K)1853 3422 y Fp(d)1853 3473 y(!)1901 3452 y Fu(\),)28 b(then)1319 3600 y Fr(u)1367 3566 y Fm(\003)1424 3600 y Fu(+)18 b Fr(\036)h Fq(\000)f Fr(u)k Fq(2)i Fr(W)1897 3566 y Fs(1)p Fp(;)p Fs(2)1987 3600 y Fu(\()p FA(K)2094 3566 y Fp(d)2094 3620 y(!)2142 3600 y Fu(\))18 b Fq(\\)h Fr(L)2323 3566 y Fm(1)2393 3600 y Fu(\()p FA(K)2500 3566 y Fp(d)2500 3620 y(!)2548 3600 y Fu(\))456 3744 y(and)27 b(therefore,)g(b)n(y)h(\(4.4\))g(and)f (\(4.10\))o(,)h(w)n(e)f(gather)g(that)1401 3888 y Fq(E)1445 3900 y Fp(!)1493 3888 y Fu(\()p Fr(u)1573 3853 y Fm(\003)1630 3888 y Fu(+)18 b Fr(\036)p Fu(\))23 b Fq(\025)g(E)1949 3900 y Fp(!)1997 3888 y Fu(\()p Fr(u)p Fu(\))g(=)g Fq(E)2264 3900 y Fp(!)2312 3888 y Fu(\()p Fr(u)2392 3853 y Fm(\003)2430 3888 y Fu(\))14 b Fr(:)456 4031 y Fu(Analogously)-7 b(,)28 b Fq(E)989 4043 y Fp(!)1037 4031 y Fu(\()p Fr(u)1117 4043 y Fm(\003)1175 4031 y Fu(+)19 b Fr(\036)p Fu(\))28 b Fq(\025)e(E)1503 4043 y Fp(!)1551 4031 y Fu(\()p Fr(u)1631 4043 y Fm(\003)1669 4031 y Fu(\),)31 b(for)e(an)n(y)g Fr(\036)e Fq(2)g Fr(C)2266 4001 y Fm(1)2260 4052 y Fs(0)2336 4031 y Fu(\()p FA(K)2443 4001 y Fp(d)2443 4052 y(!)2491 4031 y Fu(\).)44 b(Th)n(us)29 b Fr(u)2850 4001 y Fm(\003)2888 4031 y Fr(;)f(u)2987 4043 y Fm(\003)3054 4031 y Fu(are)h(critical)456 4131 y(for)e Fq(E)627 4143 y Fp(!)674 4131 y Fu(.)37 b(By)28 b(Corollary)d(A.3)i(and)g(the)h(fact)g(that)g Fr(u)2082 4101 y Fm(\003)2143 4131 y Fq(\025)22 b Fr(u)2278 4143 y Fm(\003)2316 4131 y Fu(,)28 b(w)n(e)f(gather)f(that)i(either)f Fr(u)3210 4143 y Fm(\003)3271 4131 y Fu(=)c Fr(u)3407 4101 y Fm(\003)456 4231 y Fu(or)29 b Fr(u)608 4243 y Fm(\003)674 4231 y Fr(<)f(u)815 4200 y Fm(\003)852 4231 y Fu(.)46 b(The)31 b(\014rst)f(p)r(ossibilit)n(y)g(is)h(ruled)f(out)h (b)n(y)g(\(4.6\).)46 b(The)31 b(second)e(b)n(y)j(\(4.7\).)46 b(This)456 4330 y(ends)27 b(the)h(pro)r(of)f(of)34 b(\(4.5\).)555 4474 y(With)40 b(this,)h(w)n(e)d(no)n(w)g(pro)n(v)n(e)f(the)i(desired)f (result.)69 b(T)-7 b(ak)n(e)2488 4452 y(\026)2486 4474 y Fr(k)44 b Fq(2)e FA(Z)2728 4444 y Fp(d)2805 4474 y Fu(with)3008 4452 y(\026)3005 4474 y Fr(k)i Fu(=)d Fr(\013!)h Fu(for)456 4574 y(some)34 b Fr(\013)i(>)f Fu(0)f(and)h(with)h Fr(\013)f Fu(as)f(small)h(as)f(p)r(ossible.)59 b(Due)35 b(to)g(\(4.5\),)i(w)n(e)e(ha)n(v)n(e)e(that)j(either)456 4673 y Fr(u)p Fu(\()p Fq(\001)20 b Fu(+)666 4651 y(\026)664 4673 y Fr(k)r Fu(\))28 b Fq(\025)g Fr(u)p Fu(\()p Fq(\001)p Fu(\))i(or)g Fr(u)p Fu(\()p Fq(\001)20 b Fu(+)1342 4651 y(\026)1340 4673 y Fr(k)r Fu(\))28 b Fq(\024)g Fr(u)p Fu(\()p Fq(\001)p Fu(\).)45 b(W)-7 b(e)31 b(assume)e(that)i(the)g (\014rst)f(p)r(ossibilit)n(y)g(holds)g(\(the)456 4773 y(other)d(case)h(b)r(eing)g(analogous\))e(and)j(w)n(e)f(then)g(sho)n(w) g(that)g Fr(u)p Fu(\()p Fq(\001)19 b Fu(+)g Fr(k)s Fu(\))24 b Fq(\025)g Fr(u)p Fu(\()p Fq(\001)p Fu(\))k(for)g(an)n(y)g Fr(k)f Fq(2)d FA(Z)3405 4743 y Fp(d)456 4873 y Fu(so)j(that)g Fr(!)22 b Fq(\001)c Fr(k)26 b Fq(\025)d Fu(0.)555 4972 y(Indeed,)30 b(tak)n(e)f(suc)n(h)g(a)g Fr(k)s Fu(.)43 b(Assume)29 b(b)n(y)h(con)n(tradiction)e(that)i Fr(u)p Fu(\()5 b(\026)-47 b Fr(x)20 b Fu(+)f Fr(k)s Fu(\))26 b Fr(<)g(u)p Fu(\()5 b(\026)-47 b Fr(x)p Fu(\))30 b(for)f(some)461 5072 y(\026)-47 b Fr(x)p Fu(.)37 b(Then,)28 b(b)n(y)g(\(4.5\))g(and)f (Corollary)e(A.3,)456 5216 y(\(4.11\))998 b Fr(u)p Fu(\()p Fq(\001)19 b Fu(+)f Fr(k)s Fu(\))23 b Fr(<)g(u)p Fu(\()p Fq(\001)p Fu(\))14 b Fr(:)p eop %%Page: 13 13 13 12 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(13)456 450 y Fu(Also,)27 b(there)g(exist)h Fr(a)23 b Fq(2)g FA(N)18 b Fq(\000)g(f)p Fu(0)p Fq(g)p Fu(,)27 b Fr(b)22 b Fq(2)i FA(Z)p Fu(,)k Fr(\013)23 b Fq(2)h FA(R)j Fu(and)2236 428 y(^)2233 450 y Fr(k)f Fq(2)e FA(Z)2439 420 y Fp(d)2496 450 y Fq(\\)19 b Fr(!)2625 420 y Fm(?)2708 450 y Fu(so)27 b(that)456 621 y(\(4.12\))1048 b Fr(ak)26 b Fu(=)c Fr(b)1955 599 y Fu(\026)1953 621 y Fr(k)f Fu(+)2102 599 y(^)2100 621 y Fr(k)c(:)456 783 y Fu(Explicitly)-7 b(,)27 b(to)h(con\014rm)f (\(4.12\))o(,)h(giv)n(en)1611 1026 y Fr(k)e Fu(=)1810 922 y Fp(d)1768 947 y Fo(X)1770 1124 y Fp(j)s Fs(=1)1902 1026 y Fr(k)1945 1038 y Fp(j)1980 1026 y Fr(e)2019 1038 y Fp(j)2076 1026 y Fq(2)e FA(Z)2213 992 y Fp(d)2266 1026 y Fr(;)456 1272 y Fu(one)j(ma)n(y)g(tak)n(e)905 1511 y Fr(a)83 b Fu(:=)1245 1407 y Fp(d)1203 1432 y Fo(X)1209 1609 y Fp(i)p Fs(=1)1338 1489 y Fu(\026)1336 1511 y Fr(k)1382 1477 y Fs(2)1379 1531 y Fp(i)1442 1511 y Fq(2)24 b FA(N)18 b Fq(\000)g(f)p Fu(0)p Fq(g)c Fr(;)814 1728 y(h)862 1694 y Fs(\()p Fp(j)s Fs(\))1032 1728 y Fu(:=)83 b Fr(ae)1286 1740 y Fp(j)1338 1728 y Fq(\000)18 b Fu(\()1455 1706 y(\026)1453 1728 y Fr(k)k Fq(\001)d Fr(e)1599 1740 y Fp(j)1633 1728 y Fu(\))1667 1706 y(\026)1665 1728 y Fr(k)26 b Fq(2)e FA(Z)1871 1694 y Fp(d)1928 1728 y Fq(\\)2004 1706 y Fu(\026)2002 1728 y Fr(k)2048 1694 y Fm(?)2127 1728 y Fu(=)f FA(Z)2273 1694 y Fp(d)2330 1728 y Fq(\\)c Fr(!)2459 1694 y Fm(?)2529 1728 y Fr(;)41 b Fu(for)27 b Fr(j)h Fu(=)23 b(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(d)p Fu(,)905 1916 y(^)903 1938 y Fr(k)86 b Fu(:=)1245 1834 y Fp(d)1203 1859 y Fo(X)1205 2036 y Fp(j)s Fs(=1)1336 1938 y Fr(k)1379 1950 y Fp(j)1414 1938 y Fr(h)1462 1903 y Fs(\()p Fp(j)s Fs(\))1572 1938 y Fq(2)24 b FA(Z)1709 1903 y Fp(d)1766 1938 y Fq(\\)19 b Fr(!)1895 1903 y Fm(?)738 2157 y Fu(and)41 b Fr(b)83 b Fu(:=)g Fr(k)21 b Fq(\001)1310 2135 y Fu(\026)1308 2157 y Fr(k)26 b Fq(2)e FA(Z)456 2319 y Fu(and)j(these)h(c)n(hoices)e(easily)h(giv)n(e)g(\(4.12\))o(.) 555 2419 y(Then,)h(b)n(y)h(\(4.12\))o(,)1443 2547 y(0)22 b Fq(\024)h Fr(a!)e Fq(\001)d Fr(k)26 b Fu(=)d Fr(b)1948 2525 y Fu(\026)1946 2547 y Fr(k)e Fq(\001)d Fr(!)26 b Fu(=)d Fr(\013b)14 b Fq(j)p Fr(!)s Fq(j)2421 2512 y Fs(2)456 2691 y Fu(and)27 b(so)456 2853 y(\(4.13\))1169 b Fr(b)22 b Fq(\025)h Fu(0)14 b Fr(:)456 3015 y Fu(Also,)27 b(b)n(y)g(the)h(p)r (erio)r(dicit)n(y)g(of)f Fr(u)p Fu(,)1248 3181 y Fr(u)p Fu(\()p Fr(x)19 b Fu(+)f Fr(ak)s Fu(\))23 b(=)f Fr(u)p Fu(\()p Fr(x)d Fu(+)f Fr(b)1976 3159 y Fu(\026)1974 3181 y Fr(k)j Fu(+)2123 3159 y(^)2121 3181 y Fr(k)s Fu(\))i(=)g Fr(u)p Fu(\()p Fr(x)18 b Fu(+)g Fr(b)2576 3159 y Fu(\026)2574 3181 y Fr(k)s Fu(\))456 3342 y(for)27 b(an)n(y)f Fr(x)p Fu(.)38 b(F)-7 b(rom)27 b(this,)h(\(4.13\))f(and)g(our)g(assumptions)g (on)2388 3321 y(\026)2386 3342 y Fr(k)s Fu(,)h(w)n(e)f(get)g(that)456 3504 y(\(4.14\))976 b Fr(u)p Fu(\()p Fq(\001)19 b Fu(+)f Fr(ak)s Fu(\))23 b Fq(\025)f Fr(u)p Fu(\()p Fq(\001)p Fu(\))14 b Fr(:)456 3666 y Fu(On)28 b(the)i(other)e(hand,)i(from)e(the) i(fact)f(that)g Fr(a)d Fq(2)g FA(N)19 b Fq(\000)g(f)p Fu(0)p Fq(g)27 b Fu(\(and)j(so)e Fr(a)d Fq(\025)g Fu(1\))k(and)g (\(4.11\))o(,)h(w)n(e)456 3766 y(deduce)d(that)1345 3892 y Fr(u)p Fu(\()p Fq(\001)18 b Fu(+)g Fr(ak)s Fu(\))24 b Fr(<)e Fq(\001)14 b(\001)g(\001)23 b Fr(<)g(u)p Fu(\()p Fq(\001)18 b Fu(+)g Fr(k)s Fu(\))23 b Fr(<)g(u)p Fu(\()p Fq(\001)p Fu(\))14 b Fr(;)456 4037 y Fu(in)27 b(con)n(tradiction)g (with)h(\(4.14\))o(.)1898 b Fj(\003)555 4225 y Fu(One)28 b(of)f(the)i(\014rst)e(consequences)g(of)g(the)i(Birkho\013)e(prop)r (ert)n(y)f(and)i(of)g(the)g(no-symmetry-)456 4325 y(breaking)35 b(is)h(that)g(global)f(minimizers)h(\(when)h(their)f(domain)g(is)g (unfolded)h(to)f(the)h(whole)456 4424 y(space)26 b FA(R)749 4394 y Fp(d)787 4424 y Fu(\))h(are)f(lo)r(cal)h(minimizers)f(in)h(an)n (y)g(domain)f(of)h FA(R)2312 4394 y Fp(d)2350 4424 y Fu(.)37 b(More)26 b(precisely)-7 b(,)26 b(the)i(follo)n(wing)456 4524 y(result)f(easily)g(follo)n(ws)f(from)i(Corollary)d(4.3)h(and)i (Prop)r(osition)e(2.2:)456 4657 y FA(Prop)s(osition)38 b(4.4.)44 b Fn(L)l(et)35 b Fr(u)f Fq(2)g Fr(W)1584 4617 y Fs(1)p Fp(;)p Fs(2)1572 4682 y(lo)r(c)1674 4657 y Fu(\()p FA(K)1781 4627 y Fp(d)1781 4677 y(!)1830 4657 y Fu(\))23 b Fq(\\)g Fr(L)2020 4627 y Fm(1)2090 4657 y Fu(\()p FA(K)2197 4627 y Fp(d)2197 4677 y(!)2245 4657 y Fu(\))36 b Fn(b)l(e)g(such)g (that)g Fq(E)2836 4669 y Fp(!)2884 4657 y Fu(\()p Fr(u)p Fu(\))f Fr(<)e Fu(+)p Fq(1)j Fn(and)456 4767 y Fq(E)500 4779 y Fp(!)547 4767 y Fu(\()p Fr(u)p Fu(\))d Fq(\024)e(E)832 4779 y Fp(!)880 4767 y Fu(\()p Fr(u)22 b Fu(+)g Fr(')p Fu(\))35 b Fn(for)g(any)g Fr(')e Fq(2)f Fr(W)1755 4737 y Fs(1)p Fp(;)p Fs(2)1845 4767 y Fu(\()p FA(K)1952 4737 y Fp(d)1952 4788 y(!)2000 4767 y Fu(\))23 b Fq(\\)f Fr(L)2189 4737 y Fm(1)2259 4767 y Fu(\()p FA(K)2366 4737 y Fp(d)2366 4788 y(!)2414 4767 y Fu(\))p Fn(.)54 b(L)l(et)39 b Fu(~)-47 b Fr(u)32 b Fq(2)g Fr(W)2930 4728 y Fs(1)p Fp(;)p Fs(2)2918 4793 y(lo)r(c)3020 4767 y Fu(\()p FA(R)3124 4737 y Fp(d)3162 4767 y Fu(\))j Fn(b)l(e)g(the)456 4871 y FA(K)531 4841 y Fp(d)569 4871 y Fn(-p)l(erio)l(dic)d(extension)d(of)i Fr(u)e Fn(to)h(the)f(whole)j FA(R)1991 4841 y Fp(d)2029 4871 y Fn(.)39 b(Then)456 5033 y Fu(\(4.15\))919 b Fq(E)1632 5045 y Fp(B)1689 5033 y Fu(\()5 b(~)-47 b Fr(u)18 b Fu(+)g Fr(\036)p Fu(\))24 b Fq(\025)e(E)2106 5045 y Fp(B)2163 5033 y Fu(\()5 b(~)-47 b Fr(u)p Fu(\))14 b Fr(;)456 5195 y Fn(for)30 b(any)g(b)l(al)t(l)h Fr(B)c Fq(\032)c FA(R)1150 5165 y Fp(d)1218 5195 y Fn(and)30 b(any)g Fr(\036)24 b Fq(2)f Fr(C)1754 5165 y Fm(1)1748 5215 y Fs(0)1825 5195 y Fu(\()p Fr(B)t Fu(\))p Fn(.)p eop %%Page: 14 14 14 13 bop 456 251 a Fs(14)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fu(4.2.)40 b FA(Pro)s(of)32 b(of)g(Theorem)e(2.3.)40 b Fu(Let)28 b Fr(u)g Fu(b)r(e)g(as)f(in)h(Theorem)e(A.8)i(\(and)g(recall)f(also)f (Corol-)456 550 y(lary)37 b(4.2\).)71 b(The)39 b(last)f(prop)r(ert)n(y) g(of)h Fr(u)f Fu(needed)h(for)g(pro)n(ving)e(Theorem)h(2.3)g(is)h(the)g (mini-)456 649 y(mizing)33 b(prop)r(ert)n(y)g(of)g Fr(u)g Fu(under)h(p)r(erturbations)f(in)h Fr(W)2217 619 y Fs(1)p Fp(;)p Fs(2)2307 649 y Fu(\()p FA(K)2414 619 y Fp(d)2414 670 y(!)2462 649 y Fu(\))23 b Fq(\\)f Fr(L)2651 619 y Fm(1)2721 649 y Fu(\()p FA(K)2828 619 y Fp(d)2828 670 y(!)2876 649 y Fu(\).)56 b(F)-7 b(or)33 b(that,)i(let)456 749 y Fr(\036)23 b Fq(2)h Fr(W)697 719 y Fs(1)p Fp(;)p Fs(2)787 749 y Fu(\()p FA(K)894 719 y Fp(d)894 770 y(!)942 749 y Fu(\))18 b Fq(\\)h Fr(L)1123 719 y Fm(1)1193 749 y Fu(\()p FA(K)1300 719 y Fp(d)1300 770 y(!)1348 749 y Fu(\))28 b(and)g(let)g(us)f(sho)n(w)g(that)456 902 y(\(4.16\))928 b Fq(E)1641 914 y Fp(!)1689 902 y Fu(\()p Fr(u)p Fu(\))23 b Fq(\024)f(E)1955 914 y Fp(!)2003 902 y Fu(\()p Fr(u)d Fu(+)f Fr(\036)p Fu(\))c Fr(:)456 1055 y Fu(Let)23 b Fr(\032)643 1067 y Fp(n)712 1055 y Fu(b)r(e)h(a)f (standard)g(molli\014er)g(\(see,)h(e.g.,)g(Theorem)f(1.6.1)f(in)i ([Zie89)o(]\))g(and)f Fr(\036)3042 1067 y Fp(n)3111 1055 y Fu(:=)g Fr(\036)10 b Fq(\003)g Fr(\032)3376 1067 y Fp(n)3421 1055 y Fu(.)456 1155 y(Then,)26 b(up)g(to)g(subsequences,)g Fr(\036)1485 1167 y Fp(n)1557 1155 y Fu(con)n(v)n(erges)d(to)j Fr(\036)g Fu(almost)g(ev)n(erywhere)e(and)i(in)g Fr(W)3144 1124 y Fs(1)p Fp(;)p Fs(2)3234 1155 y Fu(\()p FA(K)3341 1124 y Fp(d)3341 1175 y(!)3389 1155 y Fu(\).)555 1254 y(F)-7 b(or)27 b(an)n(y)g Fr(R)d(>)e Fu(0,)28 b(let)1315 1428 y Fr(G)1380 1440 y Fp(R)1458 1428 y Fu(:=)1569 1335 y Fo(n)1624 1428 y Fr(x)c Fq(2)f FA(K)1848 1393 y Fp(d)1848 1448 y(!)1924 1428 y Fu(s.t.)2072 1332 y Fo(\014)2072 1382 y(\014)2072 1432 y(\014)2132 1371 y Fr(!)p 2109 1408 101 4 v 2109 1484 a Fq(j)p Fr(!)s Fq(j)2238 1428 y(\001)c Fr(x)2327 1332 y Fo(\014)2327 1382 y(\014)2327 1432 y(\014)2378 1428 y Fq(\024)k Fr(R)2530 1335 y Fo(o)456 1629 y Fu(and)32 b(let)g Fr(\037)798 1641 y Fp(R)884 1629 y Fq(2)f Fr(C)1035 1599 y Fm(1)1029 1650 y Fs(0)1105 1629 y Fu(\()p Fr(G)1202 1641 y Fp(R)p Fs(+1)1341 1629 y Fu(\))i(with)g Fq(kr)p Fr(\037)1763 1641 y Fp(R)1817 1629 y Fq(k)d(\024)h Fu(10,)h(0)f Fq(\024)f Fr(\037)2344 1641 y Fp(R)2430 1629 y Fq(\024)g Fu(1)i(and)g Fr(\037)2817 1641 y Fp(R)2871 1629 y Fu(\()p Fr(x)p Fu(\))g(=)f(1)h(for)f(an)n(y)456 1745 y Fr(x)23 b Fq(2)h Fr(G)670 1757 y Fp(R)724 1745 y Fu(.)37 b(Set)28 b(also)f Fr(\036)1143 1702 y Fs(\()p Fp(R)p Fs(\))1143 1755 y Fp(n)1273 1745 y Fu(:=)22 b Fr(\036)1432 1757 y Fp(n)1478 1745 y Fr(\037)1530 1757 y Fp(R)1612 1745 y Fu(and)27 b Fr(\036)1822 1715 y Fs(\()p Fp(R)p Fs(\))1952 1745 y Fu(:=)c Fr(\036\037)2164 1757 y Fp(R)2219 1745 y Fu(.)37 b(Notice)27 b(that,)h(b)n(y)g(construction,) 456 1901 y(\(4.17\))483 b Fq(k)p Fr(\036)1243 1866 y Fs(\()p Fp(R)p Fs(\))1243 1921 y Fp(n)1349 1901 y Fq(k)1391 1917 y Fp(L)1437 1901 y Fh(1)1497 1917 y Fs(\()p Fl(K)1581 1901 y Fk(d)1581 1934 y(!)1624 1917 y Fs(\))1677 1901 y Fq(\024)22 b(k)p Fr(\036)1855 1913 y Fp(n)1901 1901 y Fq(k)1943 1917 y Fp(L)1989 1901 y Fh(1)2048 1917 y Fs(\()p Fl(K)2132 1901 y Fk(d)2132 1934 y(!)2175 1917 y Fs(\))2228 1901 y Fq(\024)h(k)p Fr(\036)p Fq(k)2449 1917 y Fp(L)2495 1901 y Fh(1)2554 1917 y Fs(\()p Fl(K)2638 1901 y Fk(d)2638 1934 y(!)2681 1917 y Fs(\))2725 1901 y Fr(:)555 2054 y Fu(Let)28 b(us)g(no)n(w)e(mak)n(e)h(some)g(elemen)n (tary)g(observ)-5 b(ations.)35 b(First)28 b(of)f(all,)g(for)h(an)n(y)e Fr(a;)i(b)23 b Fq(2)g FA(R)p Fu(,)1210 2227 y Fq(j)p Fr(a)c Fu(+)f Fr(b)p Fq(j)g(\000)g Fu(1)23 b Fq(\024)f(j)p Fr(a)p Fq(j)d Fu(+)f Fq(j)p Fr(b)p Fq(j)g(\000)g Fu(1)k Fq(\024)2218 2131 y Fo(\014)2218 2181 y(\014)2218 2231 y(\014)2246 2227 y Fq(j)p Fr(a)p Fq(j)c(\000)g Fu(1)2479 2131 y Fo(\014)2479 2181 y(\014)2479 2231 y(\014)2525 2227 y Fu(+)g Fq(j)p Fr(b)p Fq(j)456 2400 y Fu(and)1160 2523 y Fq(j)p Fr(a)g Fu(+)g Fr(b)p Fq(j)g(\000)g Fu(1)23 b Fq(\025)f(j)p Fr(a)p Fq(j)d(\000)f(j)p Fr(b)p Fq(j)g(\000)g Fu(1)23 b Fq(\025)f(\000)2232 2427 y Fo(\014)2232 2477 y(\014)2232 2527 y(\014)2260 2523 y Fq(j)p Fr(a)p Fq(j)c(\000)g Fu(1)2493 2427 y Fo(\014)2493 2477 y(\014)2493 2527 y(\014)2539 2523 y Fq(\000)g(j)p Fr(b)p Fq(j)c Fr(;)456 2678 y Fu(that)27 b(is,)456 2852 y(\(4.18\))1428 2756 y Fo(\014)1428 2806 y(\014)1428 2856 y(\014)1455 2852 y Fu(1)18 b Fq(\000)g(j)p Fr(a)h Fu(+)f Fr(b)p Fq(j)1826 2756 y Fo(\014)1826 2806 y(\014)1826 2856 y(\014)1876 2852 y Fq(\024)1964 2756 y Fo(\014)1964 2806 y(\014)1964 2856 y(\014)1992 2852 y Fu(1)f Fq(\000)h(j)p Fr(a)p Fq(j)2224 2756 y Fo(\014)2224 2806 y(\014)2224 2856 y(\014)2271 2852 y Fu(+)g Fq(j)p Fr(b)p Fq(j)c Fr(:)456 3038 y Fu(Also,)24 b(since)h Fr(F)35 b Fq(2)23 b Fr(C)1095 3008 y Fs(2)1133 3038 y Fu(\()p FA(R)1237 3008 y Fp(d)1287 3038 y Fq(\002)12 b FA(R)p Fu(\))25 b(and)f Fr(@)1695 3050 y Fp(\026)1740 3038 y Fr(F)12 b Fu(\()p Fr(x;)i Fq(\006)p Fu(1\))23 b(=)f(0,)j(w)n(e)f(ha)n (v)n(e)f(that)i(there)f(exists)h Fr(M)31 b(>)23 b Fu(0,)456 3137 y(whic)n(h)k(ma)n(y)g(dep)r(end)h(on)g Fq(k)p Fr(\036)p Fq(k)1410 3154 y Fp(L)1456 3138 y Fh(1)1515 3154 y Fs(\()p Fl(K)1599 3138 y Fk(d)1599 3171 y(!)1642 3154 y Fs(\))1700 3137 y Fu(so)f(that)456 3328 y(\(4.19\))809 b Fq(j)p Fr(@)1545 3340 y Fp(\026)1590 3328 y Fr(F)12 b Fu(\()p Fr(x;)i(\026)p Fu(\))p Fq(j)24 b(\024)e Fr(M)2091 3233 y Fo(\014)2091 3282 y(\014)2091 3332 y(\014)2118 3328 y Fu(1)c Fq(\000)g(j)p Fr(\026)p Fq(j)2357 3233 y Fo(\014)2357 3282 y(\014)2357 3332 y(\014)2399 3328 y Fr(;)456 3514 y Fu(for)27 b(an)n(y)f Fr(x)e Fq(2)f FA(K)963 3484 y Fp(d)963 3535 y(!)1039 3514 y Fu(and)k(an)n(y)g Fq(j)p Fr(\026)p Fq(j)c(\024)g Fu(1)18 b(+)g Fq(k)p Fr(\036)p Fq(k)1840 3531 y Fp(L)1886 3514 y Fh(1)1946 3531 y Fs(\()p Fl(K)2030 3514 y Fk(d)2030 3548 y(!)2072 3531 y Fs(\))2103 3514 y Fu(.)36 b(Fixed)28 b Fr(t)23 b Fq(2)h Fu([0)p Fr(;)14 b Fu(1],)26 b(let)i(no)n(w)1190 3687 y Fr(U)1247 3699 y Fp(R)1301 3687 y Fu(\()p Fr(x;)14 b(t)p Fu(\))24 b(:=)f Fr(u)p Fu(\()p Fr(x)p Fu(\))c(+)f Fr(t\036)1954 3653 y Fs(\()p Fp(R)p Fs(\))2061 3687 y Fu(\()p Fr(x)p Fu(\))h(+)f(\(1)g Fq(\000)g Fr(t)p Fu(\))p Fr(\036)p Fu(\()p Fr(x)p Fu(\))c Fr(:)456 3840 y Fu(By)27 b(construction,)g Fq(j)p Fr(U)1165 3852 y Fp(R)1219 3840 y Fq(j)c(\024)g Fu(1)18 b(+)g Fq(k)p Fr(\036)p Fq(k)1629 3857 y Fp(L)1675 3840 y Fh(1)1734 3857 y Fs(\()p Fl(K)1818 3840 y Fk(d)1818 3874 y(!)1861 3857 y Fs(\))1891 3840 y Fu(,)28 b(therefore,)f(from)g (\(4.19\))g(and)g(\(4.18\))o(,)617 3935 y Fo(\014)617 3985 y(\014)617 4035 y(\014)645 4031 y Fr(@)689 4043 y Fp(\026)733 4031 y Fr(F)798 3939 y Fo(\020)848 4031 y Fr(x;)14 b(U)989 4043 y Fp(R)1043 4031 y Fu(\()p Fr(x;)g(t)p Fu(\))1221 3939 y Fo(\021)1272 3935 y(\014)1272 3985 y(\014)1272 4035 y(\014)1382 4031 y Fq(\024)83 b Fr(M)1633 3935 y Fo(\014)1633 3985 y(\014)1633 4035 y(\014)1661 4031 y Fu(1)18 b Fq(\000)g(j)p Fr(U)1884 4043 y Fp(R)1938 4031 y Fu(\()p Fr(x;)c(t)p Fu(\))p Fq(j)2139 3935 y Fo(\014)2139 3985 y(\014)2139 4035 y(\014)2191 4031 y Fq(\024)1382 4213 y(\024)83 b Fr(M)1633 4121 y Fo(\020)1683 4118 y(\014)1683 4168 y(\014)1683 4218 y(\014)1711 4213 y Fu(1)18 b Fq(\000)g(j)p Fr(u)p Fu(\()p Fr(x)p Fu(\))p Fq(j)2059 4118 y Fo(\014)2059 4168 y(\014)2059 4218 y(\014)2105 4213 y Fu(+)g Fq(j)p Fr(t)p Fu(\()p Fr(\036)2322 4179 y Fs(\()p Fp(R)p Fs(\))2430 4213 y Fu(\()p Fr(x)p Fu(\))h Fq(\000)f Fr(\036)p Fu(\()p Fr(x)p Fu(\)\))i(+)e Fr(\036)p Fu(\()p Fr(x)p Fu(\))p Fq(j)3121 4121 y Fo(\021)3195 4213 y Fq(\024)1382 4396 y(\024)83 b Fr(M)1633 4304 y Fo(\020)1683 4301 y(\014)1683 4350 y(\014)1683 4400 y(\014)1711 4396 y Fu(1)18 b Fq(\000)g(j)p Fr(u)p Fu(\()p Fr(x)p Fu(\))p Fq(j)2059 4301 y Fo(\014)2059 4350 y(\014)2059 4400 y(\014)2105 4396 y Fu(+)g Fq(j)p Fr(\036)2260 4362 y Fs(\()p Fp(R)p Fs(\))2367 4396 y Fu(\()p Fr(x)p Fu(\))i Fq(\000)e Fr(\036)p Fu(\()p Fr(x)p Fu(\))p Fq(j)i Fu(+)e Fq(j)p Fr(\036)p Fu(\()p Fr(x)p Fu(\))p Fq(j)3073 4304 y Fo(\021)3138 4396 y Fr(:)456 4569 y Fu(Since,)27 b(b)n(y)h(Theorem)f(A.8)g(and)g(Corollary)f(4.2,) 1443 4806 y Fq(j)p Fu(1)18 b Fq(\000)g(j)p Fr(u)p Fu(\()p Fr(x)p Fu(\))p Fq(j)24 b(\024)f Fr(C)1985 4818 y Fs(1)2036 4806 y Fr(e)2075 4743 y Fm(\000)p Fp(C)2175 4751 y Fi(2)2207 4647 y Fo(\014)2207 4697 y(\014)2207 4747 y(\014)2263 4720 y Fk(!)p 2244 4729 77 3 v 2244 4764 a Fh(j)p Fk(!)q Fh(j)2331 4743 y Fm(\001)p Fp(x)2389 4647 y Fo(\014)2389 4697 y(\014)2389 4747 y(\014)2434 4806 y Fr(;)456 4959 y Fu(for)k(suitable)g(constan)n(ts)g Fr(C)1319 4971 y Fs(1)1384 4959 y Fu(and)g Fr(C)1604 4971 y Fs(2)1642 4959 y Fu(,)h(w)n(e)f(gather)g(that)745 5099 y Fo(\014)745 5149 y(\014)745 5199 y(\014)772 5195 y Fr(@)816 5207 y Fp(\026)861 5195 y Fr(F)926 5103 y Fo(\020)975 5195 y Fr(x;)14 b(U)1116 5207 y Fp(R)1171 5195 y Fu(\()p Fr(x;)g(t)p Fu(\))1349 5103 y Fo(\021)1399 5099 y(\014)1399 5149 y(\014)1399 5199 y(\014)1450 5195 y Fq(\024)23 b Fr(C)1597 5207 y Fs(3)1648 5103 y Fo(\020)1698 5195 y Fr(e)1737 5132 y Fm(\000)p Fp(C)1837 5140 y Fi(2)1868 5036 y Fo(\014)1868 5086 y(\014)1868 5136 y(\014)1925 5110 y Fk(!)p 1906 5119 V 1906 5153 a Fh(j)p Fk(!)q Fh(j)1992 5132 y Fm(\001)p Fp(x)2050 5036 y Fo(\014)2050 5086 y(\014)2050 5136 y(\014)2100 5195 y Fu(+)18 b Fq(j)p Fr(\036)2255 5161 y Fs(\()p Fp(R)p Fs(\))2362 5195 y Fu(\()p Fr(x)p Fu(\))h Fq(\000)f Fr(\036)p Fu(\()p Fr(x)p Fu(\))p Fq(j)i Fu(+)e Fq(j)p Fr(\036)p Fu(\()p Fr(x)p Fu(\))p Fq(j)3067 5103 y Fo(\021)3132 5195 y Fr(;)p eop %%Page: 15 15 15 14 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(15)456 450 y Fu(for)27 b(some)g Fr(C)850 462 y Fs(3)887 450 y Fu(,)h(whic)n(h)g (ma)n(y)e(dep)r(end)j(on)e Fq(k)p Fr(\036)p Fq(k)1892 467 y Fp(L)1938 450 y Fh(1)1998 467 y Fs(\()p Fl(K)2082 450 y Fk(d)2082 484 y(!)2124 467 y Fs(\))2154 450 y Fu(.)37 b(Therefore,)783 559 y Fo(\014)783 609 y(\014)783 659 y(\014)783 709 y(\014)783 759 y(\014)810 592 y(Z)856 780 y Fl(K)914 764 y Fk(d)914 797 y(!)975 705 y Fr(F)12 b Fu(\()p Fr(x;)i(u)19 b Fu(+)f Fr(\036)p Fu(\))33 b Fq(\000)f Fr(F)12 b Fu(\()p Fr(x;)i(u)k Fu(+)g Fr(\036)1896 670 y Fs(\()p Fp(R)p Fs(\))2003 705 y Fu(\))c Fr(dx)2139 559 y Fo(\014)2139 609 y(\014)2139 659 y(\014)2139 709 y(\014)2139 759 y(\014)2190 705 y Fq(\024)635 972 y(\024)783 859 y Fo(Z)866 879 y Fs(1)829 1047 y(0)917 859 y Fo(Z)963 1047 y Fl(K)1021 1031 y Fk(d)1021 1064 y(!)1082 972 y Fq(j)p Fr(@)1149 984 y Fp(\026)1193 972 y Fr(F)e Fu(\()p Fr(x;)i(U)1431 984 y Fp(R)1486 972 y Fu(\()p Fr(x;)g(t)p Fu(\))p Fq(j)g(j)p Fr(\036)1773 937 y Fs(\()p Fp(R)p Fs(\))1881 972 y Fu(\()p Fr(x)p Fu(\))19 b Fq(\000)f Fr(\036)p Fu(\()p Fr(x)p Fu(\))p Fq(j)c Fr(dx)g(dt)25 b Fq(\024)635 1255 y(\024)83 b Fr(C)842 1267 y Fs(3)907 1142 y Fo(Z)953 1331 y Fl(K)1011 1314 y Fk(d)1011 1348 y(!)1072 1163 y Fo(\020)1121 1255 y Fr(e)1160 1193 y Fm(\000)p Fp(C)1260 1201 y Fi(2)1292 1097 y Fo(\014)1292 1147 y(\014)1292 1197 y(\014)1349 1170 y Fk(!)p 1330 1179 77 3 v 1330 1213 a Fh(j)p Fk(!)q Fh(j)1416 1193 y Fm(\001)p Fp(x)1474 1097 y Fo(\014)1474 1147 y(\014)1474 1197 y(\014)1523 1255 y Fu(+)18 b Fq(j)p Fr(\036)1678 1221 y Fs(\()p Fp(R)p Fs(\))1785 1255 y Fu(\()p Fr(x)p Fu(\))i Fq(\000)e Fr(\036)p Fu(\()p Fr(x)p Fu(\))p Fq(j)i Fu(+)e Fq(j)p Fr(\036)p Fu(\()p Fr(x)p Fu(\))p Fq(j)2491 1163 y Fo(\021)2556 1255 y Fq(j)p Fr(\036)2628 1221 y Fs(\()p Fp(R)p Fs(\))2735 1255 y Fu(\()p Fr(x)p Fu(\))h Fq(\000)f Fr(\036)p Fu(\()p Fr(x)p Fu(\))p Fq(j)c Fr(dx)25 b Fq(\024)635 1459 y(\024)83 b Fr(C)842 1471 y Fs(4)893 1459 y Fq(k)p Fr(\036)984 1424 y Fs(\()p Fp(R)p Fs(\))1109 1459 y Fq(\000)18 b Fr(\036)p Fq(k)1283 1476 y Fp(L)1329 1459 y Fi(2)1361 1476 y Fs(\()p Fl(K)1445 1459 y Fk(d)1445 1492 y(!)1487 1476 y Fs(\))635 1588 y Fq(\024)83 b Fu(2)p Fr(C)884 1600 y Fs(4)935 1588 y Fq(k)p Fr(\036)p Fq(k)1068 1605 y Fp(L)1114 1588 y Fi(2)1145 1605 y Fs(\()p Fl(K)1229 1588 y Fk(d)1229 1621 y(!)1272 1605 y Fm(\000)p Fp(G)1376 1613 y Fk(R)1422 1605 y Fs(\))1466 1588 y Fr(;)456 1759 y Fu(with)27 b Fr(C)703 1771 y Fs(4)767 1759 y Fu(p)r(ossibly)f(dep)r (ending)h(on)g Fq(k)p Fr(\036)p Fq(k)1731 1776 y Fp(L)1777 1759 y Fh(1)1836 1776 y Fs(\()p Fl(K)1920 1759 y Fk(d)1920 1793 y(!)1963 1776 y Fs(\))1993 1759 y Fu(.)37 b(In)27 b(particular,)e(since)i(the)g(ab)r(o)n(v)n(e)e(quan)n(tit)n(y)456 1859 y(tends)i(to)h(zero)e(for)h Fr(R)d Fq(\000)-14 b(!)23 b Fu(+)p Fq(1)p Fu(,)k(w)n(e)h(get)f(that)456 2064 y(\(4.20\))394 b(lim)1004 2117 y Fp(R)p Fm(!)p Fs(+)p Fm(1)1251 1951 y Fo(Z)1297 2139 y Fl(K)1355 2123 y Fk(d)1355 2156 y(!)1416 2064 y Fr(F)12 b Fu(\()p Fr(x;)i(u)19 b Fu(+)f Fr(\036)1796 2030 y Fs(\()p Fp(R)p Fs(\))1903 2064 y Fu(\))c Fr(dx)37 b Fu(=)2178 1951 y Fo(Z)2224 2139 y Fl(K)2282 2123 y Fk(d)2282 2156 y(!)2343 2064 y Fr(F)12 b Fu(\()p Fr(x;)i(u)k Fu(+)g Fr(\036)p Fu(\))c Fr(dx)g(:)456 2290 y Fu(Also,)27 b(b)n(y)g(insp)r(ection,)h(one)f(sees)g(that)1169 2459 y Fq(kr)p Fu(\()p Fr(\036)1361 2425 y Fs(\()p Fp(R)p Fs(\))1361 2480 y Fp(n)1486 2459 y Fq(\000)18 b Fr(\036)1618 2425 y Fs(\()p Fp(R)p Fs(\))1725 2459 y Fu(\))p Fq(k)1799 2476 y Fp(L)1845 2460 y Fi(2)1877 2476 y Fs(\()p Fl(K)1961 2460 y Fk(d)1961 2493 y(!)2004 2476 y Fs(\))2057 2459 y Fq(\024)k Fr(C)f Fq(k)p Fr(\036)2315 2471 y Fp(n)2378 2459 y Fq(\000)d Fr(\036)p Fq(k)2552 2476 y Fp(W)2623 2460 y Fi(1)p Fk(;)p Fi(2)2703 2476 y Fs(\()p Fl(K)2787 2460 y Fk(d)2787 2493 y(!)2830 2476 y Fs(\))2874 2459 y Fr(;)-2441 b Fu(\(4.21\))456 2627 y(for)33 b(a)g(suitable)g(p)r (ositiv)n(e)g(univ)n(ersal)g(constan)n(t)f Fr(C)6 b Fu(,)36 b(and)d(the)h(latter)f(quan)n(tit)n(y)h(go)r(es)e(to)i(zero)456 2726 y(when)27 b Fr(n)c Fq(\000)-14 b(!)23 b Fu(+)p Fq(1)p Fu(.)37 b(Moreo)n(v)n(er,)25 b(from)i(\(4.17\))g(and)g(the)h(Bounded)f (Con)n(v)n(ergence)f(Theorem,)1059 2936 y(lim)1005 2986 y Fp(n)p Fm(!)p Fs(+)p Fm(1)1243 2823 y Fo(Z)1289 3012 y Fp(G)1341 3020 y Fk(R)p Fi(+1)1477 2936 y Fr(F)12 b Fu(\()p Fr(x;)i(u)k Fu(+)g Fr(\036)1856 2902 y Fs(\()p Fp(R)p Fs(\))1856 2956 y Fp(n)1963 2936 y Fu(\))23 b(=)2106 2823 y Fo(Z)2152 3012 y Fp(G)2204 3020 y Fk(R)p Fi(+1)2340 2936 y Fr(F)12 b Fu(\()p Fr(x;)i(u)k Fu(+)g Fr(\036)2719 2902 y Fs(\()p Fp(R)p Fs(\))2826 2936 y Fu(\))c Fr(:)456 3189 y Fu(Since)27 b(b)r(oth)h Fr(\036)917 3159 y Fs(\()p Fp(R)p Fs(\))1052 3189 y Fu(and)f Fr(\036)1262 3146 y Fs(\()p Fp(R)p Fs(\))1262 3199 y Fp(n)1397 3189 y Fu(v)-5 b(anish)27 b(outside)h Fr(G)2005 3201 y Fp(R)p Fs(+1)2143 3189 y Fu(,)g(the)g(ab)r(o)n(v)n(e)e(reads)456 3398 y(\(4.22\))459 b(lim)1074 3447 y Fp(n)p Fm(!)p Fs(+)p Fm(1)1312 3285 y Fo(Z)1358 3473 y Fl(K)1416 3457 y Fk(d)1416 3490 y(!)1477 3398 y Fr(F)12 b Fu(\()p Fr(x;)i(u)k Fu(+)g Fr(\036)1856 3363 y Fs(\()p Fp(R)p Fs(\))1856 3418 y Fp(n)1963 3398 y Fu(\))23 b(=)2106 3285 y Fo(Z)2152 3473 y Fl(K)2210 3457 y Fk(d)2210 3490 y(!)2271 3398 y Fr(F)12 b Fu(\()p Fr(x;)i(u)k Fu(+)g Fr(\036)2650 3363 y Fs(\()p Fp(R)p Fs(\))2757 3398 y Fu(\))c Fr(:)456 3652 y Fu(Since)35 b Fr(\036)729 3609 y Fs(\()p Fp(R)p Fs(\))729 3662 y Fp(n)871 3652 y Fq(2)h Fr(C)1027 3622 y Fm(1)1021 3673 y Fs(0)1098 3652 y Fu(\()p FA(K)1205 3622 y Fp(d)1205 3673 y(!)1253 3652 y Fu(\),)h(w)n(e)e(kno)n(w)f(from)h(Theorem)f(A.8)h (that)h Fq(E)2655 3664 y Fp(!)2702 3652 y Fu(\()p Fr(u)p Fu(\))g Fq(\024)f(E)2994 3664 y Fp(!)3042 3652 y Fu(\()p Fr(u)23 b Fu(+)g Fr(\036)3282 3609 y Fs(\()p Fp(R)p Fs(\))3282 3662 y Fp(n)3389 3652 y Fu(\).)456 3752 y(Th)n(us,)i(taking)f(the)i (limit)g(as)e Fr(n)f Fq(\000)-14 b(!)23 b Fu(+)p Fq(1)h Fu(and)h(exploiting)h(\(4.21\))e(and)h(\(4.22\))o(,)h(one)f(sees)f (that)456 3926 y(\(4.23\))875 b Fq(E)1588 3938 y Fp(!)1635 3926 y Fu(\()p Fr(u)p Fu(\))24 b Fq(\024)e(E)1902 3938 y Fp(!)1950 3926 y Fu(\()p Fr(u)c Fu(+)g Fr(\036)2180 3892 y Fs(\()p Fp(R)p Fs(\))2287 3926 y Fu(\))c Fr(:)456 4094 y Fu(F)-7 b(urthermore,)26 b(for)h(some)g Fr(C)j(>)22 b Fu(0,)1245 4264 y Fq(kr)p Fu(\()p Fr(\036)1437 4229 y Fs(\()p Fp(R)p Fs(\))1562 4264 y Fq(\000)c Fr(\036)p Fu(\))p Fq(k)1768 4280 y Fp(L)1814 4264 y Fi(2)1847 4280 y Fs(\()p Fl(K)1931 4264 y Fk(d)1931 4297 y(!)1973 4280 y Fs(\))2026 4264 y Fq(\024)23 b Fr(C)d Fq(k)p Fr(\036)p Fq(k)2326 4280 y Fp(W)2397 4264 y Fi(1)p Fk(;)p Fi(2)2477 4280 y Fs(\()p Fl(K)2561 4264 y Fk(d)2561 4297 y(!)2603 4280 y Fm(\000)p Fp(G)2707 4288 y Fk(R)2754 4280 y Fs(\))2798 4264 y Fr(;)-2365 b Fu(\(4.24\))456 4435 y(whic)n(h)27 b(tends)h(to)g(zero)e(as)h Fr(R)d Fq(\000)-14 b(!)23 b Fu(+)p Fq(1)p Fu(.)37 b(Hence,)28 b(taking)f(the)h(limit)h(as)e Fr(R)d Fq(\000)-14 b(!)23 b Fu(+)p Fq(1)k Fu(in)h(\(4.23\))456 4535 y(and)i(using)h(\(4.24\))e(and)h(\(4.20\),)h(w)n(e)f(ha)n(v)n(e)f (that)h(\(4.16\))g(follo)n(ws,)g(th)n(us)g(ending)g(the)h(pro)r(of)f (of)456 4634 y(Theorem)c(2.3.)2446 b Ff(2)456 4917 y Fu(4.3.)40 b FA(Pro)s(of)g(of)g(Theorem)e(2.5.)i Fu(W)-7 b(e)35 b(no)n(w)f(address)f(the)i(pro)r(of)f(of)h(Theorem)f(2.5.)56 b(T)-7 b(ak)n(e)456 5016 y(a)34 b(p)r(oin)n(t)40 b(\026)-48 b Fr(x)35 b Fu(so)f(that)h Fq(j)p Fr(u)p Fu(\()5 b(\026)-47 b Fr(x)p Fu(\))p Fq(j)35 b Fr(<)f(\022)1511 5028 y Fs(0)1548 5016 y Fu(.)58 b(F)-7 b(rom)34 b(Theorem)g(A.7,)i(w)n(e)e(obtain)g(t)n (w)n(o)g(balls)g Fr(B)3220 5028 y Fm(\025)3311 5016 y Fu(and)456 5116 y Fr(B)519 5128 y Fm(\024)606 5116 y Fu(of)d(radius)f Fr(\032)p Fu(,)i(one)e(con)n(tained)h(in)g Fq(f)p Fr(u)d Fq(\025)g Fr(\022)1940 5128 y Fs(0)1977 5116 y Fq(g)j Fu(and)g(the)g(other)g(in)g Fq(f)p Fr(u)d Fq(\024)g(\000)p Fr(\022)2997 5128 y Fs(0)3034 5116 y Fq(g)p Fu(,)k(pro)n(vided)456 5216 y(that)e Fr(\024)686 5185 y Fm(\000)p Fs(1)775 5216 y Fr(\032)e Fq(\025)f Fr(r)975 5228 y Fs(0)1012 5216 y Fu(.)45 b(Both)30 b(balls)g(are)g (also)f(con)n(tained)g(in)i Fr(B)2338 5231 y Fp(\024)2377 5215 y Fh(\000)p Fi(1)2454 5231 y Fp(\032)2493 5216 y Fu(.)45 b(Let)30 b(us)g(denote)g(b)n(y)g Fr(T)3258 5228 y Fp(k)3329 5216 y Fu(the)p eop %%Page: 16 16 16 15 bop 456 251 a Fs(16)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fu(translation)35 b(b)n(y)i(a)g(v)n(ector)f Fr(k)41 b Fq(2)e FA(Z)1584 420 y Fp(d)1623 450 y Fu(.)66 b(By)37 b(the)g(Birkho\013)f(prop)r(ert)n (y)g(of)h(minimizers)g(\(recall)456 550 y(Corollary)25 b(4.3\),)i(w)n(e)g(deduce)h(that)1474 741 y Fr(H)1543 753 y Fm(\025)1622 741 y Fu(:=)1780 662 y Fo([)1733 841 y Fp(k)q Fm(\001)p Fp(!)r Fm(\025)p Fs(0)1932 741 y Fr(T)1981 753 y Fp(k)2022 741 y Fu(\()p Fr(B)2117 753 y Fm(\025)2173 741 y Fu(\))37 b Fq(\022)g(f)p Fr(u)22 b Fq(\025)g Fr(\022)2582 753 y Fs(0)2619 741 y Fq(g)1221 980 y Fu(and)119 b Fr(H)1543 992 y Fm(\024)1622 980 y Fu(:=)1780 901 y Fo([)1733 1080 y Fp(k)q Fm(\001)p Fp(!)r Fm(\024)p Fs(0)1932 980 y Fr(T)1981 992 y Fp(k)2022 980 y Fu(\()p Fr(B)2117 992 y Fm(\024)2173 980 y Fu(\))37 b Fq(\022)g(f)p Fr(u)22 b Fq(\024)g(\000)p Fr(\022)2647 992 y Fs(0)2684 980 y Fq(g)14 b Fr(:)456 1240 y Fu(Assuming)37 b Fr(\032)h Fu(suitably)g(big)f(with)i(resp)r (ect)e(to)h Fr(d)p Fu(,)j(it)d(follo)n(ws)f(that)h Fr(H)2751 1252 y Fm(\025)2845 1240 y Fu(and)f Fr(H)3085 1252 y Fm(\024)3179 1240 y Fu(con)n(tain)456 1339 y(a)c(half-space.)55 b(Also,)36 b(the)e(slab)f(left)i(outside)f(these)g(half-spaces)f(is)h (parallel)e(to)i(the)h(plane)456 1439 y Fq(f)p Fr(!)23 b Fq(\001)f Fr(x)32 b Fu(=)f(0)p Fq(g)p Fu(,)i(has)f(a)g(width)h(dep)r (ending)g(only)f(on)g Fr(\024)p Fu(,)i Fr(r)2253 1451 y Fs(0)2323 1439 y Fu(and)e Fr(d)h Fu(\(and)g(so)e(it)i(is)g(a)f(univ)n (ersal)456 1539 y(constan)n(t\))27 b(and)g(con)n(tains)g(the)h(set)f Fq(fj)p Fr(u)p Fq(j)c Fr(<)f(\022)1867 1551 y Fs(0)1904 1539 y Fq(g)p Fu(.)37 b(This)28 b(ends)f(the)h(pro)r(of)f(of)h(Theorem) e(2.5.)62 b Ff(2)563 1897 y Fu(5.)41 b Fv(Constr)n(uction)32 b(of)f(other)h(critical)f(points.)41 b(Pr)n(oof)31 b(of)g(Theorem)h (3.2)555 2046 y Fu(The)i(pro)r(of)e(w)n(e)h(will)h(carry)d(out)i(will)h (ha)n(v)n(e)e(the)h(same)g(steps)g(whether)g Fr(!)j Fu(is)d(rational)f (or)456 2146 y(not.)k(Some)28 b(of)f(the)h(steps)g(will)g(require)e (more)h(delicate)g(argumen)n(ts)g(when)g Fr(!)k Fu(is)c(irrational.)456 2375 y(5.1.)40 b FA(Notation)31 b(and)h(preliminaries.)38 b Fu(W)-7 b(e)27 b(recall)f(that)h(our)g(starting)f(p)r(oin)n(t)h(is)g (t)n(w)n(o)f(lo)r(cal)456 2475 y(minimizers)i Fr(u)925 2445 y Fs(\(0\))1038 2475 y Fr(<)c(u)1175 2445 y Fs(\(1\))1292 2475 y Fu(whic)n(h)29 b(are)e(in)i(the)g(same)f(lamination)g(and)g(are) g(at)g(the)h(edges)e(of)i(a)456 2574 y(gap.)555 2674 y(W)-7 b(e)30 b(consider)e(the)i(con)n(v)n(ex)d(com)n(bination)i(of)g Fr(u)2062 2644 y Fs(\(0\))2180 2674 y Fu(and)g Fr(u)2391 2644 y Fs(\(1\))2480 2674 y Fu(,)g(that)h(is,)f(for)g(an)n(y)g Fr(s)c Fq(2)h Fu([0)p Fr(;)14 b Fu(1],)456 2774 y(w)n(e)27 b(set)1467 2913 y Fr(u)1515 2879 y Fs(\()p Fp(s)p Fs(\))1625 2913 y Fu(:=)c Fr(su)1823 2879 y Fs(\(1\))1930 2913 y Fu(+)18 b(\(1)g Fq(\000)g Fr(s)p Fu(\))p Fr(u)2307 2879 y Fs(\(0\))2410 2913 y Fr(:)456 3070 y Fu(Let)27 b(also)456 3245 y(\(5.1\))664 b Fr(v)1334 3210 y Fs(\()p Fp(s)p Fs(\))1458 3245 y Fu(:=)37 b Fr(s)p Fu(\()p Fr(u)1702 3210 y Fs(\(1\))1809 3245 y Fq(\000)18 b Fr(u)1940 3210 y Fs(\(0\))2029 3245 y Fu(\))37 b(=)f Fr(u)2247 3210 y Fs(\()p Fp(s)p Fs(\))2353 3245 y Fq(\000)18 b Fr(u)2484 3210 y Fs(\(0\))2586 3245 y Fr(:)456 3422 y Fu(Notice)32 b(that)g Fr(u)953 3392 y Fs(\()p Fp(s)p Fs(\))1071 3422 y Fq(2)e Fr(C)1221 3392 y Fs(2)1259 3422 y Fu(\()p FA(K)1366 3392 y Fp(d)1366 3443 y(!)1414 3422 y Fu(\),)k Fq(j)p Fr(u)p Fq(j)c(\024)g Fu(1)i(and,)h(from)f(Theorem)f(2.3,)h(w)n(e)g (gather)f(the)h(deca)n(y)456 3522 y(estimate)456 3701 y(\(5.2\))869 b Fq(j)p Fr(v)1562 3667 y Fs(\()p Fp(s)p Fs(\))1650 3701 y Fu(\()p Fr(x)p Fu(\))p Fq(j)24 b(\024)f Fr(C)1955 3713 y Fs(1)1992 3701 y Fr(e)2031 3661 y Fm(\000)p Fp(C)2131 3669 y Fi(2)2163 3667 y Fq(j)2215 3639 y Fk(!)p 2196 3648 77 3 v 2196 3682 a Fh(j)p Fk(!)q Fh(j)2282 3661 y Fm(\001)p Fp(x)2340 3667 y Fq(j)2381 3701 y Fr(;)456 3875 y Fu(for)k(an)n(y)f Fr(s)d Fq(2)h Fu([0)p Fr(;)14 b Fu(1],)27 b(for)g(suitable)g(univ)n(ersal)f(constan)n(ts)h Fr(C)2311 3887 y Fs(1)2376 3875 y Fu(and)h Fr(C)2597 3887 y Fs(2)2634 3875 y Fu(.)555 4128 y(W)-7 b(e)38 b(consider)e(the)h (set)h(of)f(functions)g(trapp)r(ed)g(b)r(et)n(w)n(een)h Fr(u)2504 4098 y Fs(\(0\))2629 4128 y Fu(and)f Fr(u)2848 4098 y Fs(\(1\))2937 4128 y Fu(:)56 b(for)37 b(this,)j(w)n(e)456 4227 y(de\014ne)737 4402 y Fq(C)88 b Fu(:=)83 b Fq(f)p Fr(v)25 b Fq(2)f Fr(C)6 b Fu(\()p FA(K)1398 4367 y Fp(d)1398 4422 y(!)1446 4402 y Fu(\))38 b(s.t.)f Fr(u)1712 4372 y Fs(\(0\))1819 4402 y Fu(+)18 b Fr(v)31 b Fu(enjo)n(ys)26 b(the)i(Birkho\013)f(prop)r(ert)n(y)g(and)1040 4534 y Fr(u)1088 4499 y Fs(\(0\))1199 4534 y Fq(\024)c Fr(u)1335 4499 y Fs(\(0\))1442 4534 y Fu(+)18 b Fr(v)26 b Fq(\024)d Fr(u)1727 4499 y Fs(\(1\))1816 4534 y Fq(g)14 b Fr(:)-1439 b Fu(\(5.3\))456 4711 y(Notice)30 b(that,)i(due)f(to)g(\(5.2\),)g Fr(v)1459 4681 y Fs(\()p Fp(s)p Fs(\))1575 4711 y Fq(2)e(C)35 b Fu(for)c(an)n(y)f Fr(s)e Fq(2)h Fu([0)p Fr(;)14 b Fu(1].)45 b(Note)31 b(that)g(the)g(functions)g Fr(u)3357 4681 y Fs(\()p Fp(s)p Fs(\))456 4814 y Fu(are)26 b(Birkho\013)h(if)h Fr(u)1040 4784 y Fs(\(0\))1129 4814 y Fr(;)14 b(u)1214 4784 y Fs(\(1\))1330 4814 y Fu(are.)555 4914 y(Some)33 b(w)n(ords)e(ma)n(y)h(b)r(e)h(needed)g(to)f(justify)i(the)f(notation)f (c)n(hosen)g(in)h(\(5.3\):)47 b(in)33 b(the)g(fol-)456 5016 y(lo)n(wing)26 b(argumen)n(ts,)g(our)g(ob)5 b(jects)27 b(of)g(primary)f(in)n(terest)h(will)h(b)r(e)f(the)h(incremen)n(ts)f (from)g Fr(u)3356 4986 y Fs(\(0\))456 5116 y Fu(rather)k(than)h(the)h (function)g(describing)e(the)i(state.)51 b(Hence,)34 b(w)n(e)e(will)g(dev)n(elop)g(functionals)456 5216 y(etc.)37 b(adapted)27 b(to)g(this)h(notation.)37 b(This)27 b(simpli\014es)h (some)f(of)g(the)h(calculations)f(later.)p eop %%Page: 17 17 17 16 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(17)456 450 y Fu(5.2.)40 b FA(An)47 b(in)m(tegral)g(a-priori)f(b)s(ound)h(for)g (Birkho\013)f(functions.)41 b Fu(The)g(follo)n(wing)e(a-)456 550 y(priori)26 b(in)n(tegral)g(b)r(ound)i(will)g(b)r(e)g(crucial)f (for)g(our)g(w)n(ork.)456 680 y FA(Lemma)34 b(5.1.)42 b Fn(L)l(et)33 b Fr(u)1186 650 y Fs(\(0\))1307 680 y Fn(and)h Fr(u)1520 650 y Fs(\(1\))1642 680 y Fn(b)l(e)f(Birkho\013)i (functions)f(at)f(the)g(e)l(dge)h(of)g(the)g(gaps)g(of)g(a)456 780 y(lamination)d(invariant)f(under)g(inte)l(ger)g(tr)l(anslations.)38 b(Then,)456 979 y Fu(\(5.4\))1581 866 y Fo(Z)1627 1054 y Fl(K)1685 1038 y Fk(d)1685 1071 y(!)1746 979 y Fq(j)p Fr(u)1817 944 y Fs(\(1\))1924 979 y Fq(\000)18 b Fr(u)2055 944 y Fs(\(0\))2144 979 y Fq(j)23 b(\024)f Fu(2)555 1198 y(Note)h(that)g(an)g(immediate)g(consequence)e(of)i(Lemma)g(5.1)f(is)g (that)h(for)g(all)f(functions)h Fr(v)j Fq(2)e(C)456 1298 y Fu(w)n(e)j(ha)n(v)n(e)1758 1333 y Fo(Z)1804 1521 y Fl(K)1862 1505 y Fk(d)1862 1538 y(!)1923 1446 y Fr(v)f Fq(\024)d Fu(2)p Fr(:)456 1692 y Fn(Pr)l(o)l(of.)43 b Fu(The)29 b(idea)g(of)g(the)g(pro)r(of)g(is)g(v)n(ery)f(simple.)41 b(Because)29 b(of)g(the)g(Birkho\013)g(prop)r(ert)n(y)-7 b(,)28 b(the)456 1791 y(translations)h(of)i(the)g(gaps)f(to)h(a)g (\014xed)g(unit)g(cub)r(e)h(times)f([)p Fq(\000)p Fu(1)p Fr(;)14 b Fu(1])30 b(cannot)g(o)n(v)n(erlap,)g(so)g(that)456 1891 y(the)d(total)h(v)n(olume)e(should)h(b)r(e)h(less)f(than)h(the)g (v)n(olume)e(of)i([0)p Fr(;)14 b Fu(1])2495 1861 y Fp(d)2551 1891 y Fq(\002)j Fu([)p Fq(\000)p Fu(1)p Fr(;)d Fu(1])26 b(\(and)i(thence)g(2\).)555 1990 y(Let)g Fq(Q)23 b Fu(:=)g([0)p Fr(;)14 b Fu(1])1073 1960 y Fp(d)1138 1990 y Fu(and)27 b(let)h Fq(Z)i(\032)23 b FA(Z)1655 1960 y Fp(d)1722 1990 y Fu(b)r(e)28 b(a)f(minimal)h(set)f(of)h(in)n(teger)e(v)n(ectors)g(so)h (that)1764 2089 y Fo([)1743 2268 y Fp(k)q Fm(2Z)1877 2168 y Fu(\()p Fr(k)22 b Fu(+)c Fq(Q)p Fu(\))456 2421 y(co)n(v)n(ers)25 b(the)j(lift)g(of)g FA(K)1147 2391 y Fp(d)1147 2441 y(!)1223 2421 y Fu(to)f FA(R)1396 2391 y Fp(d)1434 2421 y Fu(.)37 b(F)-7 b(or)27 b(an)n(y)g Fr(k)f Fq(2)d FA(Z)2005 2391 y Fp(d)2045 2421 y Fu(,)k(let)853 2607 y Fq(A)919 2619 y Fp(k)997 2607 y Fu(:=)1122 2514 y Fo(n)1177 2607 y Fu(\()p Fr(x;)14 b(y)s Fu(\))24 b Fq(2)f(Q)c(\002)f FA(R)36 b Fu(s)p Fr(:)p Fu(t)p Fr(:)h(u)1945 2572 y Fs(\(0\))2034 2607 y Fu(\()p Fr(x)19 b Fu(+)f Fr(k)s Fu(\))23 b Fr(<)g(y)j Fq(\024)c Fr(u)2606 2572 y Fs(\(1\))2695 2607 y Fu(\()p Fr(x)d Fu(+)f Fr(k)s Fu(\))2954 2514 y Fo(o)3023 2607 y Fr(:)456 2788 y Fu(Thanks)27 b(to)g(the)h(Birkho\013)f(prop)r(ert)n(y)-7 b(,)27 b(w)n(e)g(ha)n(v)n (e)f(that)1687 2949 y Fq(A)1753 2961 y Fp(h)1815 2949 y Fq(\\)19 b(A)1955 2961 y Fp(k)2033 2949 y Fu(=)36 b Fq(;)14 b Fr(;)456 3110 y Fu(if)28 b Fr(h;)14 b(k)25 b Fq(2)f(Z)34 b Fu(with)28 b Fr(h)23 b Fq(6)p Fu(=)g Fr(k)s Fu(.)37 b(Therefore,)1073 3196 y Fo(Z)1119 3384 y Fl(K)1177 3368 y Fk(d)1177 3401 y(!)1238 3309 y Fq(j)p Fr(u)1309 3274 y Fs(\(1\))1416 3309 y Fq(\000)18 b Fr(u)1547 3274 y Fs(\(0\))1636 3309 y Fq(j)23 b Fu(=)1777 3230 y Fo(X)1770 3409 y Fp(k)q Fm(2Z)1917 3196 y Fo(Z)1964 3384 y Fp(k)q Fs(+)p Fm(Q)2124 3309 y Fr(u)2172 3274 y Fs(\(1\))2260 3309 y Fu(\()p Fr(x)p Fu(\))d Fq(\000)e Fr(u)2522 3274 y Fs(\(0\))2610 3309 y Fu(\()p Fr(x)p Fu(\))c Fr(dx)1682 3542 y Fu(=)1777 3463 y Fo(X)1770 3642 y Fp(k)q Fm(2Z)1917 3542 y Fq(jA)2006 3554 y Fp(k)2048 3542 y Fq(j)1682 3775 y Fu(=)23 b Fq(j)1827 3696 y Fo([)1807 3874 y Fp(k)q Fm(2Z)1954 3775 y Fq(A)2020 3787 y Fp(k)2061 3775 y Fq(j)14 b Fr(:)555 4019 y Fu(Since)28 b Fq(A)838 4031 y Fp(k)902 4019 y Fq(\022)23 b(Q)18 b(\002)g Fu([)p Fq(\000)p Fu(1)p Fr(;)c Fu(1])26 b(w)n(e)i(th)n(us)f(deduce)h(that)1374 4110 y Fo(Z)1420 4298 y Fl(K)1478 4282 y Fk(d)1478 4315 y(!)1539 4223 y Fq(j)p Fr(u)1610 4188 y Fs(\(1\))1717 4223 y Fq(\000)18 b Fr(u)1848 4188 y Fs(\(0\))1937 4223 y Fq(j)23 b(\024)2071 4127 y Fo(\014)2071 4177 y(\014)2071 4227 y(\014)2098 4223 y Fq(Q)c(\002)f Fu([)p Fq(\000)p Fu(1)p Fr(;)c Fu(1])2500 4127 y Fo(\014)2500 4177 y(\014)2500 4227 y(\014)1983 4405 y Fu(=)23 b(2)14 b Fr(:)3380 4565 y Fj(\003)555 4753 y Fu(W)-7 b(e)26 b(note)e(that)h(the)h(pro)r(of)e (of)31 b(\(5.4\))25 b(only)g(uses)f(that)h(the)g Fr(u)2421 4722 y Fs(\(0\))2535 4753 y Fu(and)g Fr(u)2742 4722 y Fs(\(1\))2855 4753 y Fu(are)f(Birkho\013)h(and)456 4852 y(are)i(inside)i(a)g(lamination.)40 b(The)29 b(minimization)g(prop)r (erties)f(do)g(not)h(pla)n(y)f(an)n(y)g(role.)40 b(In)29 b(the)456 4952 y(case)d(that)i(the)g Fr(u)p Fu('s)f(are)g(p)r(erio)r (dic,)g(w)n(e)h(obtain)f(equalit)n(y)g(in)h(\(5.4\))o(.)555 5116 y(If)21 b Fr(u)679 5086 y Fs(\(0\))788 5116 y Fu(and)g Fr(u)991 5086 y Fs(\(1\))1100 5116 y Fu(satisfy)f(the)h(elliptic)g (equation)f(\(2.1\),)i(recalling)d(that)i(they)f(are)g(b)r(ounded,)456 5216 y(w)n(e)27 b(can)g(obtain)g(from)h(\(5.1\))f(the)h(follo)n(wing)e (result:)p eop %%Page: 18 18 18 17 bop 456 251 a Fs(18)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y FA(Lemma)29 b(5.2.)41 b Fn(F)-6 b(or)30 b Fr(r)c Fu(=)d(0)p Fr(;)14 b Fu(1)p Fr(;)g(:)g(:)g(:)42 b Fn(as)30 b(lar)l(ge)h(as)f(we)h(wish,)g (ther)l(e)f(exists)f(a)i(p)l(ositive)g(c)l(onstant)456 550 y Fr(C)6 b Fn(,)30 b(p)l(ossibly)h(dep)l(ending)h(on)d Fr(r)r Fn(,)i(so)f(that)456 779 y Fu(\(5.5\))1464 666 y Fo(Z)1511 855 y Fl(K)1569 838 y Fk(d)1569 871 y(!)1629 779 y Fq(j)p Fr(D)1723 745 y Fp(r)1760 779 y Fu(\()p Fr(u)1840 745 y Fs(\(0\))1948 779 y Fq(\000)18 b Fr(u)2079 745 y Fs(\(1\))2167 779 y Fu(\))p Fq(j)24 b(\024)e Fr(C)e(:)456 1085 y Fn(Pr)l(o)l(of.)43 b Fu(W)-7 b(e)28 b(note)f(that)h Fr(u)1271 1054 y Fs(\(1\))1378 1085 y Fq(\000)18 b Fr(u)1509 1054 y Fs(\(0\))1626 1085 y Fu(satis\014es)456 1279 y(\(5.6\))503 b Fr(L)p Fu(\()p Fr(u)1267 1244 y Fs(\(0\))1374 1279 y Fq(\000)18 b Fr(u)1505 1244 y Fs(\(1\))1594 1279 y Fu(\))23 b(=)g Fr(f)9 b Fu(\()p Fr(x;)14 b(u)1951 1244 y Fs(\(0\))2040 1279 y Fu(\()p Fr(x)p Fu(\)\))20 b Fq(\000)e Fr(f)9 b Fu(\()p Fr(x;)14 b(u)2500 1244 y Fs(\(1\))2588 1279 y Fu(\()p Fr(x)p Fu(\)\))g Fr(:)456 1470 y Fu(F)-7 b(urthermore,)456 1664 y(\(5.7\))297 b Fq(j)p Fr(f)9 b Fu(\()p Fr(x;)14 b(u)1161 1630 y Fs(\(0\))1250 1664 y Fu(\()p Fr(x)p Fu(\)\))20 b Fq(\000)e Fr(f)9 b Fu(\()p Fr(x;)14 b(u)1710 1630 y Fs(\(1\))1799 1664 y Fu(\()p Fr(x)p Fu(\)\))p Fq(j)24 b(\024)37 b Fu(const)13 b Fq(j)p Fr(u)2365 1630 y Fs(\(1\))2454 1664 y Fu(\()p Fr(x)p Fu(\))19 b Fq(\000)f Fr(u)2715 1630 y Fs(\(0\))2804 1664 y Fu(\()p Fr(x)p Fu(\))p Fq(j)c Fr(:)456 1856 y Fu(Th)n(us,)43 b(the)e(RHS)g(of)46 b(\(5.6\))40 b(is)h(in)f Fr(L)1658 1826 y Fs(1)1722 1856 y Fq(\\)27 b Fr(L)1861 1826 y Fm(1)1931 1856 y Fu(.)75 b(By)40 b(Sc)n(hauder)g(estimates)g(\(see,)j(e.g.,)g (for-)456 1955 y(m)n(ula)27 b(\(6.23\))g(in)h([GT83)o(]\),)g(w)n(e)f (obtain)g(that)1321 2154 y Fq(k)p Fr(D)1434 2120 y Fp(r)1470 2154 y Fu(\()p Fr(u)1550 2120 y Fs(\(1\))1658 2154 y Fq(\000)18 b Fr(u)1789 2120 y Fs(\(0\))1877 2154 y Fu(\))p Fq(k)1951 2171 y Fp(L)1997 2154 y Fh(1)2057 2171 y Fs(\()p Fl(K)2141 2154 y Fk(d)2141 2188 y(!)2184 2171 y Fs(\))2251 2154 y Fq(\024)37 b Fu(const)13 b Fr(:)456 2350 y Fu(Com)n(bining)31 b(this)h(with)h(\(5.6\),)f(\(5.7\))f(and)h(the)g(Calder\023)-42 b(on-Zygm)n(und)30 b(inequalit)n(y)i(\(see,)h(e.g.,)456 2453 y(Theorem)22 b(9.13)g(of)i([GT83)o(]\),)h(w)n(e)e(obtain)g(that)g (the)h Fr(L)2141 2423 y Fs(2)2178 2453 y Fu(-norm)e(of)i Fr(D)2580 2423 y Fp(r)2617 2453 y Fu(\()p Fr(u)2697 2423 y Fs(\(1\))2796 2453 y Fq(\000)10 b Fr(u)2919 2423 y Fs(\(0\))3007 2453 y Fu(\))24 b(in)f(a)g(ball)h(of)456 2556 y(radius)k(1)g(cen)n(tered)g(at)h Fr(x)1258 2568 y Fs(0)1325 2556 y Fu(is)f(b)r(ounded)h(b)n(y)g(the)g Fr(L)2067 2525 y Fs(2)2133 2556 y Fu(norm)f(of)h Fr(f)9 b Fu(\()p Fr(x;)14 b(u)2661 2525 y Fs(\(0\))2750 2556 y Fu(\()p Fr(x)p Fu(\)\))20 b Fq(\000)f Fr(f)9 b Fu(\()p Fr(x;)14 b(u)3211 2525 y Fs(\(1\))3300 2556 y Fu(\()p Fr(x)p Fu(\)\))456 2655 y(in)27 b(a)h(ball)f(cen)n(tered)g(at)h Fr(x)1262 2667 y Fs(0)1327 2655 y Fu(and)f(of)h(radius)f(2.)555 2755 y(Consequen)n(tly)-7 b(,)27 b(b)n(y)g(H\177)-42 b(older's)27 b(inequalit)n(y)g(and)h(b)n(y)f(the)h(fact)g(that)g Fq(j)p Fr(u)p Fq(j)22 b(\024)h Fu(1,)710 2896 y Fo( )776 2925 y(Z)822 3114 y Fp(B)872 3122 y Fi(1)904 3114 y Fs(\()p Fp(x)968 3122 y Fi(0)1000 3114 y Fs(\))1044 3038 y Fq(j)p Fr(D)1138 3004 y Fp(r)1175 3038 y Fu(\()p Fr(u)1255 3004 y Fs(\(0\))1363 3038 y Fq(\000)18 b Fr(u)1494 3004 y Fs(\(1\))1582 3038 y Fu(\))p Fq(j)1637 2896 y Fo(!)1703 2913 y Fs(2)719 3337 y Fu(=)807 3195 y Fo( )873 3224 y(Z)919 3413 y Fp(B)969 3421 y Fi(1)1001 3413 y Fs(\()p Fp(x)1065 3421 y Fi(0)1097 3413 y Fs(\))1141 3337 y Fq(j)p Fr(D)1235 3303 y Fp(r)1272 3337 y Fu(\()p Fr(u)1352 3303 y Fs(\(0\))1459 3337 y Fq(\000)g Fr(u)1590 3303 y Fs(\(1\))1679 3337 y Fu(\))p Fq(j)1734 3303 y Fs(1)p Fp(=)p Fs(2)1857 3337 y Fq(\001)h(j)p Fr(D)1993 3303 y Fp(r)2030 3337 y Fu(\()p Fr(u)2110 3303 y Fs(\(0\))2217 3337 y Fq(\000)f Fr(u)2348 3303 y Fs(\(1\))2437 3337 y Fu(\))p Fq(j)2492 3303 y Fs(1)p Fp(=)p Fs(2)2596 3195 y Fo(!)2662 3212 y Fs(2)719 3643 y Fq(\024)807 3501 y Fo( )873 3530 y(Z)919 3718 y Fp(B)969 3726 y Fi(1)1001 3718 y Fs(\()p Fp(x)1065 3726 y Fi(0)1097 3718 y Fs(\))1141 3643 y Fq(j)p Fr(D)1235 3608 y Fp(r)1272 3643 y Fu(\()p Fr(u)1352 3608 y Fs(\(0\))1459 3643 y Fq(\000)g Fr(u)1590 3608 y Fs(\(1\))1679 3643 y Fu(\))p Fq(j)1734 3608 y Fs(2)p Fp(=)p Fs(3)1839 3501 y Fo(!)1905 3518 y Fs(3)p Fp(=)p Fs(2)2027 3643 y Fq(\001)2069 3501 y Fo( )2135 3530 y(Z)2181 3718 y Fp(B)2231 3726 y Fi(1)2263 3718 y Fs(\()p Fp(x)2327 3726 y Fi(0)2359 3718 y Fs(\))2403 3643 y Fq(j)p Fr(D)2497 3608 y Fp(r)2534 3643 y Fu(\()p Fr(u)2614 3608 y Fs(\(0\))2721 3643 y Fq(\000)h Fr(u)2853 3608 y Fs(\(1\))2941 3643 y Fu(\))p Fq(j)2996 3608 y Fs(2)3034 3501 y Fo(!)3099 3518 y Fs(1)p Fp(=)p Fs(2)719 3857 y Fq(\024)37 b Fu(const)13 b Fq(k)p Fr(D)1137 3823 y Fp(r)1173 3857 y Fu(\()p Fr(u)1253 3823 y Fs(\(0\))1361 3857 y Fq(\000)18 b Fr(u)1492 3823 y Fs(\(1\))1580 3857 y Fu(\))p Fq(k)1654 3872 y Fp(L)1700 3856 y Fi(2)1732 3872 y Fs(\()p Fp(B)1808 3880 y Fi(1)1841 3872 y Fs(\()p Fp(x)1905 3880 y Fi(0)1937 3872 y Fs(\)\))719 4003 y Fq(\024)37 b Fu(const)13 b Fq(k)p Fr(u)1114 3968 y Fs(\(0\))1221 4003 y Fq(\000)18 b Fr(u)1352 3968 y Fs(\(1\))1440 4003 y Fq(k)1482 4018 y Fp(L)1528 4001 y Fi(2)1560 4018 y Fs(\()p Fp(B)1636 4026 y Fi(2)1668 4018 y Fs(\()p Fp(x)1732 4026 y Fi(0)1764 4018 y Fs(\)\))719 4148 y Fq(\024)37 b Fu(const)13 b Fq(k)p Fr(u)1114 4113 y Fs(\(0\))1221 4148 y Fq(\000)18 b Fr(u)1352 4113 y Fs(\(1\))1440 4148 y Fq(k)1482 4163 y Fp(L)1528 4146 y Fi(1)1560 4163 y Fs(\()p Fp(B)1636 4171 y Fi(2)1668 4163 y Fs(\()p Fp(x)1732 4171 y Fi(0)1764 4163 y Fs(\)\))1834 4148 y Fr(:)555 4356 y Fu(Hence,)23 b(the)e(desired)g(result)g(follo)n (ws)f(b)r(ecause)g(w)n(e)h(can)g(estimate)2611 4289 y Fo(R)2650 4385 y Fl(K)2708 4368 y Fk(d)2708 4402 y(!)2769 4356 y Fq(j)p Fr(D)2863 4325 y Fp(r)2899 4356 y Fu(\()p Fr(u)2979 4325 y Fs(\(1\))3074 4356 y Fq(\000)5 b Fr(u)3192 4325 y Fs(\(0\))3280 4356 y Fu(\))p Fq(j)22 b Fu(b)n(y)456 4484 y(the)27 b(sum)g(of)h(the)f(in)n(tegral)f(o)n(v)n(er)f(balls)i(of) g(radius)f(1)h(cen)n(tered)g(at)g Fr(d)2569 4454 y Fm(\000)p Fs(1)p Fp(=)p Fs(2)2725 4484 y FA(Z)2783 4454 y Fp(d)2822 4484 y Fu(.)37 b(Then,)28 b(follo)n(wing)456 4583 y(the)22 b(argumen)n(t)f(ab)r(o)n(v)n(e,)h(w)n(e)g(estimate)g(eac)n(h)f(one)h (of)g(those)g(b)n(y)g(the)g(in)n(tegral)f(of)h Fr(u)2967 4553 y Fs(\(1\))3063 4583 y Fq(\000)8 b Fr(u)3184 4553 y Fs(\(0\))3294 4583 y Fu(o)n(v)n(er)456 4683 y(a)26 b(ball)g(of)h(radius)e(2)h(cen)n(tered)g(at)h(an)f(in)n(teger)g(p)r (oin)n(t.)36 b(Since)27 b(the)g(n)n(um)n(b)r(er)f(of)h(suc)n(h)f(balls) g(that)456 4783 y(co)n(v)n(er)j(one)i(p)r(oin)n(t)h(is)f(\014nite,)i (the)f(sum)f(of)h(the)f(in)n(tegrals)f(can)h(b)r(e)h(b)r(ounded)g(b)n (y)f(the)h(in)n(tegral)456 4882 y(o)n(v)n(er)25 b FA(K)708 4852 y Fp(d)708 4903 y(!)756 4882 y Fu(.)2601 b Fj(\003)456 5116 y Fu(5.3.)40 b FA(The)e(renormalized)e(energy.)41 b Fu(Giv)n(en)33 b(that)g(w)n(e)g(ha)n(v)n(e)e(a)h(lo)r(cal)h (minimizer)f Fr(u)3223 5086 y Fs(\(0\))3345 5116 y Fu(for)456 5216 y(the)j(energy)e Fq(E)7 b Fu(,)37 b(w)n(e)d(no)n(w)g(seek)g (another)f(critical)h(p)r(oin)n(t.)58 b(T)-7 b(o)35 b(do)f(so,)i(w)n(e) e(will)h(\014nd)g(it)g(v)n(ery)p eop %%Page: 19 19 19 18 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(19)456 450 y Fu(con)n(v)n(enien)n(t)26 b(to)i(de\014ne)f(the)h(follo)n(wing)f (renormalized)f(energy:)930 613 y(~)916 634 y Fq(E)960 646 y Fp(!)1008 634 y Fu(\()p Fr(v)s Fu(\))e(=)1226 521 y Fo(Z)1272 709 y Fl(K)1330 693 y Fk(d)1330 726 y(!)1377 541 y Fo(h)1426 577 y Fu(1)p 1426 614 42 4 v 1426 690 a(2)1478 541 y Fo(\020)1528 634 y Fr(a)1572 646 y Fp(ij)1630 634 y Fu(\()p Fr(x)p Fu(\))14 b Fr(@)1799 646 y Fp(i)1827 634 y Fu(\()p Fr(u)1907 599 y Fs(\(0\))2015 634 y Fu(+)k Fr(v)s Fu(\)\()p Fr(x)p Fu(\))c Fr(@)2342 646 y Fp(j)2378 634 y Fu(\()p Fr(u)2458 599 y Fs(\(0\))2566 634 y Fu(+)k Fr(v)s Fu(\)\()p Fr(x)p Fu(\))1396 854 y Fq(\000)32 b Fr(a)1537 866 y Fp(ij)1595 854 y Fu(\()p Fr(x)p Fu(\))14 b Fr(@)1764 866 y Fp(i)1793 854 y Fr(u)1841 820 y Fs(\(0\))1929 854 y Fu(\()p Fr(x)p Fu(\))g Fr(@)2098 866 y Fp(j)2134 854 y Fr(u)2182 820 y Fs(\(0\))2271 854 y Fu(\()p Fr(x)p Fu(\))2382 762 y Fo(\021)1396 1037 y Fu(+)32 b Fr(F)12 b Fu(\()p Fr(x;)i(u)1722 1003 y Fs(\(0\))1811 1037 y Fu(\()p Fr(x)p Fu(\))19 b(+)f Fr(v)s Fu(\()p Fr(x)p Fu(\)\))34 b Fq(\000)e Fr(F)12 b Fu(\()p Fr(x;)i(u)2570 1003 y Fs(\(0\))2659 1037 y Fu(\()p Fr(x)p Fu(\)\))2802 945 y Fo(i)2857 1037 y Fr(dx)g(:)456 827 y Fu(\(5.8\))555 1216 y(Notice)39 b(that)g(in)h(the)f(case)f(that)h Fr(!)45 b Fq(2)d FA(Q)1925 1186 y Fp(d)1989 1216 y Fq(\000)26 b(f)p Fu(0)p Fq(g)p Fu(,)40 b(this)g(renormalized)d(energy)h(di\013ers)456 1315 y(from)32 b(the)g(\\regular")e(energy)h Fq(E)1488 1327 y Fp(!)1568 1315 y Fu(b)n(y)i(a)e(constan)n(t,)i(namely)f(the)h (energy)e(of)i(the)f(minimizer)456 1415 y Fr(u)504 1385 y Fs(\(0\))592 1415 y Fu(.)k(When)26 b Fr(!)i Fu(is)d(irrational)e(and) i(the)h(domain)f(of)g(in)n(tegration)f(is)h(un)n(b)r(ounded,)h(the)f (regular)456 1515 y(energy)36 b(of)i Fr(u)884 1485 y Fs(\(0\))1010 1515 y Fu(is)g(in\014nite)g(but)g(the)g(renormalized)e (energy)h(can)g(b)r(e)h(de\014ned)g(as)f(a)g(func-)456 1614 y(tional)d(for)g(all)g(functions)h(that)g(are)e(close)h({)g(in)h (some)f(appropriate)f(sense)h({)g(to)h Fr(u)3160 1584 y Fs(\(0\))3248 1614 y Fu(.)58 b(W)-7 b(e)456 1714 y(emphasize)36 b(that)i(the)g(Euler-Lagrange)33 b(equations)k(for)g(\(5.8\))g(are)f (just)i(the)g(same)e(as)h(the)456 1814 y(Euler-Lagrange)d(equations)k (for)g(the)h(standard)e(v)-5 b(ariational)37 b(principle.)69 b(Hence,)41 b(\014nding)456 1913 y(critical)27 b(p)r(oin)n(ts)g(of)34 b(\(5.8\))27 b(will)h(lead)g(to)f(critical)g(p)r(oin)n(ts)h(of)f(the)h (original)e(problem.)456 2039 y FA(Prop)s(osition)k(5.3.)40 b Fn(If)30 b Fr(u)1297 2009 y Fs(\(0\))1415 2039 y Fn(and)h Fr(u)1625 2009 y Fs(\(1\))1743 2039 y Fn(ar)l(e)f(lo)l(c)l(al)h (minimizers,)g(then)456 2187 y Fu(\(5.9\))1622 2166 y(~)1608 2187 y Fq(E)1652 2199 y Fp(!)1700 2187 y Fu(\()p Fr(u)1780 2153 y Fs(\(1\))1888 2187 y Fq(\000)18 b Fr(u)2019 2153 y Fs(\(0\))2107 2187 y Fu(\))24 b(=)e(0)456 2352 y Fn(Pr)l(o)l(of.)43 b Fu(W)-7 b(e)28 b(note)g(that,)h(b)r(ecause)e(\(5.5\))h(and)g(the)h (fact)f(that,)g(b)n(y)g(elliptic)h(regularit)n(y)d(theory)-7 b(,)456 2452 y(w)n(e)30 b(can)g(b)r(ound)h(the)f(deriv)-5 b(ativ)n(es)30 b(in)g Fr(L)1713 2422 y Fm(1)1783 2452 y Fu(,)i(w)n(e)e(obtain)g(that)h(the)f(in)n(tegral)g(in)g(the)h (de\014nition)456 2553 y(of)564 2532 y(~)550 2553 y Fq(E)594 2565 y Fp(!)642 2553 y Fu(\()p Fr(u)722 2523 y Fs(\(1\))829 2553 y Fq(\000)18 b Fr(u)960 2523 y Fs(\(0\))1049 2553 y Fu(\))28 b(con)n(v)n(erges.)555 2653 y(If)k Fr(g)682 2665 y Fp(n)727 2653 y Fu(\()p Fr(x)p Fu(\))e(=)e(\011\()p Fq(j)p Fr(x)p Fq(j)21 b(\000)g Fr(n)p Fu(\))31 b(where)g(\011)g(is)g(a) f Fr(C)1935 2623 y Fm(1)2037 2653 y Fu(function)i(taking)e(the)i(v)-5 b(alue)31 b(1)g(in)g(\()p Fq(\0001)p Fr(;)14 b Fu(1])456 2752 y(and)27 b(the)h(v)-5 b(alue)27 b(0)h(in)g([2)p Fr(;)14 b Fq(1)p Fu(\),)27 b(w)n(e)g(ha)n(v)n(e)g(that)1162 2902 y(~)1148 2923 y Fq(E)1192 2935 y Fp(!)1240 2923 y Fu(\()p Fr(u)1320 2888 y Fs(\(1\))1427 2923 y Fq(\000)18 b Fr(u)1558 2888 y Fs(\(0\))1647 2923 y Fu(\))23 b(=)52 b(lim)1790 2973 y Fp(n)p Fm(!1)1991 2902 y Fu(~)1977 2923 y Fq(E)2021 2935 y Fp(!)2069 2831 y Fo(\020)2118 2923 y Fr(g)2158 2935 y Fp(n)2222 2923 y Fq(\001)18 b Fu(\()p Fr(u)2343 2888 y Fs(\(1\))2451 2923 y Fq(\000)g Fr(u)2582 2888 y Fs(\(0\))2670 2923 y Fu(\))2702 2831 y Fo(\021)456 3112 y Fu(b)n(y)24 b(the)h(Dominated)h(Con)n(v)n(ergence) c(Theorem.)35 b(On)25 b(the)g(other)f(hand,)i(w)n(e)e(ha)n(v)n(e)g (that)3213 3091 y(~)3199 3112 y Fq(E)3243 3124 y Fp(!)3291 3112 y Fu(\()p Fr(g)3363 3124 y Fp(n)3421 3112 y Fq(\001)456 3214 y Fu(\()p Fr(u)536 3184 y Fs(\(1\))642 3214 y Fq(\000)17 b Fr(u)772 3184 y Fs(\(0\))861 3214 y Fu(\)\))24 b Fq(\025)e Fu(0)27 b(b)r(ecause)g(of)g(the)g(fact)h(that)f Fr(u)2039 3184 y Fs(\(0\))2155 3214 y Fu(is)g(a)g(lo)r(cal)f(minimizer)h(and)g Fr(g)3090 3226 y Fp(n)3153 3214 y Fq(\001)17 b Fu(\()p Fr(u)3273 3184 y Fs(\(1\))3380 3214 y Fq(\000)456 3317 y Fr(u)504 3287 y Fs(\(0\))592 3317 y Fu(\))36 b(is)e(a)h(p)r (erturbation)f(of)h Fr(u)1471 3287 y Fs(\(0\))1595 3317 y Fu(with)g(compact)g(supp)r(ort.)58 b(Hence,)37 b(b)n(y)e(the)g (Dominated)456 3417 y(Con)n(v)n(ergence)25 b(Theorem,)456 3565 y(\(5.10\))1604 3544 y(~)1590 3565 y Fq(E)1634 3577 y Fp(!)1682 3565 y Fu(\()p Fr(u)1762 3531 y Fs(\(1\))1869 3565 y Fq(\000)18 b Fr(u)2000 3531 y Fs(\(0\))2089 3565 y Fu(\))23 b Fq(\025)g Fu(0)14 b Fr(:)456 3711 y Fu(Analogously)-7 b(,)26 b(b)n(y)h(Dominated)h(Con)n(v)n(ergence)d(Theorem,)456 3876 y(\(5.11\))447 b(0)23 b(=)1282 3855 y(~)1268 3876 y Fq(E)1312 3888 y Fp(!)1360 3876 y Fu(\(0\))g(=)52 b(lim)1577 3926 y Fp(n)p Fm(!1)1778 3855 y Fu(~)1764 3876 y Fq(E)1808 3888 y Fp(!)1856 3784 y Fo(\020)1906 3876 y Fu(\(1)18 b Fq(\000)g Fr(g)2121 3888 y Fp(n)2166 3876 y Fu(\))h Fq(\001)f Fu(\()p Fr(u)2338 3842 y Fs(\(1\))2445 3876 y Fq(\000)g Fr(u)2576 3842 y Fs(\(0\))2665 3876 y Fu(\))2697 3784 y Fo(\021)2761 3876 y Fr(:)456 4063 y Fu(Using)27 b(that)g Fr(u)916 4032 y Fs(\(1\))1032 4063 y Fu(is)g(a)g(lo)r(cal)f (minimizer)i(and)f(that)g(\(1)18 b Fq(\000)f Fr(g)2321 4075 y Fp(n)2366 4063 y Fu(\))h Fq(\001)f Fu(\()p Fr(u)2536 4032 y Fs(\(1\))2643 4063 y Fq(\000)g Fr(u)2773 4032 y Fs(\(0\))2862 4063 y Fu(\))27 b(is)g(a)g(compactly)456 4165 y(supp)r(orted)g(p)r(erturbation)g(for)g Fr(u)1508 4135 y Fs(\(1\))1597 4165 y Fu(,)h(w)n(e)f(thence)h(obtain)f(that)456 4331 y(\(5.12\))1119 4310 y(~)1105 4331 y Fq(E)1149 4343 y Fp(!)1197 4239 y Fo(\020)1247 4331 y Fu(\(1)18 b Fq(\000)g Fr(g)1462 4343 y Fp(n)1507 4331 y Fu(\))h Fq(\001)f Fu(\()p Fr(u)1679 4297 y Fs(\(1\))1787 4331 y Fq(\000)g Fr(u)1918 4297 y Fs(\(0\))2006 4331 y Fu(\))2038 4239 y Fo(\021)2125 4331 y Fq(\025)2241 4310 y Fu(~)2226 4331 y Fq(E)2270 4343 y Fp(!)2318 4331 y Fu(\()p Fr(u)2398 4297 y Fs(\(1\))2506 4331 y Fq(\000)g Fr(u)2637 4297 y Fs(\(0\))2725 4331 y Fu(\))c Fr(:)456 4502 y Fu(By)27 b(collecting)g(\(5.11\))g(and)g (\(5.12\),)g(w)n(e)g(gather)g(that)456 4655 y(\(5.13\))1604 4634 y(~)1590 4655 y Fq(E)1634 4667 y Fp(!)1682 4655 y Fu(\()p Fr(u)1762 4620 y Fs(\(1\))1869 4655 y Fq(\000)18 b Fr(u)2000 4620 y Fs(\(0\))2089 4655 y Fu(\))23 b Fq(\024)g Fu(0)14 b Fr(:)456 4800 y Fu(The)27 b(pro)r(of)g(of)h(the)g(desired)f (result)g(then)h(follo)n(ws)f(from)g(\(5.10\))g(and)g(\(5.13\).)491 b Fj(\003)555 4965 y Fu(A)28 b(similar)f(pro)r(of)g(will)h(giv)n(e)e (the)i(follo)n(wing)456 5089 y FA(Prop)s(osition)k(5.4.)42 b Fn(L)l(et)31 b Fr(v)f Fq(2)d Fr(W)1553 5059 y Fs(1)p Fp(;)p Fs(2)1643 5089 y Fu(\()p FA(K)1750 5059 y Fp(d)1750 5110 y(!)1798 5089 y Fu(\))20 b Fq(\\)g Fr(L)1982 5059 y Fs(1)2019 5089 y Fu(\()p FA(K)2126 5059 y Fp(d)2126 5110 y(!)2174 5089 y Fu(\))p Fn(,)33 b Fq(j)p Fr(v)s Fq(j)27 b(\024)f Fu(1)p Fn(.)44 b(Then)2814 5068 y Fu(~)2800 5089 y Fq(E)2844 5101 y Fp(!)2892 5089 y Fu(\()p Fr(v)s Fu(\))28 b Fq(\025)e Fu(0)p Fn(.)44 b(Mor)l(e-)456 5195 y(over,)679 5174 y Fu(~)665 5195 y Fq(E)709 5207 y Fp(!)756 5195 y Fu(\()p Fr(v)s Fu(\))24 b(=)f(0)29 b Fn(if)i(and)f(only)g(if)h Fr(u)1597 5165 y Fs(\(0\))1704 5195 y Fu(+)18 b Fr(v)33 b Fn(is)d(a)g(lo)l(c)l(al)h(minimizer.)p eop %%Page: 20 20 20 19 bop 456 251 a Fs(20)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fn(Pr)l(o)l(of.)43 b Fu(With)28 b(the)g(same)f Fr(g)1321 462 y Fp(n)1394 450 y Fu(as)g(in)g(the)h(previous)f(pro)r(of,)g(w)n(e)g(ha)n(v)n(e)1543 569 y(~)1529 590 y Fq(E)1573 602 y Fp(!)1621 590 y Fu(\()p Fr(v)s Fu(\))d(=)51 b(lim)1839 640 y Fp(n)p Fm(!1)2041 569 y Fu(~)2027 590 y Fq(E)2071 602 y Fp(!)2118 590 y Fu(\()p Fr(g)2190 602 y Fp(n)2254 590 y Fq(\001)19 b Fr(v)s Fu(\))456 772 y(Using)28 b(that)g Fr(u)918 742 y Fs(\(0\))1035 772 y Fu(is)h(a)e(lo)r(cal)h(minimizer)h(w)n(e)f(ha)n (v)n(e)f(that)2284 751 y(~)2270 772 y Fq(E)2314 784 y Fp(!)2362 772 y Fu(\()p Fr(g)2434 784 y Fp(n)2498 772 y Fq(\001)19 b Fr(v)s Fu(\))25 b Fq(\025)f Fu(0,)k(whic)n(h)g (establishes)456 871 y(the)g(\014rst)f(part)g(of)h(the)g(claim.)555 972 y(T)-7 b(o)28 b(sho)n(w)g(that)h(if)g Fr(u)1191 942 y Fs(\(0\))1298 972 y Fu(+)19 b Fr(v)32 b Fu(is)c(a)g(lo)r(cal)g (minimizer)2208 951 y(~)2194 972 y Fq(E)2238 984 y Fp(!)2286 972 y Fu(\()p Fr(v)s Fu(\))d(=)f(0,)k(w)n(e)h(pro)r(ceed)e(in)i(a)f (similar)456 1072 y(w)n(a)n(y)-7 b(.)35 b(W)-7 b(e)28 b(ha)n(v)n(e:)1314 1196 y(0)23 b(=)1481 1175 y(~)1466 1196 y Fq(E)1510 1208 y Fp(!)1558 1196 y Fu(\(0\))g(=)52 b(lim)1775 1246 y Fp(n)p Fm(!1)1976 1175 y Fu(~)1962 1196 y Fq(E)2006 1208 y Fp(!)2054 1104 y Fo(\020)2104 1196 y Fu(\(1)18 b Fq(\000)g Fr(g)2319 1208 y Fp(n)2364 1196 y Fu(\))h Fq(\001)f Fr(v)2499 1104 y Fo(\021)2563 1196 y Fr(:)456 1357 y Fu(But,)28 b(if)g Fr(u)768 1327 y Fs(\(0\))875 1357 y Fu(+)18 b Fr(v)31 b Fu(is)c(a)g(lo)r(cal)g (minimizer,)h(then)1533 1476 y(~)1519 1497 y Fq(E)1563 1509 y Fp(!)1611 1497 y Fu(\(\(1)18 b Fq(\000)g Fr(g)1858 1509 y Fp(n)1903 1497 y Fu(\))h Fq(\001)f Fr(v)s Fu(\))24 b Fq(\025)2196 1476 y Fu(~)2181 1497 y Fq(E)2225 1509 y Fp(!)2273 1497 y Fu(\()p Fr(v)s Fu(\))456 1642 y(and)j(so,)g(b)n(y)g (the)h(\014rst)g(part)f(of)g(the)h(claim,)1848 1621 y(~)1834 1642 y Fq(E)1878 1654 y Fp(!)1926 1642 y Fu(\()p Fr(v)s Fu(\))c(=)e(0,)27 b(as)g(desired.)555 1747 y(T)-7 b(o)27 b(sho)n(w)f(that)h(if)1151 1726 y(~)1137 1747 y Fq(E)1181 1759 y Fp(!)1229 1747 y Fu(\()p Fr(v)s Fu(\))d(=)e(0,)27 b(then)h Fr(u)1776 1717 y Fs(\(0\))1882 1747 y Fu(+)17 b Fr(v)30 b Fu(is)d(a)g(lo)r(cal)f(minimizer,)h(w)n(e)g(note)g(that)g (for)g(a)456 1847 y(function)h Fr(\036)g Fu(with)g(supp)r(ort)f(inside) h(a)f(ball)h Fr(B)t Fu(,)g(w)n(e)f(ha)n(v)n(e:)977 1988 y Fq(E)1021 2000 y Fp(B)1078 1988 y Fu(\()p Fr(u)1158 1954 y Fs(\(0\))1266 1988 y Fu(+)18 b Fr(v)j Fu(+)d Fr(\036)p Fu(\))i Fq(\000)e(E)1721 2000 y Fp(B)1777 1988 y Fu(\()p Fr(u)1857 1954 y Fs(\(0\))1965 1988 y Fu(+)g Fr(v)s Fu(\))23 b(=)2248 1967 y(~)2234 1988 y Fq(E)2278 2000 y Fp(!)2326 1988 y Fu(\()p Fr(v)f Fu(+)c Fr(\036)p Fu(\))h Fq(\000)2700 1967 y Fu(~)2686 1988 y Fq(E)2730 2000 y Fp(!)2778 1988 y Fu(\()p Fr(v)s Fu(\))14 b Fr(:)456 2132 y Fu(Com)n(bining)26 b(this)h(with)h(the)f(assumption)f(that)2001 2111 y(~)1987 2132 y Fq(E)2031 2144 y Fp(!)2078 2132 y Fu(\()p Fr(v)s Fu(\))e(=)f(0)j(and)h(using)g(the)g(\014rst)g(part)f(of)h(the)456 2232 y(claim,)g(w)n(e)g(deduce)h(that)1052 2373 y Fq(E)1096 2385 y Fp(B)1153 2373 y Fu(\()p Fr(u)1233 2339 y Fs(\(0\))1340 2373 y Fu(+)18 b Fr(v)k Fu(+)c Fr(\036)p Fu(\))h Fq(\000)f(E)1795 2385 y Fp(B)1852 2373 y Fu(\()p Fr(u)1932 2339 y Fs(\(0\))2039 2373 y Fu(+)g Fr(v)s Fu(\))24 b(=)2323 2352 y(~)2309 2373 y Fq(E)2353 2385 y Fp(!)2400 2373 y Fu(\()p Fr(v)e Fu(+)c Fr(\036)p Fu(\))24 b Fq(\025)f Fu(0)14 b Fr(;)456 2512 y Fu(as)27 b(desired.)2544 b Fj(\003)555 2667 y Fu(Ev)n(en)30 b(if)i(w)n(e)e(will)i(not)e(use)h(it,)i(w)n(e)d(note)h (that)g(the)h(ab)r(o)n(v)n(e)d(calculations)h(sho)n(w)g(that)h(if)h(w)n (e)456 2767 y(de\014ne)24 b(the)h(renormalized)e(energy)g(b)n(y)h (subtracting)f(the)i(energy)e(densit)n(y)h(of)h(an)n(y)e(other)h(lo)r (cal)456 2866 y(minimizer)j(remaining)g(at)g(a)h(\014nite)g(distance)f (from)g Fr(u)2181 2836 y Fs(\(0\))2298 2866 y Fu(w)n(e)g(obtain)g(the)h (same)f(functional.)456 3024 y(5.4.)40 b FA(The)33 b(heat)g(equation.) 41 b Fu(The)28 b(heat)g(\015o)n(w)g(is)g(the)g(gradien)n(t)f(\015o)n(w) h(of)g(our)f(energy)g(and)h(it)456 3123 y(approac)n(hes)22 b(critical)j(p)r(oin)n(ts,)g(as)g(w)n(e)f(will)i(discuss)e(in)h (detail.)37 b(A)25 b(detailed)g(study)g(of)g(the)h(heat)456 3223 y(\015o)n(w)32 b(prop)r(erties)g(will)h(b)r(e)h(the)f(to)r(ol)g (to)g(o)n(v)n(ercome)e(the)i(di\016culties)h(caused)e(b)n(y)h(the)g (lac)n(k)g(of)456 3323 y(compactness)e(of)h(the)h(standard)e (Ljusternik-Sc)n(hnirelmann)h(minimax)g(metho)r(ds)h(\(see,)g(for)456 3422 y(instance,)27 b([LS34)o(])h(or)f([Szu89)o(]\).)555 3522 y(W)-7 b(e)28 b(de\014ne)g(the)g(second-order)d(uniformly)j (elliptic)g(op)r(erator)e Fr(L)h Fu(b)n(y)456 3660 y(\(5.14\))968 b Fr(Lu)22 b Fu(:=)h Fr(@)1919 3672 y Fp(i)1946 3660 y Fu(\()p Fr(a)2022 3672 y Fp(ij)2081 3660 y Fr(@)2125 3672 y Fp(j)2160 3660 y Fr(u)p Fu(\))p Fr(:)456 3799 y Fu(and)k(w)n(e)g(consider)g(the)h(semilinear)f(equation)886 3940 y Fr(@)930 3952 y Fp(t)959 3940 y Fr(V)42 b Fu(=)23 b Fr(LV)37 b Fq(\000)18 b Fr(f)9 b Fu(\()p Fr(x;)14 b(u)1576 3906 y Fs(\(0\))1683 3940 y Fu(+)k Fr(V)h Fu(\))g(+)f Fr(f)9 b Fu(\()p Fr(x;)14 b(u)2181 3906 y Fs(\(0\))2270 3940 y Fu(\))83 b Fq(8)p Fu(\()p Fr(x;)14 b(t)p Fu(\))23 b Fq(2)g FA(K)2786 3906 y Fp(d)2786 3961 y(!)2853 3940 y Fq(\002)18 b Fu(\(0)p Fr(;)c Fu(+)p Fq(1)p Fu(\))886 4082 y Fr(V)19 b Fu(\()p Fr(x;)14 b Fu(0\))23 b(=)g Fr(v)s Fu(\()p Fr(x)p Fu(\))14 b Fr(;)98 b Fq(8)p Fr(x)23 b Fq(2)g FA(K)1813 4048 y Fp(d)1813 4102 y(!)1875 4082 y Fr(:)456 4010 y Fu(\(5.15\))555 4225 y(W)-7 b(e)33 b(note)g(that,)h(formally)-7 b(,)34 b(\(5.15\))e(is)h(the)g(gradien)n (t)e(\015o)n(w)h(of)h(the)g(renormalized)e(energy)-7 b(.)456 4331 y(That)24 b(is,)i(\(5.15\))e(can)g(b)r(e)h(written)g Fr(V)1597 4343 y Fp(t)1650 4331 y Fu(=)d Fq(\000r)1885 4310 y Fu(~)1871 4331 y Fq(E)1915 4343 y Fp(!)1963 4331 y Fu(\()p Fr(V)d Fu(\))25 b(where)g Fq(r)g Fu(denotes)f(the)h (Euler-Lagrange)456 4431 y(deriv)-5 b(ativ)n(e.)36 b(Hence,)27 b(one)h(has)f(formally)f(that)1275 4552 y Fr(d)p 1260 4589 74 4 v 1260 4665 a(dt)1358 4587 y Fu(~)1344 4608 y Fq(E)1388 4620 y Fp(!)1435 4608 y Fu(\(\010)1527 4574 y Fp(t)1557 4608 y Fu(\()p Fr(v)s Fu(\)\))e(=)f Fq(\000kr)1997 4587 y Fu(~)1984 4608 y Fq(E)2028 4620 y Fp(!)2075 4608 y Fu(\(\010)2167 4574 y Fp(t)2196 4608 y Fu(\()p Fr(v)s Fu(\)\))p Fq(k)2377 4574 y Fs(2)2377 4632 y Fp(L)2423 4616 y Fi(2)2456 4632 y Fs(\()p Fl(K)2540 4616 y Fk(d)2540 4649 y(!)2583 4632 y Fs(\))2626 4608 y Fr(:)456 4770 y Fu(W)-7 b(e)28 b(will)f(giv)n(e)g(precise)g(meaning)g(to)g(the)h(ab)r (o)n(v)n(e)f(formal)f(calculations)h(in)h(Lemma)f(5.13.)555 4869 y(Standard)21 b(theory)f(of)h(semigroups)f(\(see)h(Section)g (A.2\))h(giv)n(es)e(that,)j(if)e Fr(V)42 b Fq(2)24 b Fr(L)2990 4839 y Fs(2)3027 4869 y Fu(\()p FA(K)3134 4839 y Fp(d)3134 4890 y(!)3182 4869 y Fu(\),)f(there)456 4969 y(is)k(a)g(unique)h(solution)f(for)g Fr(t)c Fq(\024)g Fr(T)12 b Fu(,)27 b(w)n(e)g(denote)g(suc)n(h)h(solution)f(b)n(y)g(\010) 2637 4939 y Fp(t)2666 4969 y Fu(\()p Fr(V)19 b Fu(\).)555 5074 y(Notice)28 b(that,)g(since)f Fr(u)1270 5044 y Fs(\(0\))1387 5074 y Fu(and)g Fr(u)1596 5044 y Fs(\(1\))1712 5074 y Fu(are)g(critical)g(for)2269 5053 y(~)2255 5074 y Fq(E)2299 5086 y Fp(!)2347 5074 y Fu(,)h(w)n(e)f(ha)n(v)n(e)f(that)456 5216 y(\(5.16\))349 b(\010)1078 5181 y Fp(t)1107 5216 y Fu(\(0\))23 b(=)g(0)165 b(and)h(\010)1891 5181 y Fp(t)1920 5216 y Fu(\()p Fr(u)2000 5181 y Fs(\(1\))2108 5216 y Fq(\000)18 b Fr(u)2239 5181 y Fs(\(0\))2327 5216 y Fu(\))24 b(=)e Fr(u)2518 5181 y Fs(\(1\))2626 5216 y Fq(\000)c Fr(u)2757 5181 y Fs(\(0\))2859 5216 y Fr(:)p eop %%Page: 21 21 21 20 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(21)555 450 y Fu(Our)27 b(next)i(goal)d(is)i(to)g(extend)g(the)g(\015o)n(w)f(\010) 1921 420 y Fp(t)1951 450 y Fu(,)h(in)g(principle)g(de\014ned)g(only)g (for)f(short)g(times)456 550 y Fr(t)c Fq(2)g Fu([0)p Fr(;)14 b(T)e Fu(],)26 b(to)i(all)f(times)g Fr(t)c Fq(2)h Fu([0)p Fr(;)14 b Fu(+)p Fq(1)p Fu(\).)36 b(This)27 b(will)h(b)r(e)g (accomplished)e(in)i(Corollary)d(5.7.)36 b(W)-7 b(e)456 649 y(\014rst)23 b(establish)h(t)n(w)n(o)f(consequences)g(of)h(the)g (comparison)f(principle)h(Prop)r(osition)e(A.6,)j(whic)n(h)456 749 y(are)h(of)i(in)n(terest)f(b)n(y)g(themselv)n(es:)456 874 y FA(Corollary)33 b(5.5.)40 b Fn(If)30 b(the)g(function)g Fr(u)1672 844 y Fs(\(0\))1779 874 y Fu(+)18 b Fr(v)33 b Fn(is)d(Birkho\013,)i(then)d Fr(u)2604 844 y Fs(\(0\))2711 874 y Fu(+)18 b(\010)2854 844 y Fp(t)2884 874 y Fu(\()p Fr(v)s Fu(\))30 b Fn(is)g(Birkho\013.)456 998 y FA(Corollary)j(5.6.)40 b Fu(\010)1132 968 y Fp(t)1161 998 y Fu(\()p Fq(C)5 b Fu(\))23 b Fq(\022)g(C)5 b Fn(,)29 b(for)i(any)f Fr(t)23 b Fq(2)h Fu([0)p Fr(;)14 b(T)e Fu(])p Fn(.)555 1123 y Fu(W)-7 b(e)27 b(just)g(note)f(that)h(b)r(oth)g(Corollary)d(5.5)h(and)i (5.6)e(follo)n(w)h(b)r(ecause)g(of)g(the)h(fact)f(that)h(the)456 1223 y(comparisons)e(that)j(are)f(true)g(for)g(the)h(initial)g(data)f (remain)g(true)h(for)f(subsequen)n(t)g(times.)456 1347 y FA(Corollary)33 b(5.7.)40 b Fn(F)-6 b(or)30 b(any)g Fr(v)c Fq(2)d(C)5 b Fn(,)30 b Fu(\010)1694 1317 y Fp(t)1723 1347 y Fu(\()p Fr(v)s Fu(\))h Fn(is)f(wel)t(l)h(de\014ne)l(d)f(for)g (any)h Fr(t)23 b Fq(\025)f Fu(0)p Fn(.)456 1518 y(Pr)l(o)l(of.)43 b Fu(Let)26 b Fr(w)g Fu(:=)c(\010)1118 1488 y Fp(T)1171 1518 y Fu(\()p Fr(v)s Fu(\).)37 b(Then,)26 b(b)n(y)g(Corollary)e(5.6,)h (w)n(e)h(ma)n(y)f(de\014ne)i Fr(z)f Fu(:=)d(\010)2985 1488 y Fp(t)3014 1518 y Fu(\()p Fr(w)r Fu(\),)28 b(for)d(an)n(y)456 1618 y Fr(t)h Fq(2)h Fu([0)p Fr(;)14 b(T)e Fu(].)41 b(By)30 b(the)g(uniqueness)f(prop)r(ert)n(y)-7 b(,)29 b Fr(z)h Fu(=)c(\010)2120 1588 y Fp(t)p Fs(+)p Fp(T)2248 1618 y Fu(\()p Fr(v)s Fu(\),)31 b(hence)f(w)n(e)f(ha)n(v)n(e)f(de\014ned)i (\010)3307 1588 y Fp(t)3337 1618 y Fu(\()p Fr(v)s Fu(\))456 1718 y(for)d(an)n(y)f Fr(v)h Fq(2)c Fu([0)p Fr(;)14 b Fu(2)p Fr(T)e Fu(].)35 b(Rep)r(eating)28 b(the)g(argumen)n(t,)e(w)n(e)i (de\014ne)f(\010)2525 1688 y Fp(t)2555 1718 y Fu(\()p Fr(v)s Fu(\))h(for)f(an)n(y)g Fr(t)c Fq(\025)g Fu(0.)200 b Fj(\003)555 1889 y Fu(Using)28 b(Corollary)d(5.6)i(and)g(5.4,)g(w)n (e)g(ha)n(v)n(e)f(that)456 2038 y(\(5.17\))848 b Fq(k)p Fu(\010)1619 2004 y Fp(t)1648 2038 y Fu(\()p Fr(v)s Fu(\))p Fq(k)1797 2055 y Fp(L)1843 2039 y Fi(2)1875 2055 y Fs(\()p Fl(K)1959 2039 y Fk(d)1959 2072 y(!)2002 2055 y Fs(\))2055 2038 y Fq(\024)36 b Fu(const)14 b Fr(;)456 2188 y Fu(for)26 b(an)n(y)h Fr(v)f Fq(2)d(C)32 b Fu(and)27 b(for)g(an)n(y)f Fr(t)d Fq(\025)g Fu(0.)36 b(W)-7 b(e)28 b(no)n(w)e(giv)n(e)h(some)f (further)h(b)r(ounds)h(on)e(the)i(Sob)r(olev)456 2288 y(norms)e(of)i(the)g(heat)f(\015o)n(w:)456 2412 y FA(Lemma)h(5.8.)39 b Fq(k)p Fu(\010)1084 2382 y Fp(t)1113 2412 y Fu(\()p Fr(v)s Fu(\))p Fq(k)1262 2429 y Fp(L)1308 2413 y Fi(2)1340 2429 y Fs(\()p Fl(K)1424 2413 y Fk(d)1424 2446 y(!)1467 2429 y Fs(\))1520 2412 y Fq(\024)22 b Fr(C)6 b Fq(k)p Fr(v)s Fq(k)1799 2429 y Fp(L)1845 2413 y Fi(2)1877 2429 y Fs(\()p Fl(K)1961 2413 y Fk(d)1961 2446 y(!)2004 2429 y Fs(\))2034 2412 y Fn(,)29 b(for)h(any)f Fr(v)d Fq(2)e(C)33 b Fn(and)c(any)g Fr(t)23 b Fq(2)h Fu([0)p Fr(;)14 b Fu(1])p Fn(,)28 b(for)i(a)456 2512 y(suitable)g(c)l(onstant)f Fr(C)g Fq(\025)23 b Fu(1)p Fn(.)456 2683 y(Pr)l(o)l(of.)43 b Fu(W)-7 b(e)25 b(recall)f(the)h(follo)n(wing)f(w)n(ell)h(kno)n(wn)f (triangle)g(inequalit)n(y)h(\(see,)g(e.g.,)h(page)e(650)f(in)456 2783 y([Ev)-5 b(a98)n(])28 b(and)f(references)g(therein\):)37 b(giv)n(en)26 b(a)i(function)g Fr(\036)p Fu(\()p Fr(x;)14 b(t)p Fu(\),)29 b(one)e(has)g(that)456 2990 y(\(5.18\))1021 2895 y Fo(\015)1021 2945 y(\015)1021 2994 y(\015)1081 2877 y(Z)1164 2898 y Fp(t)1127 3066 y Fs(0)1207 2990 y Fr(\036)p Fu(\()p Fq(\001)p Fr(;)14 b(s)p Fu(\))g Fr(ds)1515 2895 y Fo(\015)1515 2945 y(\015)1515 2994 y(\015)1562 3048 y Fp(W)1633 3032 y Fk(k)q(;)p Fi(2)1717 3048 y Fs(\()p Fl(K)1801 3032 y Fk(d)1801 3065 y(!)1844 3048 y Fs(\))1911 2990 y Fq(\024)2012 2877 y Fo(Z)2095 2898 y Fp(t)2058 3066 y Fs(0)2138 2990 y Fq(k)p Fr(\036)p Fu(\()p Fq(\001)p Fr(;)g(s)p Fu(\))p Fq(k)2434 3007 y Fp(W)2505 2990 y Fk(k)q(;)p Fi(2)2589 3007 y Fs(\()p Fl(K)2673 2990 y Fk(d)2673 3024 y(!)2716 3007 y Fs(\))2760 2990 y Fr(ds)g(;)456 3182 y Fu(for)27 b(an)n(y)f Fr(k)g Fq(2)e FA(N)p Fu(.)555 3281 y(Let)k Fr(g)s Fu(\()p Fr(t)p Fu(\))23 b(:=)g Fq(k)p Fu(\010)1077 3251 y Fp(t)1106 3281 y Fu(\()p Fr(v)s Fu(\))p Fq(k)1255 3298 y Fp(L)1301 3282 y Fi(2)1333 3298 y Fs(\()p Fl(K)1417 3282 y Fk(d)1417 3315 y(!)1460 3298 y Fs(\))1490 3281 y Fu(,)k(w)n(e)h(deduce)f(from)h(\(A.4\),)g(\(A.2\))g(and)f (\(5.18\))g(that)1253 3502 y Fr(g)s Fu(\()p Fr(t)p Fu(\))c Fq(\024)36 b Fu(const)1718 3409 y Fo(\020)1767 3502 y Fq(k)p Fr(v)s Fq(k)1894 3518 y Fp(L)1940 3502 y Fi(2)1971 3518 y Fs(\()p Fl(K)2055 3502 y Fk(d)2055 3535 y(!)2098 3518 y Fs(\))2147 3502 y Fu(+)2230 3389 y Fo(Z)2313 3409 y Fp(t)2276 3577 y Fs(0)2356 3502 y Fr(g)s Fu(\()p Fr(s)p Fu(\))14 b Fr(ds)2598 3409 y Fo(\021)456 3698 y Fu(and)30 b(so)f(the)i(claim)f(follo)n(ws)f(from)h(the)g(standard)f(Gron)n(w)n (all)g(inequalit)n(y)h(\(see,)g(e.g.,)g([T)-7 b(a)n(y96)n(])456 3797 y(on)27 b(page)g(26\).)2476 b Fj(\003)555 3969 y Fu(Our)27 b(next)h(result)f(is)h(a)f(Sob)r(olev)g(b)r(ound)h(uniform)f (in)h Fr(t)p Fu(:)456 4093 y FA(Lemma)g(5.9.)38 b Fn(Fix)29 b Fr(t)1158 4105 y Fs(0)1218 4093 y Fr(>)23 b Fu(0)28 b Fr(k)e Fq(2)d FA(N)28 b Fn(and)h Fr(v)e Fq(2)c(C)5 b Fn(.)38 b(Then,)30 b(ther)l(e)e(exists)g Fr(C)i Fu(=)22 b Fr(C)6 b Fu(\()p Fr(t)3018 4105 y Fs(0)3056 4093 y Fr(;)14 b(k)s Fu(\))29 b Fn(so)f(that)456 4193 y Fq(k)p Fu(\010)558 4163 y Fp(t)586 4193 y Fu(\()p Fr(v)s Fu(\))p Fq(k)735 4210 y Fp(W)806 4193 y Fk(k)q(;)p Fi(2)891 4210 y Fs(\()p Fl(K)975 4193 y Fk(d)975 4226 y(!)1017 4210 y Fs(\))1070 4193 y Fq(\024)23 b Fr(C)6 b Fn(,)30 b(for)h(any)f Fr(t)23 b Fq(\025)g Fr(t)1741 4205 y Fs(0)1778 4193 y Fn(.)456 4364 y(Pr)l(o)l(of.)43 b Fu(W)-7 b(e)31 b(will)g(pro)n(v)n(e)e (only)h(the)h(case)f Fr(k)g Fu(=)e(1,)j(the)g(others)f(follo)n(w)g(b)n (y)g(b)r(o)r(otstrapping)g(one)456 4464 y(further)25 b(deriv)-5 b(ativ)n(e)24 b(via)g(\(A.2\))q(.)36 b(Also,)25 b(w)n(e)f(will)i(assume)e Fr(t)f Fq(2)g FA(N)i Fu(\(the)h(general)d (case)i(follo)n(wing)456 4563 y(b)n(y)32 b(a)h(time)g(scaling\).)53 b(Observ)n(e)31 b(that,)k(in)e(the)g(ligh)n(t)g(of)40 b(\(A.2\),)34 b(applied)f(here)g(with)g Fr(r)i Fu(:=)d(0)456 4663 y(and)27 b Fr(s)c Fu(:=)g(1,)k(w)n(e)g(ha)n(v)n(e)g(that)768 4867 y Fq(k)p Fu(\010)870 4833 y Fs(1)907 4867 y Fu(\()p Fr(v)s Fu(\))p Fq(k)1056 4884 y Fp(W)1127 4868 y Fi(1)p Fk(;)p Fi(2)1207 4884 y Fs(\()p Fl(K)1291 4868 y Fk(d)1291 4901 y(!)1334 4884 y Fs(\))1387 4867 y Fq(\024)36 b Fu(const)1706 4750 y Fo(\022)1777 4801 y Fq(k)p Fr(v)s Fq(k)1904 4818 y Fp(L)1950 4802 y Fi(2)1981 4818 y Fs(\()p Fl(K)2065 4802 y Fk(d)2065 4835 y(!)2108 4818 y Fs(\))p 1777 4848 361 4 v 1889 4865 a Fq(p)p 1958 4865 68 4 v 61 x Fr(t)1988 4938 y Fs(0)2166 4867 y Fu(+)2249 4754 y Fo(Z)2332 4775 y Fs(1)2295 4943 y(0)2393 4801 y Fq(k)p Fu(\010)2495 4771 y Fp(s)2530 4801 y Fu(\()p Fr(v)s Fu(\))p Fq(k)2679 4818 y Fp(L)2725 4802 y Fi(2)2757 4818 y Fs(\()p Fl(K)2841 4802 y Fk(d)2841 4835 y(!)2884 4818 y Fs(\))p 2393 4848 521 4 v 2528 4865 a Fq(p)p 2597 4865 182 4 v 65 x Fu(1)18 b Fq(\000)g Fr(s)2938 4867 y(ds)3020 4750 y Fo(\023)3109 4867 y Fr(:)456 5062 y Fu(Lemma)27 b(5.8)g(th)n(us)g(yields)h(that)1365 5215 y Fq(k)p Fu(\010)1467 5181 y Fs(1)1504 5215 y Fu(\()p Fr(v)s Fu(\))p Fq(k)1653 5232 y Fp(W)1724 5216 y Fi(1)p Fk(;)p Fi(2)1804 5232 y Fs(\()p Fl(K)1888 5216 y Fk(d)1888 5249 y(!)1931 5232 y Fs(\))1984 5215 y Fq(\024)2091 5194 y Fu(^)2072 5215 y Fr(C)6 b Fq(k)p Fr(v)s Fq(k)2264 5232 y Fp(L)2310 5216 y Fi(2)2341 5232 y Fs(\()p Fl(K)2425 5216 y Fk(d)2425 5249 y(!)2468 5232 y Fs(\))2512 5215 y Fr(;)p eop %%Page: 22 22 22 21 bop 456 251 a Fs(22)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fu(for)27 b(a)g(suitable)980 429 y(^)961 450 y Fr(C)7 b Fu(,)27 b(whic)n(h)h(ma)n(y)f(dep)r(end)h(on)f Fr(t)1928 462 y Fs(0)1966 450 y Fu(.)37 b(By)27 b(a)g(time)h(shift,)g(w)n(e)g(th)n (us)f(ha)n(v)n(e)g(that)1244 617 y Fq(k)p Fu(\010)1346 583 y Fp(j)1380 617 y Fu(\()p Fr(v)s Fu(\))p Fq(k)1529 634 y Fp(W)1600 617 y Fi(1)p Fk(;)p Fi(2)1681 634 y Fs(\()p Fl(K)1765 617 y Fk(d)1765 651 y(!)1808 634 y Fs(\))1861 617 y Fq(\024)1967 596 y Fu(^)1948 617 y Fr(C)7 b Fq(k)p Fu(\010)2116 583 y Fp(j)s Fm(\000)p Fs(1)2235 617 y Fu(\()p Fr(v)s Fu(\))p Fq(k)2384 634 y Fp(L)2430 617 y Fi(2)2462 634 y Fs(\()p Fl(K)2546 617 y Fk(d)2546 651 y(!)2589 634 y Fs(\))2633 617 y Fr(;)456 785 y Fu(for)27 b(an)n(y)f Fr(j)i Fq(2)c FA(N)p Fu(,)j Fr(j)h Fq(\025)23 b Fu(1.)36 b(By)29 b(\(5.17\))o(,)f(w)n(e)f(th)n(us)h(get)f(that)1547 955 y Fq(k)p Fu(\010)1649 920 y Fp(j)1683 955 y Fu(\()p Fr(v)s Fu(\))p Fq(k)1832 971 y Fp(W)1903 955 y Fi(1)p Fk(;)p Fi(2)1984 971 y Fs(\()p Fl(K)2068 955 y Fk(d)2068 988 y(!)2110 971 y Fs(\))2163 955 y Fq(\024)2270 934 y Fu(~)2251 955 y Fr(C)20 b(;)456 1137 y Fu(for)27 b(a)g(suitable)980 1116 y(~)961 1137 y Fr(C)7 b Fu(,)27 b(whic)n(h)h(ma)n(y)f(dep)r(end)h (on)f Fr(t)1928 1149 y Fs(0)1966 1137 y Fu(.)1391 b Fj(\003)456 1328 y FA(Corollary)47 b(5.10.)h Fn(Fix)41 b Fr(t)1331 1340 y Fs(0)1413 1328 y Fr(>)j Fu(0)p Fn(.)73 b(Then,)46 b(for)c(any)g Fr(t)i Fq(\025)g Fr(t)2447 1340 y Fs(0)2485 1328 y Fn(,)g Fu(\010)2614 1298 y Fp(t)2644 1328 y Fu(\()p Fq(C)5 b Fu(\))41 b Fn(is)h(pr)l(e-c)l(omp)l(act)g(in)456 1427 y Fr(W)546 1397 y Fs(2)p Fp(;)p Fs(2)636 1427 y Fu(\()p FA(K)743 1397 y Fp(d)743 1448 y(!)791 1427 y Fu(\))p Fn(.)456 1619 y(Pr)l(o)l(of.)h Fu(If)20 b Fr(!)25 b Fq(2)f FA(Q)1019 1588 y Fp(d)1059 1619 y Fq(\000)r(f)p Fu(0)p Fq(g)p Fu(,)19 b(the)h(pro)r(of)f(is)g(an)g(easy)f(consequence)h (of)g(Corollary)e(5.6,)j(Lemma)f(5.9)456 1718 y(and)30 b(the)h(deca)n(y)f(estimates)g(\(4.2\){\(4.3\))o(.)46 b(In)31 b(the)g(general)e(case,)i(the)g(pro)r(of)f(b)r(ecomes)g(more) 456 1818 y(tec)n(hnical,)k(since)f(there)h(are)e(more)h(space)f (directions)h(to)g(b)r(ound,)j(and)d(it)h(is)f(based)g(on)g(the)456 1917 y(follo)n(wing)26 b(ideas:)661 2049 y Fq(\017)41 b Fu(Lemma)28 b(5.9)e(pro)n(vides)h(lo)r(cal)g(con)n(v)n(ergence;)661 2149 y Fq(\017)41 b Fu(Lemma)28 b(5.1)e(pro)n(vides)h Fr(L)1560 2119 y Fs(1)1596 2149 y Fu(\()p FA(K)1703 2119 y Fp(d)1703 2169 y(!)1752 2149 y Fu(\)-con)n(v)n(ergence)e(and)i(so)g Fr(L)2591 2119 y Fs(2)2628 2149 y Fu(\()p FA(K)2735 2119 y Fp(d)2735 2169 y(!)2783 2149 y Fu(\)-con)n(v)n(ergence;)661 2248 y Fq(\017)41 b Fu(the)28 b(parab)r(olic)e(regularit)n(y)f(theory)i (in)g(\(A.2\))h(then)g(giv)n(es)e Fr(W)2685 2218 y Fs(2)p Fp(;)p Fs(2)2775 2248 y Fu(\()p FA(K)2882 2218 y Fp(d)2882 2269 y(!)2930 2248 y Fu(\)-con)n(v)n(ergence.)456 2380 y(Let)h(us)h(no)n(w)f(discuss)g(the)h(details.)37 b(W)-7 b(e)28 b(\014x)f Fr(t)1891 2392 y Fs(0)1952 2380 y Fr(>)22 b Fu(0)27 b(and)h(tak)n(e)456 2544 y(\(5.19\))1052 b Fr(v)1761 2556 y Fp(n)1829 2544 y Fq(2)24 b Fu(\010)1968 2509 y Fp(t)1993 2517 y Fi(0)2029 2544 y Fu(\()p Fq(C)5 b Fu(\))14 b Fr(;)456 2707 y Fu(for)29 b(an)n(y)h Fr(n)d Fq(2)h FA(N)p Fu(.)45 b(W)-7 b(e)30 b(w)n(ould)g(lik)n(e)g(to)g(pro)n (v)n(e)f(that,)i(up)f(to)h(subsequences,)f Fr(v)2925 2719 y Fp(n)3001 2707 y Fu(con)n(v)n(erges)d(in)456 2807 y(the)i Fr(W)690 2777 y Fs(2)p Fp(;)p Fs(2)780 2807 y Fu(\()p FA(K)887 2777 y Fp(d)887 2827 y(!)936 2807 y Fu(\)-top)r(ology)-7 b(.)41 b(Note)29 b(that,)h(thanks)f(to)h (Corollary)d(5.6,)i(b)n(y)g(p)r(ossibly)g(replacing)456 2906 y Fr(t)486 2918 y Fs(0)552 2906 y Fu(with)h(min)p Fq(f)p Fr(t)953 2918 y Fs(0)990 2906 y Fr(;)14 b Fu(1)p Fq(g)p Fu(,)29 b(w)n(e)f(ma)n(y)h(and)g(do)g(assume)f(that)i Fr(t)2248 2918 y Fs(0)2311 2906 y Fq(2)c Fu(\(0)p Fr(;)14 b Fu(1].)41 b(By)29 b(\(5.19\),)g(there)g(exists)459 3006 y(~)-45 b Fr(v)496 3018 y Fp(n)564 3006 y Fq(2)24 b(C)32 b Fu(so)27 b(that)456 3170 y(\(5.20\))1029 b Fr(v)1738 3182 y Fp(n)1806 3170 y Fu(=)23 b(\010)1954 3135 y Fp(t)1979 3143 y Fi(0)2015 3170 y Fu(\()s(~)-45 b Fr(v)2087 3182 y Fp(n)2133 3170 y Fu(\))14 b Fr(:)456 3333 y Fu(Let)27 b(also)456 3497 y(\(5.21\))987 b(\026)-45 b Fr(v)1693 3509 y Fp(n)1761 3497 y Fu(:=)23 b(\010)1932 3463 y Fp(t)1957 3471 y Fi(0)1989 3463 y Fp(=)p Fs(2)2060 3497 y Fu(\()s(~)-45 b Fr(v)2132 3509 y Fp(n)2178 3497 y Fu(\))14 b Fr(:)456 3660 y Fu(Making)26 b(use)h(of)h(Lemma)f(5.9,)f(one)h(gets)g(that)h (there)f(exists)g(a)g(sequence)g Fr(n)2852 3672 y Fp(j)2910 3660 y Fq(\000)-14 b(!)23 b Fu(+)p Fq(1)k Fu(and)g(a)456 3760 y(suitable)g(function)k(\026)-45 b Fr(v)31 b Fu(so)c(that)456 3924 y(\(5.22\))800 b(\026)-45 b Fr(v)26 b Fu(=)d Fr(L)1677 3889 y Fm(1)1677 3944 y Fs(lo)r(c)1764 3924 y Fu(\()p FA(K)1871 3889 y Fp(d)1871 3944 y(!)1919 3924 y Fu(\))c Fq(\000)67 b Fu(lim)2053 3976 y Fp(j)s Fm(!)p Fs(+)p Fm(1)2284 3924 y Fu(\026)-45 b Fr(v)2321 3936 y Fp(n)2362 3944 y Fk(j)2411 3924 y Fr(:)555 4251 y Fu(W)-7 b(e)28 b(no)n(w)f(sho)n(w)g(that)456 4415 y(\(5.23\))825 b(\026)-45 b Fr(v)26 b Fu(=)d Fr(L)1702 4380 y Fs(1)1739 4415 y Fu(\()p FA(K)1846 4380 y Fp(d)1846 4435 y(!)1894 4415 y Fu(\))c Fq(\000)67 b Fu(lim)2028 4467 y Fp(j)s Fm(!)p Fs(+)p Fm(1)2259 4415 y Fu(\026)-45 b Fr(v)2296 4427 y Fp(n)2337 4435 y Fk(j)2386 4415 y Fr(:)456 4616 y Fu(F)-7 b(or)27 b(this,)h(\014x)f Fr(")c(>)g Fu(0.)36 b(By)27 b(Lemma)h(5.1,)e(w)n(e)i(ha)n(v)n(e)e(that)i(there)f(exists)h Fr(R)2740 4628 y Fp(")2798 4616 y Fr(>)23 b Fu(0)k(so)g(that)456 4818 y(\(5.24\))1462 4705 y Fo(Z)1508 4893 y Fl(K)1566 4877 y Fk(d)1566 4910 y(!)1609 4893 y Fm(\000)p Fp(B)1711 4901 y Fk(R)1762 4818 y Fu(\()p Fr(u)1842 4784 y Fs(\(1\))1949 4818 y Fq(\000)18 b Fr(u)2080 4784 y Fs(\(0\))2169 4818 y Fu(\))37 b Fq(\024)2351 4762 y Fr(")p 2350 4799 42 4 v 2350 4875 a Fu(2)2415 4818 y Fr(;)456 5045 y Fu(so)27 b(long)f(as)h Fr(R)d Fq(\025)f Fr(R)1077 5057 y Fp(")1112 5045 y Fu(.)37 b(Moreo)n(v)n(er,)25 b(since)31 b(\026)-45 b Fr(v)1804 5057 y Fp(n)1872 5045 y Fq(2)23 b(C)5 b Fu(,)28 b(w)n(e)f(deduce)h(from)f(\(5.22\))g(that)1508 5216 y(0)36 b Fq(\024)k Fu(\026)-46 b Fr(v)1727 5228 y Fp(n)1768 5236 y Fk(j)1804 5216 y Fr(;)40 b Fu(\026)-45 b Fr(v)26 b Fq(\024)d Fr(u)2066 5181 y Fs(\(1\))2173 5216 y Fq(\000)18 b Fr(u)2304 5181 y Fs(\(0\))p eop %%Page: 23 23 23 22 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(23)456 450 y Fu(and)27 b(so,)g(from)g(\(5.24\),)g(that)1661 603 y Fo(Z)1707 791 y Fl(K)1765 775 y Fk(d)1765 808 y(!)1808 791 y Fm(\000)p Fp(B)1910 799 y Fk(R)1975 716 y Fq(j)s Fu(\026)-45 b Fr(v)2038 728 y Fp(n)2079 736 y Fk(j)2133 716 y Fq(\000)21 b Fu(\026)-45 b Fr(v)s Fq(j)1513 954 y(\024)1661 841 y Fo(Z)1707 1030 y Fl(K)1765 1013 y Fk(d)1765 1046 y(!)1808 1030 y Fm(\000)p Fp(B)1910 1038 y Fk(R)1961 954 y Fu(\()p Fq(j)s Fu(\026)g Fr(v)2056 966 y Fp(n)2097 974 y Fk(j)2133 954 y Fq(j)18 b Fu(+)g Fq(j)s Fu(\026)-45 b Fr(v)t Fq(j)p Fu(\))1513 1193 y Fq(\024)83 b Fu(2)1730 1080 y Fo(Z)1776 1268 y Fl(K)1834 1252 y Fk(d)1834 1285 y(!)1877 1268 y Fm(\000)p Fp(B)1979 1276 y Fk(R)2030 1193 y Fu(\()p Fr(u)2110 1158 y Fs(\(1\))2218 1193 y Fq(\000)18 b Fr(u)2349 1158 y Fs(\(0\))2437 1193 y Fu(\))1513 1371 y Fq(\024)83 b Fr(")14 b(;)456 1594 y Fu(pro)n(vided)26 b(that)i Fr(R)c Fq(\025)e Fr(R)1214 1606 y Fp(")1250 1594 y Fu(.)555 1694 y(The)28 b(ab)r(o)n(v)n(e)e(inequalit)n(y)-7 b(,)27 b(together)g(with)h(\(5.22\),)f(implies)h(that)1291 1960 y(lim)1242 2012 y Fp(j)s Fm(!)p Fs(+)p Fm(1)1470 1847 y Fo(Z)1516 2035 y Fl(K)1574 2019 y Fk(d)1574 2052 y(!)1635 1960 y Fq(j)s Fu(\026)-45 b Fr(v)1698 1972 y Fp(n)1739 1980 y Fk(j)1793 1960 y Fq(\000)21 b Fu(\026)-45 b Fr(v)s Fq(j)1094 2198 y Fu(=)132 b(lim)1242 2250 y Fp(j)s Fm(!)p Fs(+)p Fm(1)1470 2085 y Fo(Z)1516 2274 y Fp(B)1566 2282 y Fk(R)1631 2198 y Fq(j)s Fu(\026)-45 b Fr(v)1694 2210 y Fp(n)1735 2218 y Fk(j)1789 2198 y Fq(\000)21 b Fu(\026)-45 b Fr(v)s Fq(j)19 b Fu(+)67 b(lim)2040 2250 y Fp(j)s Fm(!)p Fs(+)p Fm(1)2268 2085 y Fo(Z)2314 2274 y Fl(K)2372 2257 y Fk(d)2372 2290 y(!)2415 2274 y Fm(\000)p Fp(B)2517 2282 y Fk(R)2582 2198 y Fq(j)s Fu(\026)-45 b Fr(v)2645 2210 y Fp(n)2686 2218 y Fk(j)2740 2198 y Fq(\000)21 b Fu(\026)-45 b Fr(v)s Fq(j)1094 2376 y(\024)83 b Fr(")14 b(:)456 2599 y Fu(Since)27 b Fr(")h Fu(ma)n(y)f(b)r(e)h(tak)n(en)f(as)g(small)g(as)g(w)n(e)g(wish,)h (\(5.23\))e(follo)n(ws.)555 2823 y(Since)i(functions)g(in)g Fq(C)k Fu(ha)n(v)n(e)26 b(sup-norm)h(b)r(ounded)h(b)n(y)f(2,)h(w)n(e)f (gather)f(from)i(\(5.23\))f(that)456 3050 y(\(5.25\))825 b(\026)-45 b Fr(v)26 b Fu(=)d Fr(L)1702 3016 y Fs(2)1739 3050 y Fu(\()p FA(K)1846 3016 y Fp(d)1846 3071 y(!)1894 3050 y Fu(\))c Fq(\000)67 b Fu(lim)2028 3102 y Fp(j)s Fm(!)p Fs(+)p Fm(1)2259 3050 y Fu(\026)-45 b Fr(v)2296 3062 y Fp(n)2337 3070 y Fk(j)2386 3050 y Fr(:)555 3497 y Fu(W)-7 b(e)22 b(no)n(w)f(mak)n(e)g(some)g(observ)-5 b(ation)21 b(on)g(the)h(parab)r(olic)f(regularit)n(y)e(theory)-7 b(.)35 b(Giv)n(en)21 b Fr(w)r(;)38 b(z)26 b Fq(2)456 3596 y(C)32 b Fu(and)27 b Fr(t)c Fq(2)h Fu([0)p Fr(;)14 b Fu(1],)26 b(b)n(y)i(\(5.18\),)f(\(A.4\))h(and)f(\(A.2\),)h(w)n(e)g (ha)n(v)n(e)e(that)879 3819 y Fq(k)p Fu(\010)981 3785 y Fp(t)1009 3819 y Fu(\()p Fr(w)r Fu(\))20 b Fq(\000)e Fu(\010)1297 3785 y Fp(t)1326 3819 y Fu(\()p Fr(z)t Fu(\))p Fq(k)1475 3836 y Fp(L)1521 3820 y Fi(2)1552 3836 y Fs(\()p Fl(K)1636 3820 y Fk(d)1636 3853 y(!)1679 3836 y Fs(\))731 3960 y Fq(\024)83 b(k)p Fr(e)960 3926 y Fp(tL)1033 3960 y Fu(\()p Fr(w)22 b Fq(\000)c Fr(z)t Fu(\))p Fq(k)1346 3977 y Fp(L)1392 3960 y Fi(2)1423 3977 y Fs(\()p Fl(K)1507 3960 y Fk(d)1507 3993 y(!)1550 3977 y Fs(\))879 4155 y Fu(+)958 4042 y Fo(Z)1040 4062 y Fp(t)1003 4231 y Fs(0)1083 4059 y Fo(\015)1083 4109 y(\015)1083 4159 y(\015)1129 4155 y Fr(e)1168 4121 y Fs(\()p Fp(t)p Fm(\000)p Fp(s)p Fs(\))p Fp(L)1378 4063 y Fo(h)1417 4155 y Fr(f)1467 4063 y Fo(\020)1516 4155 y Fq(\001)p Fr(;)c(u)1624 4121 y Fs(\(0\))1731 4155 y Fu(+)k(\010)1874 4121 y Fp(s)1910 4155 y Fu(\()p Fr(w)r Fu(\))2035 4063 y Fo(\021)2104 4155 y Fq(\000)g Fr(f)2237 4063 y Fo(\020)2286 4155 y Fq(\001)p Fr(;)c(u)2394 4121 y Fs(\(0\))2501 4155 y Fu(+)k(\010)2644 4121 y Fp(s)2680 4155 y Fu(\()p Fr(z)t Fu(\))2787 4063 y Fo(\021)o(i)2875 4059 y(\015)2875 4109 y(\015)2875 4159 y(\015)2921 4213 y Fp(L)2967 4196 y Fi(2)2999 4213 y Fs(\()p Fl(K)3083 4196 y Fk(d)3083 4230 y(!)3126 4213 y Fs(\))3170 4155 y Fr(ds)731 4392 y Fq(\024)83 b Fr(C)938 4404 y Fs(0)989 4275 y Fo(\022)1050 4392 y Fq(k)p Fr(w)21 b Fq(\000)d Fr(z)t Fq(k)1340 4409 y Fp(L)1386 4392 y Fi(2)1416 4409 y Fs(\()p Fl(K)1500 4392 y Fk(d)1500 4425 y(!)1543 4409 y Fs(\))1592 4392 y Fu(+)1675 4279 y Fo(Z)1758 4300 y Fp(t)1721 4468 y Fs(0)1801 4392 y Fq(k)p Fu(\010)1903 4358 y Fp(s)1938 4392 y Fu(\()p Fr(w)r Fu(\))h Fq(\000)f Fu(\010)2225 4358 y Fp(s)2261 4392 y Fu(\()p Fr(z)t Fu(\))p Fq(k)2410 4409 y Fp(L)2456 4392 y Fi(2)2487 4409 y Fs(\()p Fl(K)2571 4392 y Fk(d)2571 4425 y(!)2614 4409 y Fs(\))2658 4392 y Fr(ds)2740 4275 y Fo(\023)2829 4392 y Fr(;)456 4665 y Fu(for)33 b(a)g(suitable)g(constan)n(t)g Fr(C)1379 4677 y Fs(0)1449 4665 y Fr(>)g Fu(0.)54 b(This)33 b(and)g(the)h (standard)f(Gron)n(w)n(all)f(inequalit)n(y)h(\(see,)456 4765 y(e.g.,)27 b([T)-7 b(a)n(y96)n(])28 b(on)f(page)g(26\))g(giv)n(e)f (that)456 4988 y(\(5.26\))468 b Fq(k)p Fu(\010)1239 4954 y Fp(t)1268 4988 y Fu(\()p Fr(w)r Fu(\))20 b Fq(\000)e Fu(\010)1556 4954 y Fp(t)1585 4988 y Fu(\()p Fr(z)t Fu(\))p Fq(k)1734 5005 y Fp(L)1780 4988 y Fi(2)1811 5005 y Fs(\()p Fl(K)1895 4988 y Fk(d)1895 5021 y(!)1938 5005 y Fs(\))2005 4988 y Fq(\024)36 b Fr(C)2165 5000 y Fs(1)2203 4988 y Fq(k)p Fr(w)20 b Fq(\000)e Fr(z)t Fq(k)2492 5005 y Fp(L)2538 4988 y Fi(2)2569 5005 y Fs(\()p Fl(K)2653 4988 y Fk(d)2653 5021 y(!)2696 5005 y Fs(\))2740 4988 y Fr(;)456 5216 y Fu(for)27 b(an)n(y)f Fr(w)r(;)38 b(z)26 b Fq(2)e(C)32 b Fu(and)c(an)n(y)e Fr(t)d Fq(2)h Fu([0)p Fr(;)14 b Fu(1],)27 b(for)g(a)g(suitable)g(constan)n(t)g Fr(C)2647 5228 y Fs(1)2708 5216 y Fr(>)22 b Fu(0.)p eop %%Page: 24 24 24 23 bop 456 251 a Fs(24)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)555 450 y Fu(Also,)27 b(giv)n(en)f(an)n(y)42 b(^)-59 b Fr(w)s(;)42 b Fu(^)-47 b Fr(z)26 b Fq(2)e Fu(\010)1462 420 y Fp(t)1487 428 y Fi(0)1519 420 y Fp(=)p Fs(4)1591 450 y Fu(\()p Fq(C)5 b Fu(\),)27 b(exploiting)f(\(5.18\),)g(\(A.4\),)h(\(A.2\))g(and)g (Lemma)f(5.9,)456 550 y(one)h(gathers)f(that)842 718 y Fq(k)p Fu(\010)944 684 y Fp(t)973 718 y Fu(\()17 b(^)-59 b Fr(w)s Fu(\))18 b Fq(\000)g Fu(\010)1260 684 y Fp(t)1290 718 y Fu(\()5 b(^)-47 b Fr(z)s Fu(\))p Fq(k)1438 735 y Fp(W)1509 718 y Fi(2)p Fk(;)p Fi(2)1589 735 y Fs(\()p Fl(K)1673 718 y Fk(d)1673 752 y(!)1716 735 y Fs(\))694 859 y Fq(\024)83 b(k)p Fr(e)923 824 y Fp(tL)997 859 y Fu(\()17 b(^)-59 b Fr(w)21 b Fq(\000)i Fu(^)-47 b Fr(z)t Fu(\))p Fq(k)1309 876 y Fp(W)1380 859 y Fi(2)p Fk(;)p Fi(2)1459 876 y Fs(\()p Fl(K)1543 859 y Fk(d)1543 892 y(!)1586 876 y Fs(\))842 1054 y Fu(+)921 941 y Fo(Z)1003 961 y Fp(t)967 1129 y Fs(0)1046 958 y Fo(\015)1046 1008 y(\015)1046 1058 y(\015)1093 1054 y Fr(e)1132 1019 y Fs(\()p Fp(t)p Fm(\000)p Fp(s)p Fs(\))p Fp(L)1341 961 y Fo(h)1380 1054 y Fr(f)1430 961 y Fo(\020)1480 1054 y Fq(\001)p Fr(;)14 b(u)1588 1019 y Fs(\(0\))1695 1054 y Fu(+)k(\010)1838 1019 y Fp(s)1873 1054 y Fu(\()f(^)-59 b Fr(w)s Fu(\))1999 961 y Fo(\021)2067 1054 y Fq(\000)18 b Fr(f)2200 961 y Fo(\020)2250 1054 y Fq(\001)p Fr(;)c(u)2358 1019 y Fs(\(0\))2465 1054 y Fu(+)k(\010)2608 1019 y Fp(s)2643 1054 y Fu(\()5 b(^)-47 b Fr(z)t Fu(\))2750 961 y Fo(\021)o(i)2839 958 y(\015)2839 1008 y(\015)2839 1058 y(\015)2885 1112 y Fp(W)2956 1095 y Fi(2)p Fk(;)p Fi(2)3036 1112 y Fs(\()p Fl(K)3120 1095 y Fk(d)3120 1128 y(!)3163 1112 y Fs(\))3206 1054 y Fr(ds)694 1272 y Fq(\024)83 b Fr(C)901 1284 y Fs(2)939 1180 y Fo(\020)998 1216 y Fu(1)p 998 1253 42 4 v 1004 1329 a Fr(t)1063 1272 y Fq(k)17 b Fu(^)-59 b Fr(w)21 b Fq(\000)i Fu(^)-47 b Fr(z)s Fq(k)1352 1289 y Fp(L)1398 1272 y Fi(2)1430 1289 y Fs(\()p Fl(K)1514 1272 y Fk(d)1514 1306 y(!)1556 1289 y Fs(\))456 1272 y Fu(\(5.27\))842 1491 y(+)921 1378 y Fo(Z)1003 1398 y Fp(t)967 1566 y Fs(0)1046 1395 y Fo(\015)1046 1445 y(\015)1046 1495 y(\015)1093 1491 y Fr(f)1143 1398 y Fo(\020)1192 1491 y Fq(\001)p Fr(;)14 b(u)1300 1456 y Fs(\(0\))1407 1491 y Fu(+)k(\010)1550 1456 y Fp(s)1585 1491 y Fu(\()g(^)-60 b Fr(w)s Fu(\))1711 1398 y Fo(\021)1780 1491 y Fq(\000)18 b Fr(f)1913 1398 y Fo(\020)1962 1491 y Fq(\001)p Fr(;)c(u)2070 1456 y Fs(\(0\))2177 1491 y Fu(+)k(\010)2320 1456 y Fp(s)2355 1491 y Fu(\()5 b(^)-47 b Fr(z)t Fu(\))2462 1398 y Fo(\021)2512 1395 y(\015)2512 1445 y(\015)2512 1495 y(\015)2558 1549 y Fp(W)2629 1532 y Fi(2)p Fk(;)p Fi(2)2709 1549 y Fs(\()p Fl(K)2793 1532 y Fk(d)2793 1565 y(!)2836 1549 y Fs(\))2880 1491 y Fr(ds)2962 1398 y Fo(\021)694 1709 y Fq(\024)83 b Fr(C)901 1721 y Fs(3)939 1617 y Fo(\020)998 1653 y Fu(1)p 998 1690 V 1004 1766 a Fr(t)1063 1709 y Fq(k)17 b Fu(^)-59 b Fr(w)21 b Fq(\000)i Fu(^)-47 b Fr(z)s Fq(k)1352 1726 y Fp(L)1398 1709 y Fi(2)1430 1726 y Fs(\()p Fl(K)1514 1709 y Fk(d)1514 1742 y(!)1556 1726 y Fs(\))1605 1709 y Fu(+)18 b Fr(t)1718 1617 y Fo(\021)1781 1709 y Fr(;)456 1905 y Fu(for)23 b(an)n(y)g Fr(t)g Fq(2)h Fu(\(0)p Fr(;)14 b Fu(1],)24 b(where)f(the)i Fr(C)1522 1917 y Fp(i)1550 1905 y Fu('s)f(here)f(ab)r (o)n(v)n(e)g(are)g(p)r(ositiv)n(e)h(constan)n(ts)f(p)r(ossibly)g(dep)r (end-)456 2005 y(ing)k(on)g Fr(t)739 2017 y Fs(0)777 2005 y Fu(.)555 2173 y(W)-7 b(e)24 b(no)n(w)f(apply)h(these)f (estimates)h(in)g(order)e(to)h(pro)n(v)n(e)f(that)i Fr(v)2497 2185 y Fp(n)2538 2193 y Fk(i)2593 2173 y Fu(con)n(v)n(erges)d(in)j Fr(W)3144 2143 y Fs(2)p Fp(;)p Fs(2)3234 2173 y Fu(\()p FA(K)3341 2143 y Fp(d)3341 2194 y(!)3389 2173 y Fu(\).)555 2273 y(F)-7 b(or)27 b(this,)h(\014x)g Fr(")23 b(>)f Fu(0)27 b(and)h(let)456 2486 y(\(5.28\))858 b Fr(t)1557 2498 y Fp(")1616 2486 y Fu(:=)22 b(min)1879 2369 y Fo(\032)1951 2430 y Fr(t)1981 2442 y Fs(0)p 1951 2467 68 4 v 1964 2543 a Fu(4)2042 2486 y Fr(;)2161 2430 y(")p 2112 2467 139 4 v 2112 2543 a Fu(2)p Fr(C)2213 2555 y Fs(3)2260 2369 y Fo(\033)2350 2486 y Fr(:)456 2700 y Fu(Let)27 b(also)456 2884 y(\(5.29\))1033 b Fr(\034)1738 2896 y Fp(")1797 2884 y Fu(:=)1917 2828 y Fr(t)1947 2840 y Fs(0)p 1917 2865 68 4 v 1930 2941 a Fu(2)2013 2884 y Fq(\000)18 b Fr(t)2126 2896 y Fp(")2175 2884 y Fr(:)456 3081 y Fu(It)28 b(follo)n(ws)e(from)h(\(5.28\))g(that)1725 3264 y Fr(\034)1761 3276 y Fp(")1820 3264 y Fq(2)1899 3147 y Fo(\024)1952 3208 y Fr(t)1982 3220 y Fs(0)p 1952 3245 V 1965 3321 a Fu(4)2029 3264 y Fr(;)37 b Fu(1)2131 3147 y Fo(\025)456 3460 y Fu(and)27 b(so)1591 3594 y(\010)1651 3559 y Fp(\034)1682 3567 y Fk(")1718 3594 y Fu(\()s(\026)-45 b Fr(v)1790 3606 y Fp(n)1836 3594 y Fu(\))23 b Fq(2)h Fu(\010)2030 3559 y Fp(t)2055 3567 y Fi(0)2087 3559 y Fp(=)p Fs(4)2159 3594 y Fu(\()p Fq(C)5 b Fu(\))14 b Fr(;)456 3744 y Fu(for)27 b(an)n(y)f Fr(n)d Fq(2)h FA(N)p Fu(.)36 b(Consequen)n(tly)-7 b(,)27 b(making)g(use)h(of)f(\(5.27\),)g(w)n(e)g(get)h(that)1330 3913 y Fq(k)p Fu(\010)1432 3879 y Fp(t)1461 3913 y Fu(\(\010)1553 3879 y Fp(\034)1584 3887 y Fk(")1620 3913 y Fu(\()s(\026)-45 b Fr(v)1692 3925 y Fp(n)1733 3933 y Fk(i)1764 3913 y Fu(\)\))19 b Fq(\000)f Fu(\010)1990 3879 y Fp(t)2020 3913 y Fu(\(\010)2112 3879 y Fp(\034)2143 3887 y Fk(")2179 3913 y Fu(\()s(\026)-45 b Fr(v)2251 3925 y Fp(n)2292 3933 y Fk(j)2327 3913 y Fu(\)\))p Fq(k)2433 3930 y Fp(W)2504 3913 y Fi(2)p Fk(;)p Fi(2)2585 3930 y Fs(\()p Fl(K)2669 3913 y Fk(d)2669 3946 y(!)2711 3930 y Fs(\))456 3913 y Fu(\(5.30\))1183 4089 y Fq(\024)82 b Fr(C)1389 4101 y Fs(3)1427 3997 y Fo(\020)1487 4033 y Fu(1)p 1487 4070 42 4 v 1493 4146 a Fr(t)1552 4089 y Fq(k)p Fu(\010)1654 4055 y Fp(\034)1685 4063 y Fk(")1720 4089 y Fu(\()s(\026)-45 b Fr(v)1792 4101 y Fp(n)1833 4109 y Fk(i)1864 4089 y Fu(\))19 b Fq(\000)f Fu(\010)2058 4055 y Fp(\034)2089 4063 y Fk(")2125 4089 y Fu(\()s(\026)-45 b Fr(v)2197 4101 y Fp(n)2238 4109 y Fk(j)2274 4089 y Fu(\))p Fq(k)2348 4106 y Fp(L)2394 4089 y Fi(2)2425 4106 y Fs(\()p Fl(K)2509 4089 y Fk(d)2509 4123 y(!)2552 4106 y Fs(\))2601 4089 y Fu(+)18 b Fr(t)2714 3997 y Fo(\021)2777 4089 y Fr(;)456 4285 y Fu(for)27 b(an)n(y)f Fr(t)e Fq(2)f Fu(\(0)p Fr(;)14 b Fu(1].)555 4385 y(Also,)28 b(due)f(to)h(\(5.25\),)f(there)g(exists)g Fr(n)1778 4397 y Fp(")1837 4385 y Fq(2)c FA(N)28 b Fu(so)f(that)1444 4590 y Fq(k)s Fu(\026)-45 b Fr(v)1526 4602 y Fp(n)1567 4610 y Fk(i)1615 4590 y Fq(\000)22 b Fu(\026)-46 b Fr(v)1738 4602 y Fp(n)1779 4610 y Fk(j)1815 4590 y Fq(k)1857 4607 y Fp(L)1903 4590 y Fi(2)1934 4607 y Fs(\()p Fl(K)2018 4590 y Fk(d)2018 4624 y(!)2061 4607 y Fs(\))2114 4590 y Fq(\024)2277 4534 y Fr(t)2307 4546 y Fp(")2342 4534 y Fr(")p 2212 4571 235 4 v 2212 4647 a Fu(2)p Fr(C)2313 4659 y Fs(1)2350 4647 y Fr(C)2409 4659 y Fs(3)456 4799 y Fu(so)27 b(long)f(as)h Fr(n)889 4811 y Fp(i)917 4799 y Fr(;)37 b(n)1027 4811 y Fp(j)1085 4799 y Fq(\025)22 b Fr(n)1222 4811 y Fp(")1258 4799 y Fu(.)37 b(Therefore,)26 b(b)n(y)h(\(5.26\),)456 5007 y(\(5.31\))637 b Fq(k)p Fu(\010)1408 4973 y Fp(t)1437 5007 y Fu(\()s(\026)-45 b Fr(v)1509 5019 y Fp(n)1550 5027 y Fk(i)1581 5007 y Fu(\))18 b Fq(\000)g Fu(\010)1774 4973 y Fp(t)1804 5007 y Fu(\()s(\026)-45 b Fr(v)1876 5019 y Fp(n)1917 5027 y Fk(j)1952 5007 y Fu(\))p Fq(k)2026 5024 y Fp(L)2072 5007 y Fi(2)2104 5024 y Fs(\()p Fl(K)2188 5007 y Fk(d)2188 5040 y(!)2231 5024 y Fs(\))2298 5007 y Fq(\024)2426 4951 y Fr(t)2456 4963 y Fp(")2492 4951 y Fr(")p 2409 4988 139 4 v 2409 5064 a Fu(2)p Fr(C)2510 5076 y Fs(3)2571 5007 y Fr(;)456 5216 y Fu(if)28 b Fr(n)582 5228 y Fp(i)609 5216 y Fr(;)37 b(n)719 5228 y Fp(j)777 5216 y Fq(\025)23 b Fr(n)915 5228 y Fp(")978 5216 y Fu(and)k Fr(t)c Fq(2)h Fu([0)p Fr(;)14 b Fu(1].)p eop %%Page: 25 25 25 24 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(25)555 450 y Fu(Thence,)28 b(using)f(\(5.20\),)g(\(5.21\),)g(\(5.29\),)g(\(5.30\)) g(and)g(\(5.31\),)g(w)n(e)g(deduce)h(that)1319 602 y Fq(k)p Fr(v)1401 614 y Fp(n)1442 622 y Fk(i)1491 602 y Fq(\000)18 b Fr(v)1614 614 y Fp(n)1655 622 y Fk(j)1690 602 y Fq(k)1732 619 y Fp(W)1803 602 y Fi(2)p Fk(;)p Fi(2)1883 619 y Fs(\()p Fl(K)1967 602 y Fk(d)1967 636 y(!)2010 619 y Fs(\))1172 747 y Fu(=)82 b Fq(k)p Fu(\010)1421 712 y Fp(t)1446 720 y Fi(0)1478 712 y Fp(=)p Fs(2)1549 747 y Fu(\()s(\026)-45 b Fr(v)1621 759 y Fp(n)1662 767 y Fk(i)1693 747 y Fu(\))19 b Fq(\000)f Fu(\010)1887 712 y Fp(t)1912 720 y Fi(0)1945 712 y Fp(=)p Fs(2)2016 747 y Fu(\()s(\026)-45 b Fr(v)2088 759 y Fp(n)2129 767 y Fk(j)2165 747 y Fu(\))p Fq(k)2239 764 y Fp(W)2310 747 y Fi(2)p Fk(;)p Fi(2)2389 764 y Fs(\()p Fl(K)2473 747 y Fk(d)2473 780 y(!)2516 764 y Fs(\))1172 883 y Fu(=)82 b Fq(k)p Fu(\010)1421 849 y Fp(t)1446 857 y Fk(")1482 883 y Fu(\(\010)1574 849 y Fp(\034)1605 857 y Fk(")1641 883 y Fu(\()s(\026)-45 b Fr(v)1713 895 y Fp(n)1754 903 y Fk(i)1785 883 y Fu(\)\))19 b Fq(\000)f Fu(\010)2011 849 y Fp(t)2036 857 y Fk(")2073 883 y Fu(\(\010)2165 849 y Fp(\034)2196 857 y Fk(")2232 883 y Fu(\()s(\026)-45 b Fr(v)2304 895 y Fp(n)2345 903 y Fk(j)2381 883 y Fu(\)\))p Fq(k)2487 900 y Fp(W)2558 884 y Fi(2)p Fk(;)p Fi(2)2638 900 y Fs(\()p Fl(K)2722 884 y Fk(d)2722 917 y(!)2764 900 y Fs(\))1172 1060 y Fq(\024)82 b Fr(C)1378 1072 y Fs(3)1416 968 y Fo(\020)1487 1004 y Fu(1)p 1475 1041 66 4 v 1475 1117 a Fr(t)1505 1129 y Fp(")1565 1060 y Fq(k)p Fu(\010)1667 1025 y Fp(\034)1698 1033 y Fk(")1733 1060 y Fu(\()s(\026)-45 b Fr(v)1805 1072 y Fp(n)1846 1080 y Fk(i)1877 1060 y Fu(\))19 b Fq(\000)f Fu(\010)2071 1025 y Fp(\034)2102 1033 y Fk(")2138 1060 y Fu(\()s(\026)-45 b Fr(v)2210 1072 y Fp(n)2251 1080 y Fk(j)2286 1060 y Fu(\))p Fq(k)2360 1077 y Fp(L)2406 1060 y Fi(2)2438 1077 y Fs(\()p Fl(K)2522 1060 y Fk(d)2522 1093 y(!)2565 1077 y Fs(\))2613 1060 y Fu(+)18 b Fr(t)2726 1072 y Fp(")2762 968 y Fo(\021)1172 1258 y Fq(\024)82 b Fr(C)1378 1270 y Fs(3)1416 1166 y Fo(\020)1525 1202 y Fr(")p 1475 1239 139 4 v 1475 1315 a Fu(2)p Fr(C)1576 1327 y Fs(3)1642 1258 y Fu(+)18 b Fr(t)1755 1270 y Fp(")1790 1166 y Fo(\021)1854 1258 y Fr(;)456 1450 y Fu(if)28 b Fr(n)582 1462 y Fp(i)609 1450 y Fr(;)37 b(n)719 1462 y Fp(j)777 1450 y Fq(\025)23 b Fr(n)915 1462 y Fp(")950 1450 y Fu(.)37 b(Therefore,)26 b(from)i(\(5.28\),)1501 1602 y Fq(k)p Fr(v)1583 1614 y Fp(n)1624 1622 y Fk(i)1673 1602 y Fq(\000)18 b Fr(v)1796 1614 y Fp(n)1837 1622 y Fk(j)1872 1602 y Fq(k)1914 1619 y Fp(W)1985 1603 y Fi(2)p Fk(;)p Fi(2)2065 1619 y Fs(\()p Fl(K)2149 1603 y Fk(d)2149 1636 y(!)2192 1619 y Fs(\))2259 1602 y Fq(\024)36 b Fr(")456 1759 y Fu(pro)n(vided)25 b(that)i Fr(n)1025 1771 y Fp(i)1052 1759 y Fr(;)37 b(n)1162 1771 y Fp(j)1220 1759 y Fq(\025)23 b Fr(n)1358 1771 y Fp(")1393 1759 y Fu(.)37 b(That)26 b(is,)h Fr(v)1805 1771 y Fp(n)1846 1779 y Fk(i)1903 1759 y Fu(is)g(a)f(Cauc)n(h)n(y)f (sequence)i(\(and)f(th)n(us)h(con)n(v)n(erges\))456 1861 y(in)g Fr(W)642 1831 y Fs(2)p Fp(;)p Fs(2)732 1861 y Fu(\()p FA(K)839 1831 y Fp(d)839 1882 y(!)888 1861 y Fu(\).)2437 b Fj(\003)555 2036 y Fu(As)28 b(an)f(immediate)h(corollary) d(of)j(Lemma)f(5.9,)g(w)n(e)g(obtain:)456 2162 y FA(Corollary)33 b(5.11.)39 b Fn(F)-6 b(or)30 b(any)h Fr(t)23 b Fq(\025)f Fr(t)1604 2174 y Fs(0)1664 2162 y Fr(>)h Fu(0)29 b Fn(and)i(any)f Fr(v)c Fq(2)d(C)5 b Fn(,)30 b(we)g(have)h(that)456 2315 y Fu(\(5.32\))940 b(0)23 b Fq(\024)1775 2294 y Fu(~)1761 2315 y Fq(E)1805 2327 y Fp(!)1853 2315 y Fu(\(\010)1945 2281 y Fp(t)1975 2315 y Fu(\()p Fr(v)s Fu(\)\))h Fq(\024)f Fr(C)555 2467 y Fu(W)-7 b(e)25 b(no)n(w)f(sho)n(w)f(the)i(con)n(tin)n (uit)n(y)f(from)g(initial)h(data)f(of)g(the)h(heat)f(\015o)n(w,)g(whic) n(h)h(will)f(will)h(b)r(e)456 2567 y(needed)i(in)h(the)g(sequel.)456 2693 y FA(Prop)s(osition)39 b(5.12.)44 b Fn(Fixe)l(d)38 b Fr(T)47 b(>)36 b Fu(0)g Fn(and)h Fr(\017)f(>)g Fu(0)p Fn(,)j(ther)l(e)e(exists)f Fr(\016)2660 2705 y Fs(0)2697 2693 y Fu(\()p Fr(T)7 b(;)14 b(\017)p Fu(\))36 b Fr(>)g Fu(0)g Fn(so)i(that)e(if)456 2792 y Fr(v)s(;)28 b(w)d Fq(2)f(C)34 b Fn(and)c Fq(k)p Fr(v)21 b Fq(\000)d Fr(w)r Fq(k)1241 2809 y Fp(W)1312 2792 y Fi(2)p Fk(;)p Fi(2)1393 2809 y Fs(\()p Fl(K)1477 2792 y Fk(d)1477 2826 y(!)1519 2809 y Fs(\))1572 2792 y Fq(\024)23 b Fr(\016)1697 2804 y Fs(0)1734 2792 y Fu(\()p Fr(T)7 b(;)14 b(\017)p Fu(\))p Fn(,)30 b(then)1407 2956 y Fq(k)p Fu(\010)1509 2922 y Fp(t)1538 2956 y Fu(\()p Fr(v)s Fu(\))19 b Fq(\000)f Fu(\010)1807 2922 y Fp(t)1836 2956 y Fu(\()p Fr(w)r Fu(\))p Fq(k)2003 2973 y Fp(W)2074 2957 y Fi(2)p Fk(;)p Fi(2)2155 2973 y Fs(\()p Fl(K)2239 2957 y Fk(d)2239 2990 y(!)2282 2973 y Fs(\))2335 2956 y Fq(\024)k Fr(\017)14 b(;)456 3113 y Fn(for)30 b(any)g Fr(t)23 b Fq(2)h Fu([0)p Fr(;)14 b(T)e Fu(])p Fn(.)456 3288 y(Pr)l(o)l(of.)43 b Fu(First,)29 b(w)n(e)f(deal)h(with)g(the)h(case)e Fr(t)d Fq(2)g Fu([0)p Fr(;)14 b Fu(1].)40 b(Let)29 b Fr(U)9 b Fu(\()p Fr(x;)14 b(t)p Fu(\))26 b(:=)f(\010)2713 3257 y Fp(t)2742 3288 y Fu(\()p Fr(v)s Fu(\))20 b Fq(\000)f Fu(\010)3013 3257 y Fp(t)3042 3288 y Fu(\()p Fr(w)r Fu(\).)42 b(Then,)456 3387 y(b)n(y)28 b(\(A.4\),)876 3543 y Fr(U)9 b Fu(\()p Fr(x;)14 b(t)p Fu(\))83 b(=)g Fr(e)1390 3508 y Fp(tL)1464 3543 y Fu(\()p Fr(v)22 b Fq(\000)c Fr(w)r Fu(\))1351 3725 y Fq(\000)1430 3612 y Fo(Z)1512 3632 y Fp(t)1475 3800 y Fs(0)1555 3725 y Fr(e)1594 3691 y Fs(\()p Fp(t)p Fm(\000)p Fp(s)p Fs(\))p Fp(L)1804 3633 y Fo(h)1843 3725 y Fr(f)1906 3633 y Fo(\020)1956 3725 y Fr(x;)c(u)2088 3691 y Fs(\(0\))2177 3725 y Fu(\()p Fr(x)p Fu(\))20 b(+)e(\010)2451 3691 y Fp(t)2480 3725 y Fu(\()p Fr(w)r Fu(\))h(+)f Fr(U)9 b Fu(\()p Fr(x;)14 b(s)p Fu(\))2960 3633 y Fo(\021)456 3725 y Fu(\(5.33\))1351 3929 y Fq(\000)p Fr(f)1479 3837 y Fo(\020)1528 3929 y Fr(x;)g(u)1660 3895 y Fs(\(0\))1749 3929 y Fu(\()p Fr(x)p Fu(\))20 b(+)e(\010)2023 3895 y Fp(t)2052 3929 y Fu(\()p Fr(w)r Fu(\))2177 3837 y Fo(\021)d(i)2295 3929 y Fr(ds)f(:)456 4102 y Fu(De\014ne)28 b(also)1491 4219 y Fr(g)s Fu(\()p Fr(t)p Fu(\))23 b(:=)f Fq(k)p Fr(U)9 b Fu(\()p Fq(\001)p Fr(;)14 b(t)p Fu(\))p Fq(k)2065 4236 y Fp(W)2136 4219 y Fi(2)p Fk(;)p Fi(2)2216 4236 y Fs(\()p Fl(K)2300 4219 y Fk(d)2300 4252 y(!)2342 4236 y Fs(\))2386 4219 y Fr(:)456 4358 y Fu(Then,)27 b(b)n(y)i(\(A.2\),)f(\(5.18\))o(,)g (\(5.33\))f(and)g(Lemma)h(5.9,)e(w)n(e)i(ha)n(v)n(e)e(that)1095 4535 y Fr(g)s Fu(\()p Fr(t)p Fu(\))d Fq(\024)14 b Fu(const)1537 4443 y Fo(\020)1601 4535 y Fq(k)p Fr(v)21 b Fq(\000)d Fr(w)r Fq(k)1890 4552 y Fp(W)1961 4535 y Fi(2)p Fk(;)p Fi(2)2041 4552 y Fs(\()p Fl(K)2125 4535 y Fk(d)2125 4568 y(!)2168 4552 y Fs(\))1338 4750 y Fu(+)1422 4637 y Fo(Z)1505 4658 y Fp(t)1468 4826 y Fs(0)1548 4750 y Fq(k)p Fr(f)9 b Fu(\()p Fq(\001)p Fr(;)14 b(u)1780 4716 y Fs(\(0\))1886 4750 y Fu(+)k(\010)2029 4716 y Fp(t)2058 4750 y Fu(\()p Fr(w)r Fu(\))i(+)e Fr(U)9 b Fu(\()p Fq(\001)p Fr(;)14 b(s)p Fu(\)\))1338 4955 y Fq(\000)19 b Fr(f)9 b Fu(\()p Fq(\001)p Fr(;)14 b(u)1612 4920 y Fs(\(0\))1718 4955 y Fu(+)k(\010)1861 4920 y Fp(t)1891 4955 y Fu(\()p Fr(w)r Fu(\)\))p Fq(k)2090 4971 y Fp(W)2161 4955 y Fi(2)p Fk(;)p Fi(2)2242 4971 y Fs(\()p Fl(K)2326 4955 y Fk(d)2326 4988 y(!)2368 4971 y Fs(\))2412 4955 y Fr(ds)2508 4862 y Fo(\021)2581 4955 y Fq(\024)1255 5170 y(\024)c Fu(const)1551 5053 y Fo(\022)1612 5170 y Fq(k)p Fr(v)21 b Fq(\000)d Fr(w)r Fq(k)1901 5187 y Fp(W)1972 5170 y Fi(2)p Fk(;)p Fi(2)2053 5187 y Fs(\()p Fl(K)2137 5170 y Fk(d)2137 5203 y(!)2179 5187 y Fs(\))2228 5170 y Fu(+)2311 5057 y Fo(Z)2394 5077 y Fp(t)2357 5246 y Fs(0)2437 5170 y Fr(g)s Fu(\()p Fr(s)p Fu(\))c Fr(ds)2693 5053 y Fo(\023)2782 5170 y Fr(;)456 4865 y Fu(\(5.34\))p eop %%Page: 26 26 26 25 bop 456 251 a Fs(26)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fu(for)38 b(an)n(y)h Fr(t)j Fq(2)h Fu([0)p Fr(;)14 b Fu(1].)71 b(Therefore,)41 b(exploiting)e(the)h(standard)e(Gron)n(w)n(all)g (inequalit)n(y)h(\(see,)456 550 y(e.g.,)27 b([T)-7 b(a)n(y96)n(])28 b(on)f(page)g(26\),)456 690 y(\(5.35\))731 b Fr(g)s Fu(\()p Fr(t)p Fu(\))24 b Fq(\024)36 b Fu(const)13 b Fq(k)p Fr(v)22 b Fq(\000)c Fr(w)r Fq(k)2155 707 y Fp(W)2226 690 y Fi(2)p Fk(;)p Fi(2)2306 707 y Fs(\()p Fl(K)2390 690 y Fk(d)2390 723 y(!)2433 707 y Fs(\))2477 690 y Fr(;)456 835 y Fu(for)25 b(an)n(y)h Fr(t)d Fq(2)g Fu([0)p Fr(;)14 b Fu(1],)26 b(whic)n(h)h(implies)f(the)h(desired)f(claim,)g(for)g Fr(T)34 b Fq(\024)23 b Fu(1.)36 b(If,)27 b(on)f(the)h(other)e(hand,)456 934 y Fr(T)34 b(>)22 b Fu(1,)28 b(b)n(y)f(iterating)h(\(5.35\))o(,)g(w) n(e)f(deduce)h(that)884 1078 y Fq(k)p Fu(\010)986 1044 y Fp(t)1015 1078 y Fu(\()p Fr(v)s Fu(\))19 b Fq(\000)f Fu(\010)1284 1044 y Fp(t)1313 1078 y Fu(\()p Fr(w)r Fu(\))p Fq(k)1480 1095 y Fp(W)1551 1078 y Fi(2)p Fk(;)p Fi(2)1632 1095 y Fs(\()p Fl(K)1716 1078 y Fk(d)1716 1111 y(!)1759 1095 y Fs(\))1812 1078 y Fq(\024)36 b Fu(const)14 b Fr(e)2167 1044 y Fs(const)d Fp(T)2382 1078 y Fq(k)p Fr(v)21 b Fq(\000)d Fr(w)r Fq(k)2671 1095 y Fp(W)2742 1078 y Fi(2)p Fk(;)p Fi(2)2822 1095 y Fs(\()p Fl(K)2906 1078 y Fk(d)2906 1111 y(!)2949 1095 y Fs(\))2993 1078 y Fr(;)456 1223 y Fu(for)27 b(an)n(y)f Fr(t)e Fq(2)f Fu([0)p Fr(;)14 b(T)e Fu(],)26 b(whence)i(the)g(claim.)1623 b Fj(\003)456 1380 y Fu(5.5.)40 b FA(Con)m(v)m(ergence)26 b(of)f(the)h(heat)f(\015o)m(w.)41 b Fu(W)-7 b(e)23 b(no)n(w)e(in)n(v)n(estigate)g(some)g(con)n(v)n (ergence)e(prop-)456 1480 y(erties)33 b(for)g(heat)h(\015o)n(w)f(on)g (\(the)i Fr(W)1580 1450 y Fs(2)p Fp(;)p Fs(2)1670 1480 y Fu(-closure)d(of)6 b(\))34 b Fq(C)5 b Fu(.)55 b(F)-7 b(rom)33 b(Corollary)f(5.10,)i(w)n(e)f(already)456 1580 y(kno)n(w)22 b(that,)i(up)f(to)g(subsequences,)g(the)g(heat)g(\015o)n (w)f(con)n(v)n(erges)e(in)j(the)h Fr(W)2774 1549 y Fs(2)p Fp(;)p Fs(2)2864 1580 y Fu(-closure)d(of)i Fq(C)5 b Fu(;)24 b(w)n(e)456 1679 y(will)h(sho)n(w)g(in)g(Prop)r(osition)f(5.15)g(here)h (b)r(elo)n(w)f(that)i(the)g Fn(al)t(l)g Fu(the)g(limits)g(that)f(w)n(e) g(ma)n(y)g(obtain)456 1779 y(b)n(y)i(taking)g(subsequences)g(are)f (critical)h(p)r(oin)n(ts)h(of)f Fq(E)2132 1791 y Fp(!)2180 1779 y Fu(.)555 1960 y(Fixed)h Fr(t)817 1972 y Fs(0)877 1960 y Fr(>)23 b Fu(0)k(as)g(in)h(Lemma)f(5.9,)g(giv)n(en)g Fr(v)f Fq(2)e(C)5 b Fu(,)27 b(w)n(e)g(de\014ne,)h(for)f(an)n(y)g Fr(t)c(>)g(t)2987 1972 y Fs(0)1590 2101 y Fr(E)1651 2113 y Fp(v)1691 2101 y Fu(\()p Fr(t)p Fu(\))h(:=)1933 2080 y(~)1919 2101 y Fq(E)1963 2113 y Fp(!)2011 2101 y Fu(\(\010)2103 2067 y Fp(t)2133 2101 y Fu(\()p Fr(v)s Fu(\)\))14 b Fr(:)555 2242 y Fu(Notice)28 b(that,)g(b)n(y)f(Prop)r(osition)f(5.4,)456 2382 y(\(5.36\))1089 b Fr(E)1819 2394 y Fp(v)1859 2382 y Fu(\()p Fr(t)p Fu(\))23 b Fq(\025)g Fu(0)14 b Fr(:)456 2522 y Fu(No)n(w,)27 b(w)n(e)g(turn)h(to)f(estimate)h(the)g(deriv)-5 b(ativ)n(es)26 b(of)i Fr(E)2121 2534 y Fp(v)2161 2522 y Fu(.)456 2642 y FA(Lemma)f(5.13.)38 b Fn(With)28 b(the)g(notations)g (ab)l(ove,)i(we)e(have)h(that)f Fr(E)2521 2654 y Fp(v)2561 2642 y Fu(\()p Fr(t)p Fu(\))g Fn(is)h(twic)l(e)f(di\013er)l(entiable) 456 2742 y(and,)i(mor)l(e)l(over:)456 2921 y Fu(\(5.37\))729 2864 y Fr(dE)833 2876 y Fp(v)p 729 2902 144 4 v 764 2978 a Fr(dt)882 2921 y Fu(\()p Fr(t)p Fu(\))24 b(=)f Fq(\000)1167 2825 y Fo(\015)1167 2875 y(\015)1167 2925 y(\015)1212 2921 y Fr(@)1256 2933 y Fp(i)1298 2853 y Fo(\000)1336 2921 y Fr(a)1380 2933 y Fp(ij)1438 2921 y Fr(@)1482 2933 y Fp(j)1531 2853 y Fo(\000)1569 2921 y Fu(\010)1629 2886 y Fp(t)1658 2921 y Fu(\()p Fr(v)s Fu(\))1765 2853 y Fo(\001)q(\001)1861 2921 y Fq(\000)18 b Fr(f)2007 2829 y Fo(\020)2057 2921 y Fq(\001)p Fr(;)c(u)2165 2886 y Fs(\(0\))2272 2921 y Fu(+)k(\010)2415 2886 y Fp(t)2444 2921 y Fu(\()p Fr(v)s Fu(\))2551 2829 y Fo(\021)2620 2921 y Fu(+)g Fr(f)2766 2829 y Fo(\020)2816 2921 y Fq(\001)p Fr(;)c(u)2924 2886 y Fs(\(0\))3012 2829 y Fo(\021)3062 2825 y(\015)3062 2875 y(\015)3062 2925 y(\015)3108 2846 y Fs(2)3108 2979 y Fp(L)3154 2962 y Fi(2)3186 2979 y Fs(\()p Fl(K)3270 2962 y Fk(d)3270 2995 y(!)3313 2979 y Fs(\))3371 2921 y Fr(:)555 3102 y Fn(Ther)l(e)31 b(exists)e(a)h(p)l(ositive)i (universal)e(c)l(onstant)f Fr(C)36 b Fn(so)30 b(that)456 3291 y Fu(\(5.38\))1263 3170 y Fo(\014)1263 3220 y(\014)1263 3270 y(\014)1263 3320 y(\014)1301 3235 y Fr(d)1344 3205 y Fs(2)1381 3235 y Fr(E)1442 3247 y Fp(v)p 1301 3272 182 4 v 1336 3348 a Fr(dt)1409 3324 y Fs(2)1492 3291 y Fu(\()p Fr(t)p Fu(\))1586 3170 y Fo(\014)1586 3220 y(\014)1586 3270 y(\014)1586 3320 y(\014)1637 3291 y Fq(\024)23 b Fr(C)d(;)184 b Fq(8)p Fr(v)25 b Fq(2)e(C)5 b Fr(;)99 b Fq(8)p Fr(t)22 b(>)g Fu(0)14 b Fr(:)555 3481 y Fu(An)28 b(immediate)g(consequence)f(of)34 b(\(5.37\))27 b(is)g(that)456 3664 y(\(5.39\))1736 3608 y Fr(dE)1840 3620 y Fp(v)p 1736 3645 144 4 v 1772 3721 a Fr(dt)1890 3664 y Fu(\()p Fr(t)p Fu(\))d Fq(\024)e Fu(0)14 b Fr(;)456 3828 y Fu(whic)n(h)27 b(implies)h(that)456 3971 y(\(5.40\))604 b Fq(E)1317 3983 y Fp(!)1365 3971 y Fu(\()p Fr(u)1445 3937 y Fs(\(0\))1552 3971 y Fu(+)18 b(\010)1695 3937 y Fp(T)1748 3971 y Fu(\()p Fr(v)s Fu(\)\))24 b Fq(\024)e(E)2042 3983 y Fp(!)2090 3971 y Fu(\()p Fr(u)2170 3937 y Fs(\(0\))2278 3971 y Fu(+)c(\010)2421 3937 y Fp(t)2450 3971 y Fu(\()p Fr(v)s Fu(\)\))c Fr(;)456 4111 y Fu(for)27 b(an)n(y)f Fr(T)35 b Fq(\025)22 b Fr(t)h Fq(\025)g Fu(0.)456 4269 y Fn(Pr)l(o)l(of.)43 b Fu(The)33 b(fact)h(that)g(the)f(energy)g(is)g(t) n(wice)g(di\013eren)n(tiable)g(follo)n(ws)g(from)g(the)h(theory)f(of) 456 4369 y(semigroups.)h(It)24 b(su\016ces)g(to)f(study)h(equation)g (\(A.4\))g(to)g(obtain)g(that)g(the)g(\015o)n(w)f(giv)n(es)g(a)h(t)n (wice)456 4468 y(di\013eren)n(tiable)f(curv)n(e)f(in)i Fr(W)1350 4438 y Fs(1)p Fp(;)p Fs(2)1440 4468 y Fu(\()p FA(K)1547 4438 y Fp(d)1547 4489 y(!)1595 4468 y Fu(\).)36 b(Since)24 b(the)g(energy)e(is)i(di\013eren)n(tiable)f(as)g(a)g (function)h(on)456 4568 y Fr(W)546 4538 y Fs(1)p Fp(;)p Fs(2)636 4568 y Fu(\()p FA(K)743 4538 y Fp(d)743 4588 y(!)791 4568 y Fu(\),)33 b(w)n(e)e(obtain)h(the)g(desired)f(result)g (\(see)h(Lemma)f(A.4\).)50 b(W)-7 b(e)32 b(also)e(note)i(that)g(the)456 4667 y(in)n(tegration)26 b(b)n(y)h(parts)g(needed)h(to)f(write)h (\(5.37\))e(is)i(b)r(ona-\014de)f(since)910 4815 y Fr(D)979 4827 y Fp(t)1008 4815 y Fu(\010)1068 4781 y Fp(t)1097 4815 y Fu(\()p Fr(v)s Fu(\))d(=)f Fr(@)1360 4827 y Fp(i)1387 4815 y Fu(\()p Fr(a)1463 4827 y Fp(ij)1522 4815 y Fr(@)1566 4827 y Fp(j)1601 4815 y Fu(\010)1661 4781 y Fp(t)1690 4815 y Fu(\()p Fr(v)s Fu(\)\))d Fq(\000)e Fr(f)9 b Fu(\()p Fq(\001)p Fr(;)14 b(u)2122 4781 y Fs(\(0\))2229 4815 y Fu(+)k(\010)2372 4781 y Fp(t)2401 4815 y Fu(\()p Fr(v)s Fu(\)\))h(+)f Fr(f)9 b Fu(\()p Fq(\001)p Fr(;)14 b(u)2832 4781 y Fs(\(0\))2921 4815 y Fu(\))g Fr(;)456 4958 y Fu(whic)n(h)27 b(b)r(elongs)g(to)h Fr(W)1183 4928 y Fp(k)q(;)p Fs(2)1276 4958 y Fu(\()p FA(K)1383 4928 y Fp(d)1383 4979 y(!)1431 4958 y Fu(\))g(b)n(y)g(Lemma)f(5.9.)1347 b Fj(\003)555 5116 y Fu(The)28 b(next)g(result)f(sho)n(ws)f(that)i(\010)1622 5086 y Fp(t)1651 5116 y Fu(\()p Fr(v)s Fu(\))h(gets)e(closer)f(and)h (closer)f(to)i(b)r(eing)f(a)g(solution,)h(as)e Fr(t)456 5216 y Fu(increases.)p eop %%Page: 27 27 27 26 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(27)456 450 y FA(Corollary)33 b(5.14.)702 676 y Fu(lim)636 726 y Fp(t)p Fm(\000)-12 b(!)p Fs(+)p Fm(1)898 581 y Fo(\015)898 630 y(\015)898 680 y(\015)945 676 y Fr(@)989 688 y Fp(i)1030 609 y Fo(\000)1068 676 y Fr(a)1112 688 y Fp(ij)1171 676 y Fr(@)1215 688 y Fp(j)1263 609 y Fo(\000)1301 676 y Fu(\010)1361 642 y Fp(t)1391 676 y Fu(\()p Fr(v)s Fu(\))1498 609 y Fo(\001)q(\001)1593 676 y Fq(\000)18 b Fr(f)1739 584 y Fo(\020)1789 676 y Fq(\001)p Fr(;)c(u)1897 642 y Fs(\(0\))2004 676 y Fu(+)k(\010)2147 642 y Fp(t)2176 676 y Fu(\()p Fr(v)s Fu(\))2283 584 y Fo(\021)2352 676 y Fu(+)g Fr(f)2499 584 y Fo(\020)2548 676 y Fq(\001)p Fr(;)c(u)2656 642 y Fs(\(0\))2745 584 y Fo(\021)2794 581 y(\015)2794 630 y(\015)2794 680 y(\015)2841 734 y Fp(L)2887 718 y Fi(2)2918 734 y Fs(\()p Fl(K)3002 718 y Fk(d)3002 751 y(!)3045 734 y Fs(\))3098 676 y Fu(=)23 b(0)14 b Fr(:)456 973 y Fn(Pr)l(o)l(of.)43 b Fu(Giv)n(en)29 b(the)i(previous)d(results)h(\(5.37\))g(and)h(\(5.38\))f(this)h(is)g (just)g(an)g(elemen)n(tary)e(real)456 1072 y(analysis)e(argumen)n(t.) 555 1172 y(By)j(\(5.37\))o(\))f(it)g(is)f(enough)g(to)h(sho)n(w)e(that) 1662 1421 y(lim)1595 1471 y Fp(t)p Fm(\000)-12 b(!)p Fs(+)p Fm(1)1868 1365 y Fr(dE)1972 1377 y Fp(v)p 1868 1402 144 4 v 1903 1478 a Fr(dt)2022 1421 y Fu(\()p Fr(t)p Fu(\))23 b(=)g(0)14 b Fr(:)456 1663 y Fu(Assume,)28 b(b)n(y)g(con)n (tradiction,)f(that)i(this)g(is)f(not)g(true.)39 b(Then,)29 b(b)n(y)g(\(5.39\))o(,)g(there)f(exist)g Fr(\016)f(>)d Fu(0)456 1762 y(and)j(a)g(sequence)g Fr(t)1060 1774 y Fp(k)1124 1762 y Fq(\000)-14 b(!)23 b Fu(+)p Fq(1)k Fu(so)g(that)1684 1950 y Fr(dE)1788 1962 y Fp(v)p 1684 1987 V 1720 2063 a Fr(dt)1838 2006 y Fu(\()p Fr(t)1900 2018 y Fp(k)1941 2006 y Fu(\))d Fq(\024)e(\000)p Fr(\016)17 b(:)456 2240 y Fu(By)28 b(\(5.36\))f(and)g(\(5.39\))o(,)h(w)n(e)f(can)h(also)e(set) 1427 2446 y Fr(`)d Fu(:=)89 b(lim)1595 2495 y Fp(t)p Fm(\000)-11 b(!)p Fs(+)p Fm(1)1858 2446 y Fr(E)1919 2458 y Fp(v)1959 2446 y Fu(\()p Fr(t)p Fu(\))23 b Fq(2)h Fu([0)p Fr(;)14 b Fu(+)p Fq(1)p Fu(\))g Fr(:)456 2692 y Fu(Let)28 b Fr(C)35 b Fu(b)r(e)29 b(as)e(in)i(Lemma)f(5.13.)38 b(T)-7 b(ak)n(e)28 b Fr(t)c Fu(:=)g Fr(\016)s(=)p Fu(\(2)p Fr(C)6 b Fu(\))28 b(and)h Fr(")24 b Fu(:=)g Fr(\016)2547 2662 y Fs(2)2584 2692 y Fr(=)p Fu(\(5)p Fr(C)6 b Fu(\).)39 b(Then,)29 b(b)n(y)f(taking)456 2792 y Fr(k)i Fu(large)c(enough,)h(and) h(b)n(y)f(using)g(again)g(\(5.39\))o(,)h(w)n(e)f(gather)f(that)627 3056 y Fr(`)d Fq(\024)g Fr(E)834 3068 y Fp(v)873 3056 y Fu(\()p Fr(t)935 3068 y Fp(k)995 3056 y Fu(+)18 b Fr(t)p Fu(\))23 b(=)g Fr(E)1312 3068 y Fp(v)1352 3056 y Fu(\()p Fr(t)1414 3068 y Fp(k)1455 3056 y Fu(\))18 b(+)1598 2999 y Fr(dE)1702 3011 y Fp(v)p 1598 3037 V 1634 3113 a Fr(dt)1752 3056 y Fu(\()p Fr(t)1814 3068 y Fp(k)1855 3056 y Fu(\))c Fr(t)19 b Fu(+)2033 2943 y Fo(Z)2116 2963 y Fp(t)2141 2972 y Fk(k)2177 2963 y Fs(+)p Fp(t)2079 3131 y(t)2104 3140 y Fk(k)2271 2939 y Fo(\022)2342 2999 y Fr(dE)2446 3011 y Fp(v)p 2342 3037 V 2378 3113 a Fr(dt)2496 3056 y Fu(\()p Fr(\034)9 b Fu(\))20 b Fq(\000)2718 2999 y Fr(dE)2822 3011 y Fp(v)p 2718 3037 V 2753 3113 a Fr(dt)2872 3056 y Fu(\()p Fr(t)2934 3068 y Fp(k)2975 3056 y Fu(\))3007 2939 y Fo(\023)3096 3056 y Fr(d\034)33 b Fq(\024)1525 3397 y(\024)23 b Fr(`)18 b Fu(+)g Fr(")g Fq(\000)g Fr(\016)s(t)g Fu(+)g Fr(C)6 b(t)2155 3363 y Fs(2)2216 3397 y Fr(<)23 b(`)14 b(;)456 3585 y Fu(whic)n(h)27 b(is,)h(of)f(course,)g(a)g(con)n (tradiction.)1637 b Fj(\003)555 3837 y Fu(W)-7 b(e)31 b(no)n(w)f(sho)n(w)f(that)i Fn(al)t(l)h Fu(the)f Fr(W)1624 3806 y Fs(2)p Fp(;)p Fs(2)1714 3837 y Fu(-limit)f(p)r(oin)n(ts)h(of)f (the)h(heat)f(\015o)n(w)g(are)g(critical)g(p)r(oin)n(ts)456 3936 y(for)d(the)h(phase)f(transition)g(equation;)g(more)f(precisely)-7 b(,)27 b(w)n(e)h(ha)n(v)n(e:)456 4089 y FA(Prop)s(osition)i(5.15.)40 b Fn(L)l(et)29 b Fr(v)d Fq(2)e(C)5 b Fn(.)38 b(Assume)29 b(that,)h(for)h Fr(t)2278 4101 y Fp(n)2346 4089 y Fq(\000)-15 b(!)24 b Fu(+)p Fq(1)p Fn(,)1433 4294 y Fu(\010)1493 4260 y Fp(t)1518 4268 y Fk(n)1562 4294 y Fu(\()p Fr(v)s Fu(\))g Fq(\000)-14 b(!)23 b Fr(v)1893 4260 y Fp(?)1961 4294 y Fn(in)30 b Fr(W)2153 4260 y Fs(2)p Fp(;)p Fs(2)2243 4294 y Fu(\()p FA(K)2350 4260 y Fp(d)2350 4315 y(!)2398 4294 y Fu(\))14 b Fr(:)456 4500 y Fn(Then,)456 4726 y Fu(\(5.41\))390 b Fr(@)1103 4738 y Fp(i)1144 4726 y Fu(\()p Fr(a)1220 4738 y Fp(ij)1279 4726 y Fr(@)1323 4738 y Fp(j)1358 4726 y Fr(v)1401 4692 y Fp(?)1439 4726 y Fu(\))19 b Fq(\000)f Fr(f)1636 4634 y Fo(\020)1686 4726 y Fr(x;)c(u)1818 4692 y Fs(\(0\))1926 4726 y Fu(+)k Fr(v)2052 4692 y Fp(?)2090 4634 y Fo(\021)2158 4726 y Fu(+)g Fr(f)2304 4634 y Fo(\020)2354 4726 y Fr(x;)c(u)2486 4692 y Fs(\(0\))2575 4634 y Fo(\021)2661 4726 y Fu(=)37 b(0)14 b Fr(:)456 5007 y Fn(Pr)l(o)l(of.)43 b Fu(Notice)27 b(that,)h(since)g Fr(v)j Fu(and)c Fr(v)1658 4977 y Fp(?)1724 5007 y Fu(are)g(in)h(the)g Fr(W)2193 4977 y Fs(2)p Fp(;)p Fs(2)2283 5007 y Fu(-closure)e(of)h Fq(C)5 b Fu(,)27 b(w)n(e)h(ha)n(v)n(e)e(that)1472 5216 y Fq(j)p Fr(u)1543 5181 y Fs(\(0\))1650 5216 y Fu(+)18 b Fr(v)s Fq(j)c Fr(;)37 b Fq(j)p Fr(u)1944 5181 y Fs(\(0\))2051 5216 y Fu(+)18 b Fr(v)2177 5181 y Fp(?)2216 5216 y Fq(j)23 b(\024)g Fu(1)14 b Fr(:)p eop %%Page: 28 28 28 27 bop 456 251 a Fs(28)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fu(Hence,)27 b(if)i Fr(C)g Fq(\025)23 b Fu(1)k(is)g(large)f(enough,)h(w)n(e)h (deduce)f(from)h(Corollary)d(5.14)h(that)559 618 y(0)82 b(=)137 b(lim)831 668 y Fp(n)p Fm(!)p Fs(+)p Fm(1)1069 523 y Fo(\015)1069 573 y(\015)1069 623 y(\015)1115 618 y Fr(@)1159 630 y Fp(i)1201 551 y Fo(\000)1239 618 y Fr(a)1283 630 y Fp(ij)1341 618 y Fr(@)1385 630 y Fp(j)1434 551 y Fo(\000)1472 618 y Fu(\010)1532 584 y Fp(t)1557 592 y Fk(n)1602 618 y Fu(\()p Fr(v)s Fu(\))1709 551 y Fo(\001)q(\001)1804 618 y Fq(\000)18 b Fr(f)1951 526 y Fo(\020)2000 618 y Fq(\001)p Fr(;)c(u)2108 584 y Fs(\(0\))2216 618 y Fu(+)k(\010)2359 584 y Fp(t)2384 592 y Fk(n)2428 618 y Fu(\()p Fr(v)s Fu(\))2535 526 y Fo(\021)2604 618 y Fu(+)g Fr(f)2751 526 y Fo(\020)2800 618 y Fq(\001)p Fr(;)c(u)2908 584 y Fs(\(0\))2997 526 y Fo(\021)3047 523 y(\015)3047 573 y(\015)3047 623 y(\015)3093 677 y Fp(L)3139 660 y Fi(2)3171 677 y Fs(\()p Fl(K)3255 660 y Fk(d)3255 693 y(!)3297 677 y Fs(\))683 822 y Fq(\025)137 b Fu(lim)831 872 y Fp(n)p Fm(!)p Fs(+)p Fm(1)1069 726 y Fo(\015)1069 776 y(\015)1069 826 y(\015)1115 822 y Fr(@)1159 834 y Fp(i)1201 822 y Fu(\()p Fr(a)1277 834 y Fp(ij)1335 822 y Fr(@)1379 834 y Fp(j)1415 822 y Fr(v)1458 788 y Fp(?)1496 822 y Fu(\))19 b Fq(\000)f Fr(f)1693 730 y Fo(\020)1743 822 y Fq(\001)p Fr(;)c(u)1851 788 y Fs(\(0\))1958 822 y Fu(+)k Fr(v)2084 788 y Fp(?)2122 730 y Fo(\021)2190 822 y Fu(+)g Fr(f)2337 730 y Fo(\020)2386 822 y Fq(\001)p Fr(;)c(u)2494 788 y Fs(\(0\))2583 730 y Fo(\021)2632 726 y(\015)2632 776 y(\015)2632 826 y(\015)2679 880 y Fp(L)2725 863 y Fi(2)2756 880 y Fs(\()p Fl(K)2840 863 y Fk(d)2840 897 y(!)2883 880 y Fs(\))831 1000 y Fq(\000)910 930 y Fo(\015)910 980 y(\015)955 1000 y Fr(@)999 1012 y Fp(i)1041 933 y Fo(\000)1079 1000 y Fr(a)1123 1012 y Fp(ij)1181 1000 y Fr(@)1225 1012 y Fp(j)1274 933 y Fo(\000)1312 1000 y Fu(\010)1372 966 y Fp(t)1397 974 y Fk(n)1442 1000 y Fu(\()p Fr(v)s Fu(\))20 b Fq(\000)e Fr(v)1695 966 y Fp(?)1733 933 y Fo(\001\001)1809 930 y(\015)1809 980 y(\015)1855 1034 y Fp(L)1901 1017 y Fi(2)1933 1034 y Fs(\()p Fl(K)2017 1017 y Fk(d)2017 1050 y(!)2060 1034 y Fs(\))831 1179 y Fq(\000)910 1083 y Fo(\015)910 1133 y(\015)910 1183 y(\015)955 1179 y Fr(f)1019 1087 y Fo(\020)1068 1179 y Fq(\001)p Fr(;)c(u)1176 1145 y Fs(\(0\))1284 1179 y Fu(+)k(\010)1427 1145 y Fp(t)1452 1153 y Fk(n)1496 1179 y Fu(\()p Fr(v)s Fu(\))1603 1087 y Fo(\021)1672 1179 y Fq(\000)g Fr(f)1819 1087 y Fo(\020)1868 1179 y Fq(\001)p Fr(;)c(u)1976 1145 y Fs(\(0\))2083 1179 y Fu(+)k Fr(v)2209 1145 y Fp(?)2248 1087 y Fo(\021)2297 1083 y(\015)2297 1133 y(\015)2297 1183 y(\015)2343 1237 y Fp(L)2389 1220 y Fi(2)2421 1237 y Fs(\()p Fl(K)2505 1220 y Fk(d)2505 1254 y(!)2548 1237 y Fs(\))683 1382 y Fq(\025)137 b Fu(lim)831 1432 y Fp(n)p Fm(!)p Fs(+)p Fm(1)1069 1287 y Fo(\015)1069 1337 y(\015)1069 1386 y(\015)1115 1382 y Fr(@)1159 1394 y Fp(i)1201 1382 y Fu(\()p Fr(a)1277 1394 y Fp(ij)1335 1382 y Fr(@)1379 1394 y Fp(j)1415 1382 y Fr(v)1458 1348 y Fp(?)1496 1382 y Fu(\))19 b Fq(\000)f Fr(f)1693 1290 y Fo(\020)1743 1382 y Fq(\001)p Fr(;)c(u)1851 1348 y Fs(\(0\))1958 1382 y Fu(+)k Fr(v)2084 1348 y Fp(?)2122 1290 y Fo(\021)2190 1382 y Fu(+)g Fr(f)2337 1290 y Fo(\020)2386 1382 y Fq(\001)p Fr(;)c(u)2494 1348 y Fs(\(0\))2583 1290 y Fo(\021)2632 1287 y(\015)2632 1337 y(\015)2632 1386 y(\015)2679 1440 y Fp(L)2725 1424 y Fi(2)2756 1440 y Fs(\()p Fl(K)2840 1424 y Fk(d)2840 1457 y(!)2883 1440 y Fs(\))831 1561 y Fq(\000)p Fu(\(1)k(+)g Fr(C)6 b Fu(\))1182 1490 y Fo(\015)1182 1540 y(\015)1228 1561 y Fu(\010)1288 1526 y Fp(t)1313 1534 y Fk(n)1358 1561 y Fu(\()p Fr(v)s Fu(\))19 b Fq(\000)f Fr(v)1610 1526 y Fp(?)1649 1490 y Fo(\015)1649 1540 y(\015)1695 1594 y Fp(W)1766 1577 y Fi(2)p Fk(;)p Fi(2)1846 1594 y Fs(\()p Fl(K)1930 1577 y Fk(d)1930 1611 y(!)1973 1594 y Fs(\))683 1739 y Fu(=)831 1644 y Fo(\015)831 1694 y(\015)831 1743 y(\015)877 1739 y Fr(@)921 1751 y Fp(i)963 1739 y Fu(\()p Fr(a)1039 1751 y Fp(ij)1097 1739 y Fr(@)1141 1751 y Fp(j)1176 1739 y Fr(v)1219 1705 y Fp(?)1258 1739 y Fu(\))g Fq(\000)g Fr(f)1455 1647 y Fo(\020)1504 1739 y Fq(\001)p Fr(;)c(u)1612 1705 y Fs(\(0\))1720 1739 y Fu(+)k Fr(v)1846 1705 y Fp(?)1884 1647 y Fo(\021)1952 1739 y Fu(+)g Fr(f)2098 1647 y Fo(\020)2148 1739 y Fq(\001)p Fr(;)c(u)2256 1705 y Fs(\(0\))2345 1647 y Fo(\021)2394 1644 y(\015)2394 1694 y(\015)2394 1743 y(\015)2440 1797 y Fp(L)2486 1781 y Fi(2)2518 1797 y Fs(\()p Fl(K)2602 1781 y Fk(d)2602 1814 y(!)2645 1797 y Fs(\))2703 1739 y Fr(;)456 1928 y Fu(whic)n(h)27 b(pro)n(v)n(es)f (the)i(desired)f(claim.)1788 b Fj(\003)456 2097 y Fu(5.6.)40 b FA(Existence)d(of)h(another)g(critical)g(p)s(oin)m(t.)j Fu(W)-7 b(e)33 b(ha)n(v)n(e)f(dev)n(elop)r(ed)g(to)r(ols)g(ab)r(out)h (the)456 2197 y(con)n(v)n(ergence)25 b(of)i(the)h(heat)f(\015o)n(w)g (and)g(its)h(relation)e(with)i(the)g(renormalized)e(energy)-7 b(.)36 b(In)27 b(this)456 2296 y(section,)k(w)n(e)g(will)h(put)g(them)g (together)e(to)h(sho)n(w)f(the)i(existence)f(of)g(another)g(critical)f (p)r(oin)n(t)456 2396 y(in)f(\(the)g Fr(W)820 2366 y Fs(2)p Fp(;)p Fs(2)910 2396 y Fu(-closure)e(of)6 b(\))30 b Fq(C)5 b Fu(,)29 b(under)f(a)h(suitable)f(assumption)h(on)f(the)h (basin)g(of)f(attraction)456 2495 y(of)f(the)h(heat)g(\015o)n(w.)36 b(This)27 b(assumption)h(will)f(then)h(b)r(e)g(established)g(of)f(in)h Fq(x)g Fu(5.7.)456 2619 y FA(Theorem)i(5.16.)40 b Fn(L)l(et)29 b Fr(v)1285 2589 y Fs(\()p Fp(s)p Fs(\))1402 2619 y Fn(b)l(e)h(as)g (de\014ne)l(d)g(in)37 b Fu(\(5.1\))o Fn(.)i(F)-6 b(or)30 b Fr(i)23 b Fu(=)f(0)p Fr(;)14 b Fu(1)29 b Fn(de\014ne)2889 2587 y Fs(1)456 2775 y Fu(\(5.42\))258 b FA(B)995 2787 y Fp(i)1046 2775 y Fu(:=)23 b Fq(f)p Fr(s)f Fq(2)i Fu([0)p Fr(;)14 b Fu(1])28 b Fn(s.t.)39 b Fr(W)1775 2740 y Fs(2)p Fp(;)p Fs(2)1865 2775 y Fu(\()p FA(K)1972 2740 y Fp(d)1972 2795 y(!)2020 2775 y Fu(\))19 b Fq(\000)65 b Fu(lim)2154 2824 y Fp(t)p Fm(!)p Fs(+)p Fm(1)2376 2775 y Fu(\010)2436 2740 y Fp(t)2465 2775 y Fu(\()p Fr(v)2540 2740 y Fs(\()p Fp(s)p Fs(\))2628 2775 y Fu(\))24 b(=)e Fr(v)2814 2740 y Fs(\()p Fp(i)p Fs(\))2894 2775 y Fq(g)14 b Fr(:)456 2967 y Fn(L)l(et)34 b(us)g(assume)h(that)g Fu([0)p Fr(;)14 b Fu(1])21 b Fq(n)g Fu(\()p FA(B)1539 2979 y Fs(0)1599 2967 y Fq([)h FA(B)1744 2979 y Fs(1)1781 2967 y Fu(\))33 b Fq(6)p Fu(=)f Fq(;)p Fn(.)53 b(Then,)37 b(ther)l(e)e(exists)f Fr(u)2803 2937 y Fp(?)2873 2967 y Fq(2)f Fr(C)3026 2937 y Fs(2)3063 2967 y Fu(\()p FA(K)3170 2937 y Fp(d)3170 2987 y(!)3218 2967 y Fu(\))i Fn(such)456 3066 y(that:)661 3190 y Fq(\017)41 b Fr(@)788 3202 y Fp(i)816 3190 y Fu(\()p Fr(a)892 3202 y Fp(ij)951 3190 y Fr(@)995 3202 y Fp(j)1030 3190 y Fr(u)1078 3160 y Fp(?)1115 3190 y Fu(\))24 b(=)e Fr(f)9 b Fu(\()p Fr(x;)14 b(u)1472 3160 y Fp(?)1510 3190 y Fu(\))p Fn(;)661 3290 y Fq(\017)41 b Fr(u)792 3260 y Fp(?)860 3290 y Fn(enjoys)30 b(the)g(Birkho\013)i(pr)l(op)l(erty;)661 3389 y Fq(\017)41 b Fr(u)792 3359 y Fs(\(0\))881 3389 y Fu(\()p Fr(x)p Fu(\))24 b Fr(<)f(u)1152 3359 y Fp(?)1189 3389 y Fu(\()p Fr(x)p Fu(\))h Fr(<)f(u)1460 3359 y Fs(\(1\))1549 3389 y Fu(\()p Fr(x)p Fu(\))p Fn(.)456 3561 y(Pr)l(o)l(of.)43 b Fu(Notice)37 b(that,)k(if)d(w)n(e)f(pro)n(v)n(e)f(the)i(existence)f (of)g Fr(v)2317 3531 y Fp(?)2395 3561 y Fq(2)j(C)30 b(\000)24 b(f)p Fu(0)p Fr(;)j(u)2835 3531 y Fs(\(1\))2949 3561 y Fq(\000)e Fr(u)3087 3531 y Fs(\(0\))3175 3561 y Fq(g)37 b Fu(satis-)456 3664 y(fying)g(\(5.41\))o(,)i(then)e(w)n(e)g(ma)n(y)e (de\014ne)i Fr(u)1762 3634 y Fp(?)1838 3664 y Fu(:=)g Fr(v)2006 3634 y Fp(?)2069 3664 y Fu(+)24 b Fr(u)2206 3634 y Fs(\(0\))2331 3664 y Fu(and)37 b(w)n(e)f(are)g(done)g(\(recall)g (again)456 3764 y(Corollary)27 b(A.3\).)46 b(F)-7 b(or)29 b(\014nding)i(suc)n(h)f(a)g Fr(v)1789 3734 y Fp(?)1827 3764 y Fu(,)h(thanks)f(to)h(Prop)r(osition)d(5.15,)i(it)h(is)f(enough)g (to)456 3863 y(\014nd)h(\026)-45 b Fr(v)26 b Fq(2)d(C)33 b Fu(and)27 b(some)g(sequence)g Fr(T)1605 3875 y Fp(n)1673 3863 y Fq(\000)-14 b(!)23 b Fu(+)p Fq(1)p Fu(,)k(suc)n(h)g(that)456 4014 y(\(5.43\))457 b Fr(W)1216 3980 y Fs(2)p Fp(;)p Fs(2)1306 4014 y Fu(\()p FA(K)1413 3980 y Fp(d)1413 4035 y(!)1461 4014 y Fu(\))19 b Fq(\000)73 b Fu(lim)1595 4064 y Fp(n)p Fm(!)p Fs(+)p Fm(1)1833 4014 y Fu(\010)1893 3980 y Fp(T)1932 3988 y Fk(n)1977 4014 y Fu(\()s(\026)-45 b Fr(v)t Fu(\))23 b Fq(62)h(f)p Fu(0)p Fr(;)j(u)2369 3980 y Fs(\(1\))2476 4014 y Fq(\000)18 b Fr(u)2607 3980 y Fs(\(0\))2695 4014 y Fq(g)c Fr(:)456 4198 y Fu(Therefore,)26 b(w)n(e)h(will)h(no)n(w)f(sho)n(w)g(the)h(existence)f(of)h(a)f (function)k(\026)-45 b Fr(v)31 b Fu(for)c(whic)n(h)g(\(5.43\))g(holds.) 555 4397 y(Let)33 b(\026)-46 b Fr(s)25 b Fq(2)h Fu([0)p Fr(;)14 b Fu(1])k Fq(\000)h Fu(\()p FA(B)1219 4409 y Fs(0)1276 4397 y Fq([)g FA(B)1418 4409 y Fs(1)1455 4397 y Fu(\))30 b(and)i(\026)-46 b Fr(v)29 b Fu(:=)24 b Fr(v)1903 4367 y Fs(\()s(\026)-36 b Fp(s)q Fs(\))1991 4397 y Fu(.)41 b(Then,)29 b(from)g(Corollary)d(5.10,)i(there)h(exist)456 4497 y Fr(V)42 b Fq(2)23 b(C)32 b Fu(and)c(a)f(sequence)g Fr(t)1305 4509 y Fp(n)1373 4497 y Fq(\000)-14 b(!)23 b Fu(+)p Fq(1)k Fu(suc)n(h)g(that)1370 4645 y Fr(W)1460 4611 y Fs(2)p Fp(;)p Fs(2)1550 4645 y Fu(\()p FA(K)1657 4611 y Fp(d)1657 4666 y(!)1706 4645 y Fu(\))18 b Fq(\000)73 b Fu(lim)1839 4695 y Fp(n)p Fm(!)p Fs(+)p Fm(1)2077 4645 y Fu(\010)2137 4611 y Fp(t)2162 4619 y Fk(n)2207 4645 y Fu(\()s(\026)-45 b Fr(v)t Fu(\))23 b(=)g Fr(V)33 b(:)456 4840 y Fu(If)26 b Fr(V)42 b Fq(6)p Fu(=)23 b(0)p Fr(;)k(u)855 4810 y Fs(\(1\))958 4840 y Fq(\000)15 b Fr(u)1086 4810 y Fs(\(0\))1175 4840 y Fu(,)26 b(w)n(e)g(are)f(done.)36 b(On)25 b(the)i(other)e(hand,)h(if,)h(sa)n(y)e Fr(V)42 b Fu(=)22 b Fr(u)2899 4810 y Fs(\(1\))3003 4840 y Fq(\000)15 b Fr(u)3131 4810 y Fs(\(0\))3219 4840 y Fu(,)27 b(since)459 4940 y(\026)-45 b Fr(s)28 b Fq(62)h FA(B)675 4952 y Fs(1)712 4940 y Fu(,)j(there)e(m)n(ust)h(exist)g(another)f(sequence)g Fr(\034)2082 4952 y Fp(n)2156 4940 y Fq(\000)-14 b(!)28 b Fu(+)p Fq(1)j Fu(and)g(a)f(suitable)h Fr(\021)g(>)d Fu(0)j(suc)n(h)p 456 5034 499 4 v 555 5107 a Fs(1)588 5133 y Fz(In)c(what)f(follo)n(ws,)f(w)n(e)h(will)f(use)h(the)g (notation)i Fe(W)1962 5109 y Fd(2)p Fc(;)p Fd(2)2062 5133 y Fb(\000)17 b Fz(lim)24 b(to)i(denote)i(the)e(limit)e(in)h(the)i Fe(W)3155 5109 y Fd(2)p Fc(;)p Fd(2)3264 5133 y Fz(norm.)456 5216 y(This)c(is)g(to)h(a)n(v)n(oid)g(confusion)g(with)g(other)g(t)n (yp)r(es)h(of)e(con)n(v)n(ergence.)p eop %%Page: 29 29 29 28 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(29)456 450 y Fu(that)456 590 y(\(5.44\))1248 495 y Fo(\015)1248 545 y(\015)1248 594 y(\015)1294 590 y Fu(\010)1354 556 y Fp(\034)1385 564 y Fk(n)1430 590 y Fu(\()s(\026)-45 b Fr(v)s Fu(\))33 b Fq(\000)f Fu(\()p Fr(u)1747 556 y Fs(\(1\))1854 590 y Fq(\000)18 b Fr(u)1985 556 y Fs(\(0\))2074 590 y Fu(\))2106 495 y Fo(\015)2106 545 y(\015)2106 594 y(\015)2152 648 y Fp(W)2223 632 y Fi(2)p Fk(;)p Fi(2)2303 648 y Fs(\()p Fl(K)2387 628 y Fk(!)2387 669 y(d)2430 648 y Fs(\))2483 590 y Fq(\025)23 b Fr(\021)17 b(:)456 771 y Fu(In)32 b(the)g(ligh)n(t)g(of)g(Corollary)d(5.10,)j(p)r(ossibly) f(extracting)g(a)h(subsequence,)g(w)n(e)g(ma)n(y)f(assume)456 870 y(that)1356 972 y Fr(W)1446 937 y Fs(2)p Fp(;)p Fs(2)1536 972 y Fu(\()p FA(K)1643 937 y Fp(d)1643 992 y(!)1691 972 y Fu(\))19 b Fq(\000)73 b Fu(lim)1825 1021 y Fp(n)p Fm(!)p Fs(+)p Fm(1)2063 972 y Fu(\010)2123 937 y Fp(\034)2154 945 y Fk(n)2199 972 y Fu(\()s(\026)-45 b Fr(v)s Fu(\))23 b(=)g Fr(W)j(;)456 1138 y Fu(for)k(some)h Fr(W)41 b Fq(2)30 b(C)5 b Fu(.)48 b(F)-7 b(urthermore,)31 b(from)g(\(5.44\))o(,)h Fr(W)42 b Fq(6)p Fu(=)28 b Fr(u)2359 1108 y Fs(\(1\))2469 1138 y Fq(\000)20 b Fr(u)2602 1108 y Fs(\(0\))2691 1138 y Fu(.)48 b(If)32 b(also)e Fr(W)41 b Fq(6)p Fu(=)29 b(0,)j(the)456 1237 y(pro)r(of)e(is)g(complete.)46 b(If,)31 b(on)g(the)f(other)g (hand,)i Fr(W)40 b Fu(=)27 b(0,)k(from)f(the)h(construction)f(ab)r(o)n (v)n(e)f(w)n(e)456 1337 y(ha)n(v)n(e)d(the)i(existence)f(of)h(t)n(w)n (o)f(sequences,)g Fr(t)1825 1349 y Fp(n)1897 1337 y Fu(and)h Fr(\034)2095 1349 y Fp(n)2140 1337 y Fu(,)g(for)f(whic)n(h)1028 1476 y Fr(W)1118 1441 y Fs(2)p Fp(;)p Fs(2)1208 1476 y Fu(\()p FA(K)1315 1441 y Fp(d)1315 1496 y(!)1363 1476 y Fu(\))19 b Fq(\000)72 b Fu(lim)1497 1525 y Fp(n)p Fm(!)p Fs(+)p Fm(1)1735 1476 y Fu(\010)1795 1441 y Fp(t)1820 1449 y Fk(n)1865 1476 y Fu(\()s(\026)-45 b Fr(v)s Fu(\))84 b(=)e Fr(u)2251 1441 y Fs(\(1\))2358 1476 y Fq(\000)18 b Fr(u)2489 1441 y Fs(\(0\))2744 1476 y Fu(and)1022 1649 y Fr(W)1112 1614 y Fs(2)p Fp(;)p Fs(2)1202 1649 y Fu(\()p FA(K)1309 1614 y Fp(d)1309 1669 y(!)1357 1649 y Fu(\))h Fq(\000)73 b Fu(lim)1491 1698 y Fp(n)p Fm(!)p Fs(+)p Fm(1)1729 1649 y Fu(\010)1789 1614 y Fp(\034)1820 1622 y Fk(n)1865 1649 y Fu(\()s(\026)-45 b Fr(v)s Fu(\))84 b(=)e(0)14 b Fr(:)456 1816 y Fu(P)n(ossibly)26 b(taking)h (subsequences,)f(w)n(e)i(ma)n(y)f(assume)f(also)h(that)h Fr(t)2527 1828 y Fp(n)2595 1816 y Fr(<)23 b(\034)2719 1828 y Fp(n)2764 1816 y Fu(.)37 b(Let)28 b(us)f(de\014ne)1311 1955 y Fr(\024)83 b Fu(:=)g Fq(k)p Fr(u)1703 1920 y Fs(\(1\))1809 1955 y Fq(\000)18 b Fr(u)1940 1920 y Fs(\(0\))2029 1955 y Fq(k)2071 1970 y Fp(W)2142 1953 y Fi(2)p Fk(;)p Fi(2)2222 1970 y Fs(\()p Fl(K)2306 1950 y Fk(!)2306 1990 y(d)2348 1970 y Fs(\))2544 1955 y Fu(and)1222 2093 y Fr(g)s Fu(\()p Fr(t)p Fu(\))83 b(:=)1613 2023 y Fo(\015)1613 2073 y(\015)1659 2093 y Fu(\010)1719 2059 y Fp(t)1748 2093 y Fu(\()s(\026)-45 b Fr(v)s Fu(\))1855 2023 y Fo(\015)1855 2073 y(\015)1902 2127 y Fp(W)1973 2110 y Fi(2)p Fk(;)p Fi(2)2053 2127 y Fs(\()p Fl(K)2137 2107 y Fk(!)2137 2147 y(d)2180 2127 y Fs(\))2237 2093 y Fr(:)456 2254 y Fu(Clearly)23 b(from)i(its)g (de\014nition,)g Fr(\024)e(>)g Fu(0.)35 b(F)-7 b(or)25 b(su\016cien)n(tly)f(large)g Fr(n)p Fu(,)h(w)n(e)f(ha)n(v)n(e)g Fr(g)s Fu(\()p Fr(t)2967 2266 y Fp(n)3012 2254 y Fu(\))f Fq(\025)g Fr(\024=)p Fu(2)h(and)456 2353 y Fr(g)s Fu(\()p Fr(\034)567 2365 y Fp(n)612 2353 y Fu(\))f Fq(\024)g Fr(\024=)p Fu(8.)36 b(Using)27 b(the)g(con)n(tin)n(uit)n(y)g(prop)r (erties)f(of)i(the)f(heat)g(\015o)n(w)h(\(A.4\))g(and)f(\(A.2\),)h(one) 456 2453 y(sees)f(that)h Fr(g)j Fu(is)c(a)h(con)n(tin)n(uous)f (function.)38 b(Therefore,)27 b(there)h(exists)f Fr(T)2692 2465 y Fp(n)2761 2453 y Fq(2)c Fu([)p Fr(t)2892 2465 y Fp(n)2938 2453 y Fr(;)14 b(\034)3011 2465 y Fp(n)3056 2453 y Fu(])28 b(for)g(whic)n(h)456 2553 y Fr(g)s Fu(\()p Fr(T)580 2565 y Fp(n)624 2553 y Fu(\))c(=)e Fr(\024=)p Fu(4,)27 b(that)h(is)456 2728 y(\(5.45\))1130 2657 y Fo(\015)1130 2707 y(\015)1176 2728 y Fu(\010)1236 2694 y Fp(T)1275 2702 y Fk(n)1320 2728 y Fu(\()s(\026)-45 b Fr(v)s Fu(\))1427 2657 y Fo(\015)1427 2707 y(\015)1474 2761 y Fp(W)1545 2744 y Fi(2)p Fk(;)p Fi(2)1625 2761 y Fs(\()p Fl(K)1709 2741 y Fk(!)1709 2781 y(d)1752 2761 y Fs(\))1805 2728 y Fu(=)1902 2672 y(1)p 1902 2709 42 4 v 1902 2785 a(4)1968 2728 y Fq(k)p Fr(u)2058 2694 y Fs(\(1\))2164 2728 y Fq(\000)18 b Fr(u)2295 2694 y Fs(\(0\))2384 2728 y Fq(k)2426 2743 y Fp(W)2497 2726 y Fi(2)p Fk(;)p Fi(2)2577 2743 y Fs(\()p Fl(K)2661 2723 y Fk(!)2661 2763 y(d)2703 2743 y Fs(\))2747 2728 y Fr(:)456 2900 y Fu(By)29 b(Corollary)f(5.10,)i(up)g(to)g(subsequence,)g(w)n(e)g(ma)n(y)g(assume) f(that)h(there)g(exists)g Fr(v)3150 2870 y Fp(])3209 2900 y Fq(2)d(C)35 b Fu(so)456 3000 y(that)1360 3101 y Fr(v)1403 3066 y Fp(])1457 3101 y Fu(=)23 b Fr(W)1635 3066 y Fs(2)p Fp(;)p Fs(2)1725 3101 y Fu(\()p FA(K)1832 3066 y Fp(d)1832 3121 y(!)1880 3101 y Fu(\))18 b Fq(\000)73 b Fu(lim)2013 3151 y Fp(n)p Fm(!)p Fs(+)p Fm(1)2252 3101 y Fu(\010)2312 3066 y Fp(T)2351 3074 y Fk(n)2396 3101 y Fu(\()s(\026)-45 b Fr(v)s Fu(\))14 b Fr(:)456 3267 y Fu(F)-7 b(rom)25 b(\(5.45\))o(,)h(w)n(e)g(ha)n(v)n(e)e(that)i Fr(v)1464 3237 y Fp(])1519 3267 y Fq(6)p Fu(=)c(0)p Fr(;)28 b(u)1747 3237 y Fs(\(1\))1850 3267 y Fq(\000)14 b Fr(u)1977 3237 y Fs(\(0\))2066 3267 y Fu(,)26 b(hence)g(\(5.43\))f(follo)n(ws,)g (th)n(us)h(concluding)456 3366 y(the)i(pro)r(of)f(of)g(the)h(desired)f (result.)1818 b Fj(\003)555 3518 y Fu(Since)40 b(t)n(w)n(o)e(disjoin)n (t)h(op)r(en)h(sets)f(cannot)g(co)n(v)n(er)e(an)i(in)n(terv)-5 b(al,)41 b(w)n(e)e(obtain)g(from)g(Theo-)456 3618 y(rem)27 b(5.16)f(that:)456 3735 y FA(Corollary)39 b(5.17.)44 b Fn(If)35 b FA(B)1290 3747 y Fs(0)1362 3735 y Fn(and)h FA(B)1597 3747 y Fs(1)1669 3735 y Fn(ar)l(e)f(op)l(en)h(in)f(the)g (standar)l(d)h(top)l(olo)l(gy)g(of)g Fu([0)p Fr(;)14 b Fu(1])p Fn(,)36 b(then,)456 3835 y(the)30 b(r)l(esults)e(in)i(The)l (or)l(em)h(5.16)g(hold)h(true.)555 3953 y Fu(Our)27 b(next)h(goal)e (will)i(b)r(e)g(to)f(sho)n(w)g(that,)h(indeed,)g FA(B)2222 3965 y Fs(0)2287 3953 y Fu(and)f FA(B)2516 3965 y Fs(1)2581 3953 y Fu(are)g(op)r(en.)456 4105 y(5.7.)40 b FA(The)34 b(basin)f(of)g(attraction)h(of)g(the)f(\014xed)g(p)s(oin)m(ts)g(of)g (the)g(heat)h(\015o)m(w.)41 b Fu(W)-7 b(e)29 b(no)n(w)456 4205 y(in)n(v)n(estigate)21 b(the)j(basin)e(of)h(attraction)f(of)h(the) h(heat)e(\015o)n(w,)i(with)f(the)h(aim)e(of)h(pro)n(ving)f(that)h(the) 456 4305 y(h)n(yp)r(othesis)j(of)h(Corollary)e(5.17)h(holds)g(true.)37 b(The)27 b(idea)g(of)g(gaining)f(compactness)g(from)h(the)456 4404 y(assumption)32 b(that)h(no)g(critical)f(p)r(oin)n(ts)h(\(but)g (trivial)g(ones\))f(exist)h(has)f(b)r(een)h(recen)n(tly)g(used)456 4504 y(b)n(y)24 b(sev)n(eral)f(authors)g(\(see,)j(e.g.,)e([CZES90)o(],) h(and)g(in)g(the)g(con)n(text)f(of)h(Aubry-Mather)e(theory)456 4604 y([KdlLR97)n(])28 b(and)g([CdlL98)o(]\).)555 4703 y(F)-7 b(or)27 b(an)n(y)g Fr(r)f(>)c Fu(0,)28 b(w)n(e)f(in)n(tro)r (duce)g(the)h(\\energy)e(ball")1418 4840 y Fq(B)1473 4852 y Fp(r)1532 4840 y Fu(:=)d Fq(f)p Fr(v)j Fq(2)d(C)32 b Fu(s.t.)2068 4819 y(~)2054 4840 y Fq(E)2098 4852 y Fp(!)2145 4840 y Fu(\()p Fr(v)s Fu(\))24 b Fq(\024)f Fr(r)r Fq(g)14 b Fr(:)555 4976 y Fu(Also,)39 b(giv)n(en)d(a)g(norm)h Fq(k)24 b(\001)g(k)1463 4988 y Fp(X)1563 4976 y Fu(on)37 b(some)f(space)g Fr(X)7 b Fu(,)38 b(w)n(e)f(denote)g(b)n(y)f Fr(B)2873 4946 y Fp(X)2973 4976 y Fu(the)h(standard)456 5076 y(\(closed\))27 b(ball.)37 b(That)27 b(is,)h(w)n(e)f(set)1218 5216 y Fr(B)1285 5181 y Fp(X)1281 5236 y(r)1349 5216 y Fu(\()p Fr(x)1428 5228 y Fs(0)1466 5216 y Fu(\))c(:=)g Fq(f)p Fr(x)g Fq(2)g Fr(X)34 b Fu(s.t.)j Fq(k)p Fr(x)18 b Fq(\000)g Fr(x)2310 5228 y Fs(0)2348 5216 y Fq(k)2390 5228 y Fp(X)2476 5216 y Fq(\024)23 b Fr(r)r Fq(g)14 b Fr(:)p eop %%Page: 30 30 30 29 bop 456 251 a Fs(30)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fu(W)g(e)25 b(also)f(set)h Fr(B)954 420 y Fp(X)950 471 y(r)1040 450 y Fu(:=)e Fr(B)1218 420 y Fp(X)1214 471 y(r)1281 450 y Fu(\(0\).)36 b(W)-7 b(e)26 b(no)n(w)e(p)r(oin)n(t)h(out)g(an)g (inclusion)f(of)h(balls,)h(whic)n(h)e(is)h(needed)456 550 y(in)i(what)h(follo)n(ws:)456 668 y FA(Lemma)h(5.18.)40 b Fn(Ther)l(e)31 b(exists)e(a)h(p)l(ositive)h(universal)g(c)l(onstant)e Fr(c)g Fn(for)i(which)1573 828 y Fq(C)23 b(\\)18 b Fr(B)1780 783 y Fp(W)1851 758 y Fi(1)p Fk(;)p Fi(2)1932 783 y Fs(\()p Fl(K)2016 758 y Fk(d)2016 800 y(!)2058 783 y Fs(\))1776 860 y Fp(c)1806 819 y Fm(p)p 1861 819 33 3 v 41 x Fp(r)2112 828 y Fq(\022)k(B)2254 840 y Fp(r)2304 828 y Fr(;)456 977 y Fn(for)30 b(any)g Fr(r)c(>)d Fu(0)p Fn(.)456 1129 y(Pr)l(o)l(of.)43 b Fu(T)-7 b(ak)n(e)26 b Fr(v)h Fq(2)c(C)32 b Fu(and)c(let)g Fr(C)h Fq(\025)23 b Fu(1)k(b)r(e)h(suitably)f(large.) 36 b(Then,)470 1246 y(~)456 1267 y Fq(E)500 1279 y Fp(!)547 1267 y Fu(\()p Fr(v)s Fu(\))24 b(=)849 1434 y(=)997 1321 y Fo(Z)1043 1509 y Fl(K)1101 1493 y Fk(d)1101 1526 y(!)1171 1378 y Fu(1)p 1171 1415 42 4 v 1171 1491 a(2)1237 1434 y Fr(a)1281 1446 y Fp(i;j)1359 1434 y Fr(@)1403 1446 y Fp(i)1431 1434 y Fr(v)s(@)1518 1446 y Fp(j)1553 1434 y Fr(v)d Fu(+)d Fr(F)12 b Fu(\()p Fr(x;)i(u)1926 1399 y Fs(\(0\))2034 1434 y Fu(+)k Fr(v)s Fu(\))h Fq(\000)f Fr(F)12 b Fu(\()p Fr(x;)i(u)2523 1399 y Fs(\(0\))2612 1434 y Fu(\))19 b Fq(\000)f Fr(@)2790 1446 y Fp(\026)2835 1434 y Fr(F)12 b Fu(\()p Fr(x;)i(u)3064 1399 y Fs(\(0\))3153 1434 y Fu(\))g Fr(v)j(dx)24 b Fq(\024)849 1631 y(\024)83 b Fu(\(\003)18 b(+)g Fr(C)6 b Fu(\))p Fq(k)p Fr(v)s Fq(k)1412 1597 y Fs(2)1412 1655 y Fp(W)1483 1639 y Fi(1)p Fk(;)p Fi(2)1563 1655 y Fs(\()p Fl(K)1647 1639 y Fk(d)1647 1672 y(!)1690 1655 y Fs(\))1733 1631 y Fr(;)456 1775 y Fu(pro)n(ving)26 b(that)i(the)g(desired)f(result)g(holds.)1575 b Fj(\003)555 1927 y Fu(Let)36 b(us)g(no)n(w)f(study)h(the)g(connected)f(comp)r(onen) n(ts)g(of)h(the)g(energy)e(ball.)61 b(Here)36 b(and)f(in)456 2027 y(the)24 b(sequel,)h(\\connected)f(comp)r(onen)n(ts")f(is)i(short) e(for)h(\\)p Fn(p)l(ath-c)l(onne)l(cte)l(d)j(c)l(omp)l(onents)f(in)h (the)456 2126 y Fr(W)546 2096 y Fs(2)p Fp(;)p Fs(2)636 2126 y Fu(\()p FA(K)743 2096 y Fp(d)743 2147 y(!)791 2126 y Fu(\))p Fn({top)l(olo)l(gy)p Fu(".)456 2248 y FA(Lemma)k(5.19.)40 b Fn(L)l(et)31 b(us)f(assume)h(that)g Fu(0)f Fn(and)i Fr(u)2034 2217 y Fs(\(1\))2141 2248 y Fq(\000)19 b Fr(u)2273 2217 y Fs(\(0\))2393 2248 y Fn(ar)l(e)31 b(the)g(only)h(functions)f Fr(v)j Fn(c)l(on-)456 2348 y(taine)l(d)c(in)g Fq(C)k Fn(for)d(which)1264 2327 y Fu(~)1250 2348 y Fq(E)1294 2360 y Fp(!)1342 2348 y Fu(\()p Fr(v)s Fu(\))24 b(=)e(0)p Fn(.)555 2451 y(Then,)37 b(ther)l(e)e(exists) f Fr(r)1284 2463 y Fs(0)1354 2451 y Fr(>)d Fu(0)j Fn(so)h(that,)h(for)g (any)e Fr(r)h Fq(2)d Fu(\(0)p Fr(;)14 b(r)2446 2463 y Fs(0)2484 2451 y Fu(])p Fn(,)36 b Fu(0)e Fn(and)h Fr(u)2858 2421 y Fs(\(1\))2969 2451 y Fq(\000)21 b Fr(u)3103 2421 y Fs(\(0\))3226 2451 y Fn(ar)l(e)35 b(in)456 2551 y(di\013er)l(ent)30 b(c)l(onne)l(cte)l(d)f(c)l(omp)l(onents)h(of)g Fq(B)1761 2563 y Fp(r)1798 2551 y Fn(.)456 2703 y(Pr)l(o)l(of.)43 b Fu(The)27 b(argumen)n(t)g(is)g(b)n(y)h(con)n(tradiction.)555 2803 y(Supp)r(ose)j(that,)h(con)n(trary)c(to)j(the)g(conclusions)e(of)i (the)g(Lemma,)g(for)f(an)n(y)g Fr(h)e Fq(2)g FA(N)p Fu(,)k(there)456 2902 y(exists)27 b(a)g(path)h Fr(\015)991 2914 y Fp(h)1057 2902 y Fq(2)23 b Fr(C)6 b Fu(\([0)p Fr(;)14 b Fu(1])p Fr(;)g(W)1526 2872 y Fs(2)p Fp(;)p Fs(2)1616 2902 y Fu(\()p FA(K)1723 2872 y Fp(d)1723 2923 y(!)1771 2902 y Fu(\)\))28 b(so)f(that)1456 3039 y Fr(\015)1499 3051 y Fp(h)1542 3039 y Fu(\(0\))c(=)g(0)14 b Fr(;)1456 3171 y(\015)1499 3183 y Fp(h)1542 3171 y Fu(\(1\))23 b(=)g Fr(u)1807 3136 y Fs(\(1\))1914 3171 y Fq(\000)18 b Fr(u)2045 3136 y Fs(\(0\))2300 3171 y Fu(and)1470 3313 y(~)1456 3334 y Fq(E)1500 3346 y Fp(!)1548 3334 y Fu(\()p Fr(\015)1623 3346 y Fp(h)1666 3334 y Fu(\()p Fr(\033)s Fu(\)\))25 b Fq(\024)1937 3278 y Fu(1)p 1934 3315 48 4 v 1934 3391 a Fr(h)2006 3334 y(;)180 b Fq(8)p Fr(\033)25 b Fq(2)f Fu([0)p Fr(;)14 b Fu(1])g Fr(:)-2155 b Fu(\(5.46\))456 3504 y(Let)31 b(\000)660 3516 y Fp(h)703 3504 y Fu(\()p Fr(\033)s Fu(\))g(:=)e(\010)1025 3474 y Fs(1)1063 3504 y Fu(\()p Fr(\015)1138 3516 y Fp(h)1181 3504 y Fu(\()p Fr(\033)s Fu(\)\),)34 b(for)d(an)n(y)g Fr(\033)i Fq(2)d Fu([0)p Fr(;)14 b Fu(1].)48 b(Notice)32 b(that,)h(b)n(y)g(\(5.46\))d (and)i(\(5.40\))o(,)h(w)n(e)456 3604 y(ha)n(v)n(e)26 b(that)456 3759 y(\(5.47\))1439 3738 y(~)1425 3759 y Fq(E)1469 3771 y Fp(!)1517 3759 y Fu(\(\000)1601 3771 y Fp(h)1644 3759 y Fu(\()p Fr(\033)s Fu(\)\))e Fq(\024)1916 3738 y Fu(~)1902 3759 y Fq(E)1946 3771 y Fp(!)1994 3759 y Fu(\()p Fr(\015)2069 3771 y Fp(h)2112 3759 y Fu(\()p Fr(\033)s Fu(\)\))h Fq(\024)2383 3702 y Fu(1)p 2380 3740 V 2380 3816 a Fr(h)2452 3759 y(:)456 3923 y Fu(Exploiting)i(\(5.46\))g (and)h(\(5.16\))o(,)g(one)f(sees)g(that)1317 4067 y(\000)1369 4079 y Fp(h)1412 4067 y Fu(\(0\))c(=)g(0)k(and)g(\000)1911 4079 y Fp(h)1954 4067 y Fu(\(1\))d(=)e Fr(u)2219 4033 y Fs(\(1\))2326 4067 y Fq(\000)c Fr(u)2457 4033 y Fs(\(0\))2560 4067 y Fr(:)456 4203 y Fu(Also,)27 b(since)g Fr(\015)912 4215 y Fp(h)978 4203 y Fq(2)d Fr(C)6 b Fu(\([0)p Fr(;)14 b Fu(1])p Fr(;)g(W)1448 4173 y Fs(2)p Fp(;)p Fs(2)1537 4203 y Fu(\()p FA(K)1644 4173 y Fp(d)1644 4224 y(!)1692 4203 y Fu(\)\),)29 b(Prop)r(osition)d(5.12)g(implies)i(that)456 4344 y(\(5.48\))814 b(\000)1535 4356 y Fp(h)1601 4344 y Fq(2)24 b Fr(C)6 b Fu(\([0)p Fr(;)14 b Fu(1])p Fr(;)g(W)2071 4310 y Fs(2)p Fp(;)p Fs(2)2160 4344 y Fu(\()p FA(K)2267 4310 y Fp(d)2267 4364 y(!)2315 4344 y Fu(\)\))g Fr(:)456 4480 y Fu(Let)1321 4617 y Fr(\024)83 b Fu(:=)g Fq(k)p Fr(u)1713 4583 y Fs(\(1\))1820 4617 y Fq(\000)18 b Fr(u)1951 4583 y Fs(\(0\))2039 4617 y Fq(k)2081 4634 y Fp(W)2152 4617 y Fi(2)p Fk(;)p Fi(2)2232 4634 y Fs(\()p Fl(K)2316 4617 y Fk(d)2316 4650 y(!)2358 4634 y Fs(\))2555 4617 y Fu(and)1212 4746 y Fr(g)s Fu(\()p Fr(\033)s Fu(\))83 b(:=)g Fq(k)p Fu(\000)1717 4758 y Fp(h)1759 4746 y Fu(\()p Fr(\033)s Fu(\))p Fq(k)1915 4763 y Fp(W)1986 4746 y Fi(2)p Fk(;)p Fi(2)2067 4763 y Fs(\()p Fl(K)2151 4746 y Fk(d)2151 4779 y(!)2194 4763 y Fs(\))2237 4746 y Fr(:)456 4887 y Fu(W)-7 b(e)35 b(ha)n(v)n(e)f(that)i Fr(\024)f(>)h Fu(0,)h Fr(g)s Fu(\(0\))e(=)h(0)e(and)h Fr(g)s Fu(\(1\))h(=)f Fr(\024)p Fu(.)60 b(Also,)37 b(from)e(\(5.48\))o(,)i(it)f(follo)n(ws)e (that)456 4986 y Fr(g)25 b Fq(2)f Fr(C)6 b Fu(\([0)p Fr(;)14 b Fu(1])p Fr(;)g FA(R)p Fu(\),)27 b(therefore)f(there)i(exists) f Fr(\033)1892 4998 y Fp(h)1959 4986 y Fq(2)c Fu([0)p Fr(;)14 b Fu(1],)27 b(so)g(that)h Fr(g)s Fu(\()p Fr(\033)2658 4998 y Fp(h)2701 4986 y Fu(\))23 b(=)g Fr(\024=)p Fu(2,)k(that)g(is)456 5192 y(\(5.49\))494 b Fq(k)p Fu(\000)1257 5204 y Fp(h)1299 5192 y Fu(\()p Fr(\033)1378 5204 y Fp(h)1422 5192 y Fu(\))p Fq(k)1496 5209 y Fp(W)1567 5192 y Fi(2)p Fk(;)p Fi(2)1647 5209 y Fs(\()p Fl(K)1731 5192 y Fk(d)1731 5225 y(!)1774 5209 y Fs(\))1827 5192 y Fu(=)1924 5126 y Fq(k)p Fr(u)2014 5096 y Fs(\(1\))2121 5126 y Fq(\000)18 b Fr(u)2252 5096 y Fs(\(0\))2341 5126 y Fq(k)2383 5143 y Fp(W)2454 5126 y Fi(2)p Fk(;)p Fi(2)2533 5143 y Fs(\()p Fl(K)2617 5126 y Fk(d)2617 5159 y(!)2660 5143 y Fs(\))p 1924 5173 766 4 v 2287 5249 a Fu(2)2714 5192 y Fr(:)p eop %%Page: 31 31 31 30 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(31)456 450 y Fu(Moreo)n(v)n(er,)27 b(\000)898 462 y Fp(h)941 450 y Fu(\()p Fr(\033)1020 462 y Fp(h)1063 450 y Fu(\))g Fq(2)f Fu(\010)1263 420 y Fs(1)1300 450 y Fu(\()p Fq(C)5 b Fu(\),)30 b(whic)n(h)f(is)g(compact)g(in)h Fr(W)2313 420 y Fs(2)p Fp(;)p Fs(2)2403 450 y Fu(\()p FA(K)2510 420 y Fp(d)2510 471 y(!)2558 450 y Fu(\),)g(due)g(to)f(Corollary)e (5.10.)456 550 y(Therefore,)e(p)r(ossibly)g(taking)h(subsequences,)g(w) n(e)f(ma)n(y)h(assume)f(that)h(there)g(exists)j(^)-45 b Fr(v)26 b Fq(2)d(C)31 b Fu(so)456 649 y(that)1380 762 y(^)-45 b Fr(v)26 b Fu(=)d Fr(W)1621 728 y Fs(2)p Fp(;)p Fs(2)1711 762 y Fu(\()p FA(K)1818 728 y Fp(d)1818 783 y(!)1866 762 y Fu(\))c Fq(\000)71 b Fu(lim)2000 817 y Fp(h)p Fm(!)p Fs(+)p Fm(1)2236 762 y Fu(\000)2288 774 y Fp(h)2331 762 y Fu(\()p Fr(\033)2410 774 y Fp(h)2454 762 y Fu(\))14 b Fr(:)456 945 y Fu(Recalling)28 b(\(5.49\))o(,)g(w)n(e) f(get)g(that)k(^)-45 b Fr(v)26 b Fq(6)p Fu(=)d(0)p Fr(;)k(u)1815 914 y Fs(\(1\))1922 945 y Fq(\000)18 b Fr(u)2053 914 y Fs(\(0\))2142 945 y Fu(.)37 b(Therefore,)26 b(b)n(y)i(h)n(yp)r (othesis,)456 1100 y(\(5.50\))842 b Fq(E)1555 1112 y Fp(!)1603 1100 y Fu(\()s(^)-45 b Fr(v)22 b Fu(+)c Fr(u)1828 1065 y Fs(\(0\))1916 1100 y Fu(\))24 b Fr(>)e Fq(E)2103 1112 y Fp(!)2151 1100 y Fu(\()p Fr(u)2231 1065 y Fs(\(0\))2320 1100 y Fu(\))14 b Fr(:)456 1248 y Fu(On)27 b(the)h(other)f(hand,)h (passing)e(to)i(the)g(limit)g(in)g(\(5.47\))o(,)g(w)n(e)f(gather)f (that)1511 1403 y Fq(E)1555 1415 y Fp(!)1603 1403 y Fu(\()s(^)-45 b Fr(v)22 b Fu(+)c Fr(u)1828 1368 y Fs(\(0\))1916 1403 y Fu(\))24 b Fq(\024)e(E)2103 1415 y Fp(!)2151 1403 y Fu(\()p Fr(u)2231 1368 y Fs(\(0\))2320 1403 y Fu(\))14 b Fr(;)456 1551 y Fu(whic)n(h)27 b(con)n(tradicts)f(\(5.50\).)2024 b Fj(\003)555 1719 y Fu(W)-7 b(e)37 b(can)f(no)n(w)f(sho)n(w)h(that,)j (if)d(there)g(are)g(no)g(critical)f(p)r(oin)n(ts)i(of)f(the)g(phase)g (transition)456 1819 y(equation)24 b(with)h(energy)e(close)h(to)g(the)h (one)g(of)f(the)h(minima)g(\(except,)h(of)e(course,)g(the)h(minima)456 1918 y(themselv)n(es\),)38 b(then)e(there)g(is)g(a)g Fr(W)1616 1888 y Fs(2)p Fp(;)p Fs(2)1706 1918 y Fu(-ball)f(around)g (the)i(minima)f(whic)n(h)g(is)g(con)n(tained)f(in)456 2018 y(their)27 b(basin)g(of)h(attraction)f(under)g(the)h(heat)g (equation.)36 b(More)27 b(precisely)-7 b(,)26 b(w)n(e)i(ha)n(v)n(e)e (that:)456 2142 y FA(Prop)s(osition)32 b(5.20.)41 b Fn(L)l(et)30 b(us)h(assume)g(that)h Fu(0)f Fn(and)g Fr(u)2214 2112 y Fs(\(1\))2323 2142 y Fq(\000)19 b Fr(u)2455 2112 y Fs(\(0\))2575 2142 y Fn(ar)l(e)31 b(the)h(only)g(functions)f Fr(v)456 2241 y Fn(c)l(ontaine)l(d)c(in)g Fq(C)k Fn(for)d(which)g Fq(E)1402 2253 y Fp(!)1450 2241 y Fu(\()p Fr(u)1530 2211 y Fs(\(0\))1631 2241 y Fu(+)12 b Fr(v)s Fu(\))23 b(=)f Fq(E)1937 2253 y Fp(!)1985 2241 y Fu(\()p Fr(u)2065 2211 y Fs(\(0\))2154 2241 y Fu(\))27 b Fn(holds.)39 b(L)l(et)27 b Fr(r)2636 2253 y Fs(0)2700 2241 y Fn(b)l(e)g(as)g(in)g(L)l(emma)g (5.19)456 2341 y(and)j Fr(r)c Fq(2)d Fu(\(0)p Fr(;)14 b(r)906 2353 y Fs(0)943 2341 y Fu(])p Fn(.)39 b(L)l(et)29 b(us)h(assume)f(that:)39 b(if)31 b Fr(V)42 b Fq(2)23 b(C)g(\\)c(B)2221 2353 y Fp(r)2287 2341 y Fn(satis\014es)1073 2514 y Fr(@)1117 2526 y Fp(i)1159 2514 y Fu(\()p Fr(a)1235 2526 y Fp(ij)1293 2514 y Fr(@)1337 2526 y Fp(j)1372 2514 y Fr(V)g Fu(\))g Fq(\000)f Fr(f)1636 2421 y Fo(\020)1686 2514 y Fr(x;)c(u)1818 2479 y Fs(\(0\))1926 2514 y Fu(+)k Fr(V)2075 2421 y Fo(\021)2143 2514 y Fu(+)g Fr(f)2290 2421 y Fo(\020)2339 2514 y Fr(x;)c(u)2471 2479 y Fs(\(0\))2560 2421 y Fo(\021)2647 2514 y Fu(=)36 b(0)14 b Fr(;)456 2698 y Fn(then)33 b Fr(V)49 b Fq(2)30 b(f)p Fu(0)p Fr(;)14 b(u)995 2668 y Fs(\(1\))1104 2698 y Fq(\000)20 b Fr(u)1237 2668 y Fs(\(0\))1326 2698 y Fq(g)p Fn(.)50 b(Then,)35 b(ther)l(e)f(exists)f(a)h(universal)g(c)l(onstant)f Fr(c)g Fn(such)h(that)f(the)456 2797 y(fol)t(lowing)f(holds:)661 2945 y Fq(\017)41 b Fn(if)31 b Fr(v)26 b Fq(2)e(C)f(\\)18 b Fr(B)1177 2900 y Fp(W)1248 2875 y Fi(2)p Fk(;)p Fi(2)1329 2900 y Fs(\()p Fl(K)1413 2875 y Fk(d)1413 2916 y(!)1455 2900 y Fs(\))1173 2977 y Fp(c)1203 2935 y Fm(p)p 1258 2935 33 3 v 42 x Fp(r)1485 2945 y Fn(,)31 b(then)e Fr(W)1815 2915 y Fs(2)p Fp(;)p Fs(2)1905 2945 y Fu(\()p FA(K)2012 2915 y Fp(d)2012 2966 y(!)2060 2945 y Fu(\))19 b Fq(\000)65 b Fu(lim)2194 2995 y Fp(t)p Fm(!)p Fs(+)p Fm(1)2416 2945 y Fu(\010)2476 2911 y Fp(t)2506 2945 y Fu(\()p Fr(v)s Fu(\))23 b(=)g(0)p Fn(;)661 3118 y Fq(\017)41 b Fn(if)25 b Fr(v)h Fq(2)e(C)10 b(\\)c Fr(B)1146 3072 y Fp(W)1217 3047 y Fi(2)p Fk(;)p Fi(2)1297 3072 y Fs(\()p Fl(K)1381 3047 y Fk(d)1381 3089 y(!)1424 3072 y Fs(\))1142 3150 y Fp(c)1172 3108 y Fm(p)p 1226 3108 V 1226 3150 a Fp(r)1454 3118 y Fu(\()p Fr(u)1534 3087 y Fs(\(1\))1628 3118 y Fq(\000)g Fr(u)1747 3087 y Fs(\(0\))1835 3118 y Fu(\))p Fn(,)26 b(then)e Fr(W)2187 3087 y Fs(2)p Fp(;)p Fs(2)2277 3118 y Fu(\()p FA(K)2384 3087 y Fp(d)2384 3138 y(!)2432 3118 y Fu(\))6 b Fq(\000)51 b Fu(lim)2541 3167 y Fp(t)p Fm(!)p Fs(+)p Fm(1)2762 3118 y Fu(\010)2822 3083 y Fp(t)2851 3118 y Fu(\()p Fr(v)s Fu(\))24 b(=)f Fr(u)3118 3083 y Fs(\(1\))3212 3118 y Fq(\000)6 b Fr(u)3331 3083 y Fs(\(0\))3419 3118 y Fn(.)456 3317 y(Pr)l(o)l(of.)43 b Fu(W)-7 b(e)20 b(will)g(fo)r(cus)f(on)h(the)g(pro)r(of)f(of)g(the)h(\014rst)f(claim,)j (the)e(pro)r(of)f(of)g(the)h(second)f(one)g(b)r(eing)456 3417 y(iden)n(tical)30 b(Recalling)g(Lemma)g(5.19,)g(w)n(e)g(denote)h (b)n(y)f Fq(B)2230 3387 y Fp(?)2227 3438 y(r)2298 3417 y Fu(b)r(e)h(the)g(connected)f(comp)r(onen)n(t)h(of)456 3520 y Fq(B)511 3532 y Fp(r)576 3520 y Fu(con)n(taining)c(0)i(and)f (not)h(con)n(taining)f Fr(u)1815 3489 y Fs(\(1\))1923 3520 y Fq(\000)18 b Fr(u)2054 3489 y Fs(\(0\))2143 3520 y Fu(.)41 b(If)29 b Fr(c)f Fu(is)h(suitably)g(small,)f(Lemma)h(5.18)456 3619 y(implies)e(that)1573 3756 y Fq(C)c(\\)18 b Fr(B)1780 3711 y Fp(W)1851 3686 y Fi(2)p Fk(;)p Fi(2)1932 3711 y Fs(\()p Fl(K)2016 3686 y Fk(d)2016 3727 y(!)2058 3711 y Fs(\))1776 3788 y Fp(c)1806 3746 y Fm(p)p 1861 3746 V 42 x Fp(r)2112 3756 y Fq(\022)k(B)2254 3768 y Fp(r)2304 3756 y Fr(:)456 3940 y Fu(Also,)30 b(notice)f(that)i(0)26 b Fq(2)h(C)e(\\)20 b Fr(B)1459 3895 y Fp(W)1530 3870 y Fi(2)p Fk(;)p Fi(2)1610 3895 y Fs(\()p Fl(K)1694 3870 y Fk(d)1694 3911 y(!)1737 3895 y Fs(\))1455 3972 y Fp(c)1485 3930 y Fm(p)p 1539 3930 V 1539 3972 a Fp(r)1767 3940 y Fu(,)30 b(whic)n(h)g(is)g(a)f(con)n(v)n(ex)g(\(and)h(hence)f (connected\))h(set)456 4064 y(in)d Fr(W)642 4034 y Fs(2)p Fp(;)p Fs(2)732 4064 y Fu(\()p FA(K)839 4034 y Fp(d)839 4085 y(!)888 4064 y Fu(\),)h(therefore,)e(from)i(the)g(ab)r(o)n(v)n(e)e (inclusion,)h(w)n(e)h(get)f(that)1571 4241 y Fq(C)c(\\)c Fr(B)1779 4195 y Fp(W)1850 4170 y Fi(2)p Fk(;)p Fi(2)1930 4195 y Fs(\()p Fl(K)2014 4170 y Fk(d)2014 4212 y(!)2056 4195 y Fs(\))1775 4273 y Fp(c)1805 4231 y Fm(p)p 1859 4231 V 1859 4273 a Fp(r)2110 4241 y Fq(\022)j(B)2255 4206 y Fp(?)2252 4261 y(r)2306 4241 y Fr(:)456 4442 y Fu(T)-7 b(ak)n(e)23 b(no)n(w)g Fr(v)j Fq(2)e(C)16 b(\\)11 b Fr(B)1160 4397 y Fp(W)1231 4372 y Fi(2)p Fk(;)p Fi(2)1311 4397 y Fs(\()p Fl(K)1395 4372 y Fk(d)1395 4413 y(!)1438 4397 y Fs(\))1156 4474 y Fp(c)1186 4432 y Fm(p)p 1240 4432 V 1240 4474 a Fp(r)1468 4442 y Fu(.)36 b(By)24 b(Corollary)d(5.10) i(and)h(Prop)r(osition)e(5.15,)i(w)n(e)f(ha)n(v)n(e)g(that)456 4553 y(there)k(exist)g(a)h(sequence)f Fr(t)1308 4565 y Fp(n)1376 4553 y Fq(\000)-14 b(!)23 b Fu(+)p Fq(1)k Fu(and)h Fr(v)1913 4523 y Fm(\003)1974 4553 y Fq(2)23 b(C)5 b Fu(,)28 b(suc)n(h)f(that)1142 4701 y Fr(W)1232 4667 y Fs(2)p Fp(;)p Fs(2)1322 4701 y Fu(\()p FA(K)1429 4667 y Fp(d)1429 4722 y(!)1477 4701 y Fu(\))19 b Fq(\000)72 b Fu(lim)1611 4751 y Fp(n)p Fm(!)p Fs(+)p Fm(1)1849 4701 y Fu(\010)1909 4667 y Fp(t)1934 4675 y Fk(n)1979 4701 y Fu(\()p Fr(v)s Fu(\))37 b(=)g Fr(v)2268 4667 y Fm(\003)2472 4701 y Fu(and)-2150 b(\(5.51\))1142 4895 y Fr(@)1186 4907 y Fp(i)1227 4895 y Fu(\()q Fr(a)1304 4907 y Fp(ij)1362 4895 y Fr(@)1406 4907 y Fp(j)1441 4895 y Fr(v)1484 4861 y Fm(\003)1522 4895 y Fu(\))19 b Fq(\000)f Fr(f)1719 4803 y Fo(\020)1769 4895 y Fr(x;)c(u)1901 4861 y Fs(\(0\))2009 4895 y Fu(+)k Fr(v)2135 4861 y Fm(\003)2173 4803 y Fo(\021)2241 4895 y Fu(+)g Fr(f)2387 4803 y Fo(\020)2437 4895 y Fr(x;)c(u)2569 4861 y Fs(\(0\))2658 4803 y Fo(\021)2744 4895 y Fu(=)37 b(0)14 b Fr(:)-2469 b Fu(\(5.52\))456 5068 y(Notice)27 b(also)g(that,)h(from)f(Corollary)e(5.6)i(and)g(\(5.40\))o(,)1704 5216 y(\010)1764 5181 y Fp(t)1793 5216 y Fu(\()p Fq(B)1883 5181 y Fp(?)1880 5236 y(r)1921 5216 y Fu(\))c Fq(\022)g(B)2122 5181 y Fp(?)2119 5236 y(r)2173 5216 y Fr(;)p eop %%Page: 32 32 32 31 bop 456 251 a Fs(32)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fu(th)n(us)27 b Fr(v)681 420 y Fm(\003)743 450 y Fq(2)c(C)g(\\)c(B)1020 420 y Fp(?)1017 471 y(r)1057 450 y Fu(.)37 b(Consequen)n(tly)-7 b(,)27 b(from)g(\(5.52\))o(,)h(our)f(h)n(yp)r(otheses)g(imply)h(that) 1572 617 y Fr(v)1615 582 y Fm(\003)1676 617 y Fq(2)c(f)p Fu(0)p Fr(;)14 b(u)1924 582 y Fs(\(1\))2030 617 y Fq(\000)k Fr(u)2161 582 y Fs(\(0\))2250 617 y Fq(g)c Fr(:)456 779 y Fu(Since)27 b Fr(u)720 749 y Fs(\(1\))827 779 y Fq(\000)19 b Fr(u)959 749 y Fs(\(0\))1070 779 y Fq(62)24 b(B)1207 749 y Fp(?)1204 800 y(r)1272 779 y Fu(b)n(y)j(construction,)g(w)n(e)g (deduce)h(that)g Fr(v)2508 749 y Fm(\003)2569 779 y Fu(=)23 b(0,)k(that)h(is)1369 942 y Fr(W)1459 908 y Fs(2)p Fp(;)p Fs(2)1549 942 y Fu(\()p FA(K)1656 908 y Fp(d)1656 963 y(!)1704 942 y Fu(\))19 b Fq(\000)73 b Fu(lim)1838 992 y Fp(n)p Fm(!)p Fs(+)p Fm(1)2076 942 y Fu(\010)2136 908 y Fp(t)2161 916 y Fk(n)2206 942 y Fu(\()p Fr(v)s Fu(\))38 b(=)e(0)14 b Fr(:)456 1133 y Fu(W)-7 b(e)28 b(no)n(w)f(sho)n(w)f(that,) i(actually)-7 b(,)456 1292 y(\(5.53\))729 b Fr(W)1488 1258 y Fs(2)p Fp(;)p Fs(2)1578 1292 y Fu(\()p FA(K)1685 1258 y Fp(d)1685 1312 y(!)1733 1292 y Fu(\))18 b Fq(\000)65 b Fu(lim)1866 1342 y Fp(t)p Fm(!)p Fs(+)p Fm(1)2089 1292 y Fu(\010)2149 1258 y Fp(t)2178 1292 y Fu(\()p Fr(v)s Fu(\))38 b(=)e(0)14 b Fr(;)456 1487 y Fu(whic)n(h)25 b(will)h(indeed)h(pro)n(v)n(e)d(the)i(\014rst)g(claim.)36 b(If)c(\(5.53\))25 b(w)n(ere)g(false,)h(there)g(w)n(ould)f(exist)h (some)456 1587 y Fr(\021)g(>)d Fu(0)k(and)g(another)g(div)n(erging)f (sequence,)h(sa)n(y)f Fr(T)2065 1599 y Fp(n)2110 1587 y Fu(,)i(for)f(whic)n(h)456 1746 y(\(5.54\))864 b Fq(k)p Fu(\010)1635 1712 y Fp(T)1674 1720 y Fk(n)1718 1746 y Fu(\()p Fr(v)s Fu(\))p Fq(k)1867 1763 y Fp(W)1938 1746 y Fi(2)p Fk(;)p Fi(2)2019 1763 y Fs(\()p Fl(K)2103 1746 y Fk(d)2103 1779 y(!)2145 1763 y Fs(\))2199 1746 y Fq(\025)22 b Fr(\021)17 b(:)456 1919 y Fu(But)25 b(no)n(w,)h(as)f(b)r(efore,)h(w)n (e)f(can)g(extract)g(a)g(subsequence)g Fr(T)2319 1931 y Fp(n)2360 1940 y Fk(k)2426 1919 y Fu(for)g(whic)n(h)g(\010)2846 1887 y Fp(T)2885 1895 y Fk(n)2922 1910 y(k)2967 1919 y Fu(\()p Fr(v)s Fu(\))h(con)n(v)n(erges)456 2018 y(and,)k(exactly)g (as)g(done)g(here)f(ab)r(o)n(v)n(e,)h(w)n(e)g(can)g(sho)n(w)f(that)i (it)f(m)n(ust)h(con)n(v)n(erge)c(to)k(0,)f(con)n(tra-)456 2118 y(dicting)e(\(5.54\).)37 b(This)27 b(pro)n(v)n(es)f(\(5.53\))h (and)g(completes)g(the)h(pro)r(of)f(of)h(the)g(\(\014rst\))g(claim.)93 b Fj(\003)555 2303 y Fu(W)-7 b(e)29 b(can)f(no)n(w)f(sho)n(w)h(that)g (the)h(h)n(yp)r(otheses)e(of)h(Corollary)e(5.17)h(are)g(ful\014lled)i (in)g(the)f(case)456 2402 y(that)g(there)f(are)g(no)h(critical)f(p)r (oin)n(ts)h(of)g(the)g(phase)g(transition)f(equation)g(with)i(energy)d (close)456 2502 y(to)h(the)h(one)f(of)h(the)g(minima:)456 2632 y FA(Prop)s(osition)k(5.21.)41 b Fn(L)l(et)30 b(us)h(assume)g (that)h Fu(0)f Fn(and)g Fr(u)2214 2601 y Fs(\(1\))2323 2632 y Fq(\000)19 b Fr(u)2455 2601 y Fs(\(0\))2575 2632 y Fn(ar)l(e)31 b(the)h(only)g(functions)f Fr(v)456 2731 y Fn(c)l(ontaine)l(d)26 b(in)g Fq(C)31 b Fn(for)c(which)g Fq(E)1398 2743 y Fp(!)1446 2731 y Fu(\()p Fr(u)1526 2701 y Fs(\(0\))1626 2731 y Fu(+)10 b Fr(v)s Fu(\))23 b(=)g Fq(E)1931 2743 y Fp(!)1979 2731 y Fu(\()p Fr(u)2059 2701 y Fs(\(0\))2148 2731 y Fu(\))j Fn(holds.)39 b(L)l(et)25 b Fr(r)h(>)d Fu(0)i Fn(and)i(let)f(us)f(assume)456 2831 y(that:)38 b(if)31 b Fr(V)42 b Fq(2)23 b(B)963 2843 y Fp(r)1029 2831 y Fn(satis\014es)1073 3010 y Fr(@)1117 3022 y Fp(i)1159 3010 y Fu(\()p Fr(a)1235 3022 y Fp(ij)1293 3010 y Fr(@)1337 3022 y Fp(j)1372 3010 y Fr(V)c Fu(\))g Fq(\000)f Fr(f)1636 2918 y Fo(\020)1686 3010 y Fr(x;)c(u)1818 2976 y Fs(\(0\))1926 3010 y Fu(+)k Fr(V)2075 2918 y Fo(\021)2143 3010 y Fu(+)g Fr(f)2290 2918 y Fo(\020)2339 3010 y Fr(x;)c(u)2471 2976 y Fs(\(0\))2560 2918 y Fo(\021)2647 3010 y Fu(=)36 b(0)14 b Fr(;)456 3206 y Fn(then)33 b Fr(V)48 b Fq(2)30 b(f)p Fu(0)p Fr(;)14 b(u)994 3176 y Fs(\(1\))1103 3206 y Fq(\000)21 b Fr(u)1237 3176 y Fs(\(0\))1325 3206 y Fq(g)p Fn(.)50 b(L)l(et)32 b FA(B)1656 3218 y Fs(0)1727 3206 y Fn(and)i FA(B)1960 3218 y Fs(1)2030 3206 y Fn(b)l(e)g(as)g (de\014ne)l(d)f(in)40 b Fu(\(5.42\))o Fn(.)50 b(Then,)35 b FA(B)3242 3218 y Fs(0)3313 3206 y Fn(and)456 3306 y FA(B)524 3318 y Fs(1)590 3306 y Fn(ar)l(e)c(op)l(en)f(in)g(the)f (standar)l(d)i(top)l(olo)l(gy)g(of)g Fu([0)p Fr(;)14 b Fu(1])p Fn(.)456 3490 y(Pr)l(o)l(of.)43 b Fu(W)-7 b(e)26 b(will)f(only)h(deal)f(with)h FA(B)1620 3502 y Fs(0)1657 3490 y Fu(,)g(the)g(case)f(of)g FA(B)2181 3502 y Fs(1)2244 3490 y Fu(b)r(eing)g(analogous.)34 b(Let)29 b(\026)-45 b Fr(s)23 b Fq(2)g Fu([0)p Fr(;)14 b Fu(1])25 b(b)r(e)456 3590 y(suc)n(h)i(that)1354 3714 y Fr(W)1444 3680 y Fs(2)p Fp(;)p Fs(2)1534 3714 y Fu(\()p FA(K)1641 3680 y Fp(d)1641 3735 y(!)1689 3714 y Fu(\))19 b Fq(\000)64 b Fu(lim)1823 3764 y Fp(t)p Fm(!)p Fs(+)p Fm(1)2045 3714 y Fu(\010)2105 3680 y Fp(t)2134 3714 y Fu(\()p Fr(v)2209 3680 y Fs(\()s(\026)-36 b Fp(s)q Fs(\))2297 3714 y Fu(\))37 b(=)g(0)14 b Fr(;)456 3892 y Fu(and)27 b(let)738 3877 y(\026)737 3892 y Fr(t)h Fu(b)r(e)g(so)f(that)1551 4024 y(\010)1612 3979 y Fs(\026)1611 3990 y Fp(t)1640 4024 y Fu(\()p Fr(v)1715 3990 y Fs(\()s(\026)-36 b Fp(s)q Fs(\))1803 4024 y Fu(\))24 b Fq(2)f Fr(B)2004 3979 y Fp(W)2075 3954 y Fi(2)p Fk(;)p Fi(2)2155 3979 y Fs(\()p Fl(K)2239 3954 y Fk(d)2239 3996 y(!)2282 3979 y Fs(\))2000 4057 y Fp(c)2030 4015 y Fm(p)p 2084 4015 33 3 v 2084 4057 a Fp(r)r(=)p Fs(2)2326 4024 y Fr(;)456 4179 y Fu(with)j Fr(c)g Fu(as)g(in)g(Lemma)g(5.18.)35 b(Then,)27 b(from)e(Prop)r(osition)g(5.12,)g(and)h(Corollary)d(5.6,)j (w)n(e)g(ha)n(v)n(e)456 4279 y(that)1481 4411 y(\010)1542 4366 y Fs(\026)1541 4377 y Fp(t)1570 4411 y Fu(\()p Fr(v)1645 4377 y Fs(\()p Fp(s)p Fs(\))1733 4411 y Fu(\))d Fq(2)h(C)f(\\)18 b Fr(B)2074 4366 y Fp(W)2145 4341 y Fi(2)p Fk(;)p Fi(2)2226 4366 y Fs(\()p Fl(K)2310 4341 y Fk(d)2310 4382 y(!)2352 4366 y Fs(\))2070 4443 y Fp(c)2100 4401 y Fm(p)p 2155 4401 V 42 x Fp(r)2396 4411 y Fr(;)456 4569 y Fu(for)23 b(an)n(y)g Fr(s)g Fq(2)h Fu(\()s(\026)-45 b Fr(s)11 b Fq(\000)g Fr(\017;)17 b Fu(\026)-45 b Fr(s)11 b Fu(+)g Fr(\017)p Fu(\),)25 b(for)e(a)h(suitable)g Fr(\017)f(>)f Fu(0.)36 b(Then,)25 b(in)f(the)g(ligh)n(t)g(of)g(Prop)r(osition)f (5.20,)1354 4736 y Fr(W)1444 4702 y Fs(2)p Fp(;)p Fs(2)1534 4736 y Fu(\()p FA(K)1641 4702 y Fp(d)1641 4756 y(!)1689 4736 y Fu(\))c Fq(\000)64 b Fu(lim)1823 4786 y Fp(t)p Fm(!)p Fs(+)p Fm(1)2045 4736 y Fu(\010)2105 4702 y Fp(t)2134 4736 y Fu(\()p Fr(v)2209 4702 y Fs(\()s(\026)-36 b Fp(s)q Fs(\))2297 4736 y Fu(\))37 b(=)g(0)14 b Fr(;)456 4931 y Fu(for)27 b(an)n(y)f Fr(s)d Fq(2)h Fu(\()s(\026)-45 b Fr(s)19 b Fq(\000)f Fr(\017;)e Fu(\026)-44 b Fr(s)18 b Fu(+)g Fr(\017)p Fu(\),)27 b(hence)h FA(B)1679 4943 y Fs(0)1744 4931 y Fu(is)f(op)r(en.)1357 b Fj(\003)555 5116 y Fu(In)35 b(particular,)g(from)g(Corollary)d(A.3,)37 b(Prop)r(osition)c(5.21,)i(Corollary)d(5.17)i(and)g(Theo-)456 5216 y(rem)27 b(5.16,)f(w)n(e)h(gather:)p eop %%Page: 33 33 33 32 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(33)456 450 y FA(Corollary)42 b(5.22.)j Fn(L)l(et)38 b(us)f(assume)g(that)h Fu(0)f Fn(and)h Fr(u)2174 420 y Fs(\(1\))2287 450 y Fq(\000)23 b Fr(u)2423 420 y Fs(\(0\))2550 450 y Fn(ar)l(e)38 b(the)f(only)i (functions)e Fr(v)456 550 y Fn(c)l(ontaine)l(d)29 b(in)h Fq(C)k Fn(for)c(which)h Fq(E)1415 562 y Fp(!)1462 550 y Fu(\()p Fr(u)1542 520 y Fs(\(0\))1649 550 y Fu(+)17 b Fr(v)s Fu(\))23 b(=)g Fq(E)1961 562 y Fp(!)2009 550 y Fu(\()p Fr(u)2089 520 y Fs(\(0\))2178 550 y Fu(\))29 b Fn(holds.)40 b(L)l(et)29 b Fr(r)d(>)c Fu(0)29 b Fn(b)l(e)h(suitably)f (smal)t(l)456 649 y(and)h(let)g(us)f(assume)g(that:)39 b(if)31 b Fr(V)42 b Fq(2)23 b(C)g(\\)c(B)1781 661 y Fp(r)1847 649 y Fn(satis\014es)1073 808 y Fr(@)1117 820 y Fp(i)1159 808 y Fu(\()p Fr(a)1235 820 y Fp(ij)1293 808 y Fr(@)1337 820 y Fp(j)1372 808 y Fr(V)g Fu(\))g Fq(\000)f Fr(f)1636 715 y Fo(\020)1686 808 y Fr(x;)c(u)1818 773 y Fs(\(0\))1926 808 y Fu(+)k Fr(V)2075 715 y Fo(\021)2143 808 y Fu(+)g Fr(f)2290 715 y Fo(\020)2339 808 y Fr(x;)c(u)2471 773 y Fs(\(0\))2560 715 y Fo(\021)2647 808 y Fu(=)36 b(0)14 b Fr(;)456 982 y Fn(then)29 b Fr(V)42 b Fq(2)24 b(f)p Fu(0)p Fr(;)14 b(u)978 952 y Fs(\(1\))1084 982 y Fq(\000)k Fr(u)1215 952 y Fs(\(0\))1303 982 y Fq(g)p Fn(.)38 b(Then,)31 b(ther)l(e)f(exists)f Fr(u)2132 952 y Fp(?)2193 982 y Fq(2)24 b Fr(C)2337 952 y Fs(2)2374 982 y Fu(\()p FA(K)2481 952 y Fp(d)2481 1002 y(!)2529 982 y Fu(\))30 b Fn(such)g(that:)661 1101 y Fq(\017)41 b Fr(@)788 1113 y Fp(i)816 1101 y Fu(\()p Fr(a)892 1113 y Fp(ij)951 1101 y Fr(@)995 1113 y Fp(j)1030 1101 y Fr(u)1078 1070 y Fp(?)1115 1101 y Fu(\))24 b(=)e Fr(f)9 b Fu(\()p Fr(x;)14 b(u)1472 1070 y Fp(?)1510 1101 y Fu(\))p Fn(;)661 1200 y Fq(\017)41 b Fr(u)792 1170 y Fp(?)860 1200 y Fn(enjoys)30 b(the)g(Birkho\013)i(pr)l(op)l(erty;)661 1300 y Fq(\017)41 b Fr(u)792 1270 y Fs(\(0\))881 1300 y Fu(\()p Fr(x)p Fu(\))24 b Fr(<)f(u)1152 1270 y Fp(?)1189 1300 y Fu(\()p Fr(x)p Fu(\))h Fr(<)f(u)1460 1270 y Fs(\(1\))1549 1300 y Fu(\()p Fr(x)p Fu(\))p Fn(.)456 1459 y Fu(5.8.)40 b FA(Pro)s(of)g(of)g(Theorem)f(3.2.)h Fu(If)c(there)e(exists)h Fr(v)j Fq(2)e(C)28 b(\000)22 b(f)p Fu(0)p Fr(;)14 b(u)2663 1429 y Fs(\(1\))2774 1459 y Fq(\000)23 b Fr(u)2910 1429 y Fs(\(0\))2999 1459 y Fq(g)p Fu(,)36 b(for)f(whic)n(h)470 1544 y(~)456 1565 y Fq(E)500 1577 y Fp(!)547 1565 y Fu(\()p Fr(v)s Fu(\))24 b(=)f(0,)h(then)g(w)n(e)f(are)g(done.)35 b(The)24 b(reason)e(is)i(that,)g(w)n(e)g(ha)n(v)n(e)e(sho)n(wn)h(in)h (Lemma)f(5.4)g(that)456 1667 y(in)k(suc)n(h)h(a)f(case)g Fr(u)1033 1637 y Fs(\(0\))1140 1667 y Fu(+)18 b Fr(v)31 b Fu(is)c(a)g(lo)r(cal)g(minimizer)h(and,)g(hence,)f(it)h(satis\014es)f (\(2.1\).)555 1767 y(Hence,)35 b(w)n(e)e(will)g(assume)g(that)h(the)f (only)g Fr(v)j Fq(2)d(C)k Fu(for)c(whic)n(h)g(the)h(renormalized)e (energy)456 1867 y(v)-5 b(anishes)27 b(are)f(0)p Fr(;)14 b(u)1047 1837 y Fs(\(1\))1154 1867 y Fq(\000)k Fr(u)1285 1837 y Fs(\(0\))1373 1867 y Fu(.)555 1966 y(Also,)27 b(if,)h(for)e(an)n(y)g(small)h Fr(r)f(>)c Fu(0,)27 b(there)g(exists)f Fr(V)2112 1936 y Fs(\()p Fp(r)r Fs(\))2224 1966 y Fq(2)d(C)f(\\)17 b(B)2495 1978 y Fp(r)2549 1966 y Fq(\000)f(f)p Fu(0)p Fr(;)e(u)2799 1936 y Fs(\(1\))2904 1966 y Fq(\000)j Fr(u)3034 1936 y Fs(\(0\))3122 1966 y Fq(g)27 b Fu(so)f(that)967 2129 y Fr(@)1011 2141 y Fp(i)1053 2037 y Fo(\020)1102 2129 y Fr(a)1146 2141 y Fp(ij)1205 2129 y Fr(@)1249 2141 y Fp(j)1284 2129 y Fr(V)1351 2095 y Fs(\()p Fp(r)r Fs(\))1439 2037 y Fo(\021)1507 2129 y Fq(\000)18 b Fr(f)1654 2037 y Fo(\020)1703 2129 y Fr(x;)c(u)1835 2095 y Fs(\(0\))1943 2129 y Fu(+)k Fr(V)2093 2095 y Fs(\()p Fp(r)r Fs(\))2181 2037 y Fo(\021)2249 2129 y Fu(+)g Fr(f)2396 2037 y Fo(\020)2445 2129 y Fr(x;)c(u)2577 2095 y Fs(\(0\))2666 2037 y Fo(\021)2753 2129 y Fu(=)36 b(0)14 b Fr(;)456 2303 y Fu(then)42 b(w)n(e)f(set)g Fr(U)1004 2273 y Fs(\()p Fp(r)r Fs(\))1139 2303 y Fu(:=)46 b Fr(u)1321 2273 y Fs(\(0\))1437 2303 y Fu(+)27 b Fr(V)1596 2273 y Fs(\()p Fp(r)r Fs(\))1726 2303 y Fu(and)42 b Fr(U)1968 2273 y Fs(\()p Fp(r)r Fs(\))2098 2303 y Fu(satis\014es)e(the)i(claims)f (of)h(Theorem)e(3.2.)456 2403 y(Therefore,)34 b(w)n(e)g(ma)n(y)g (assume)f(that)i(there)f(exists)g Fr(r)i(>)e Fu(0)g(small,)i(so)d (that,)j(if)f Fr(V)53 b Fq(2)34 b(C)28 b(\\)23 b(B)3408 2415 y Fp(r)456 2503 y Fu(satis\014es)1073 2610 y Fr(@)1117 2622 y Fp(i)1159 2610 y Fu(\()p Fr(a)1235 2622 y Fp(ij)1293 2610 y Fr(@)1337 2622 y Fp(j)1372 2610 y Fr(V)c Fu(\))g Fq(\000)f Fr(f)1636 2518 y Fo(\020)1686 2610 y Fr(x;)c(u)1818 2576 y Fs(\(0\))1926 2610 y Fu(+)k Fr(V)2075 2518 y Fo(\021)2143 2610 y Fu(+)g Fr(f)2290 2518 y Fo(\020)2339 2610 y Fr(x;)c(u)2471 2576 y Fs(\(0\))2560 2518 y Fo(\021)2647 2610 y Fu(=)36 b(0)14 b Fr(;)456 2767 y Fu(then)34 b Fr(V)52 b Fq(2)33 b(f)p Fu(0)p Fr(;)14 b(u)1008 2737 y Fs(\(1\))1118 2767 y Fq(\000)22 b Fr(u)1253 2737 y Fs(\(0\))1342 2767 y Fq(g)p Fu(.)55 b(Under)33 b(these)h(assumptions,)g(Corollary)e(5.22)g (pro)n(vides)g(the)456 2866 y(desired)27 b(solution.)2334 b Ff(2)456 3023 y Fu(5.9.)40 b FA(Some)32 b(remarks)h(ab)s(out)g (energy)g(barriers.)42 b Fu(W)-7 b(e)29 b(ha)n(v)n(e)f(sho)n(wn)g (that,)i(giv)n(en)e(a)g(gap)456 3122 y(in)f(a)g(lamination)f(b)n(y)h (Birkho\013)g(minimizers,)g(w)n(e)f(can)h(\014nd)h(a)e(critical)h (function)g Fr(v)k Fu(inside)c(the)456 3222 y(gap.)555 3323 y(This)i(function)h(has)f(renormalized)f(energy)2000 3302 y(~)1986 3323 y Fq(E)2030 3335 y Fp(!)2078 3323 y Fu(\()p Fr(v)s Fu(\))e Fq(\025)g Fu(0.)41 b(If)2507 3302 y(~)2493 3323 y Fq(E)2537 3335 y Fp(!)2585 3323 y Fu(\()p Fr(v)s Fu(\))26 b(=)g(0,)j(w)n(e)g(ha)n(v)n(e)f(sho)n(wn)456 3422 y(in)22 b(Prop)r(osition)e(5.4)h(that)h Fr(v)k Fu(is)21 b(another)g(minimizer.)36 b(W)-7 b(e)22 b(can)f(consider)g(no)n(w)h (the)g(lamination)456 3522 y(generated)i(b)n(y)h(our)g(previous)f (lamination)h(and)g Fq(L)2048 3541 y Fp(u)2087 3524 y Fi(\(0\))2165 3541 y Fs(+)p Fp(v)2255 3522 y Fu(.)37 b(If)25 b(it)h(has)f(gaps,)g(w)n(e)g(can)g(apply)g(the)456 3622 y(pro)r(cedure)h(again.)555 3721 y(The)g(conclusion)f(is)h(that)g Fn(either)j(ther)l(e)f(is)h(a)f(critic)l(al)i(p)l(oint)e(with)h (strictly)f(p)l(ositive)i(r)l(enor-)456 3821 y(malize)l(d)h(ener)l(gy)f (or)g(ther)l(e)g(is)g(a)g(foliation)i(of)e(Birkho\013)i(minimizers)p Fu(.)555 3920 y(Hence,)41 b(if)e(w)n(e)f(de\014ne)g(the)g Fn(ener)l(gy)i(gap)45 b Fu(of)38 b(the)g(normal)f Fr(!)k Fu(as)d(the)g(suprem)n(um)g(of)g(the)456 4020 y(renormalized)23 b(energies)i(of)g(all)g(the)h(critical)f(p)r(oin)n(ts)g(with)h(these)f (frequency)-7 b(,)26 b(w)n(e)f(obtain)g(that)456 4120 y Fn(ther)l(e)31 b(is)i(a)f(foliation)h(by)g(minimizers)f(which)i(ar)l (e)e(Birkho\013)h(with)f(r)l(esp)l(e)l(ct)g(to)g Fr(!)i Fn(if)f(and)f(only)456 4219 y(if)e(the)g(ener)l(gy)g(gap)h(is)f(zer)l (o)p Fu(.)555 4319 y(This)i(seems)g(v)n(ery)f(reminiscen)n(t)h(of)g (the)h(criterion)e(in)i([Mat86)o(])f(for)g(the)h(existence)f(of)g(an) 456 4419 y(in)n(v)-5 b(arian)n(t)25 b(circle)h(for)h(t)n(wist)g(maps.) 36 b(Indeed,)27 b(the)g(energy)f(gap)g(de\014ned)h(ab)r(o)n(v)n(e)f (has)g(the)h(same)456 4518 y(\015a)n(v)n(or)33 b(as)i(the)g(energy)f (gap)h(de\014ned)h(in)f([Mat86)o(])h(since)f(b)r(oth)h(are)e(the)i (suprem)n(um)f(of)g(the)456 4618 y(di\013erence)28 b(of)f(energies)g (in)h(critical)g(p)r(oin)n(ts.)38 b(In)28 b([Mat86)o(])g(it)h(is)e(sho) n(wn)h(that)g(the)g(energy)f(gap)456 4717 y(dep)r(ends)j(con)n(tin)n (uously)e(with)j(resp)r(ect)e(to)h Fr(!)s Fu(.)43 b(W)-7 b(e)30 b(think)h(that)f(it)g(w)n(ould)f(b)r(e)i(in)n(teresting)e(to)456 4817 y(in)n(v)n(estigate)d(the)i(con)n(tin)n(uit)n(y)f(of)g(the)h (energy)f(gap)g(de\014ned)h(here.)555 4917 y(W)-7 b(e)27 b(also)e(note)i(that)g(the)f(pro)r(of)g(here)g(constructs)g(critical)g (p)r(oin)n(ts)g(b)n(y)g(follo)n(wing)g(the)h(heat)456 5016 y(\015o)n(w)33 b(with)h(initial)g(data)f(in)h(a)f(sp)r(eci\014c)h (family)g Fr(v)2047 4986 y Fs(\()p Fp(s)p Fs(\))2134 5016 y Fu(.)56 b(Hence,)35 b(w)n(e)e(can)h(de\014ne)g(a)f(simpli\014ed) 456 5116 y(energy)20 b(gap)h(as)g(the)h(suprem)n(um)f(of)h(the)g (renormalized)e(energies)g(for)i(the)g(critical)f(p)r(oin)n(ts)g(that) 456 5216 y(are)33 b(obtained)h(b)n(y)h(taking)f(the)g(heat)h(\015o)n(w) f(on)g(p)r(oin)n(ts)g(in)h(the)g(ab)r(o)n(v)n(e)e(family)-7 b(.)58 b(Our)34 b(results)p eop %%Page: 34 34 34 33 bop 456 251 a Fs(34)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fu(sho)n(w)24 b(that)i(there)g(is)f(a)g(foliation)g(b)n(y)h(minimizers)f(if)h(and)g (only)f(if)h(the)g(simpli\014ed)g(energy)e(gap)456 550 y(is)i(zero.)35 b(W)-7 b(e)27 b(do)e(not)i(kno)n(w)e(whether)h(the)h (energy)e(gap)g(and)h(the)g(simpli\014ed)h(energy)e(gap)g(are)456 649 y(the)f(same.)35 b(The)24 b(simpli\014ed)g(energy)e(gap)h(seems)h (more)f(amenable)g(to)g(computation)h(b)r(ecause)456 749 y(the)k(range)e(o)n(v)n(er)g(whic)n(h)h(w)n(e)g(are)g(tak)n(en)g (the)h(suprem)n(um)f(is)h(explicitly)f(giv)n(en.)555 849 y(One)c(w)n(ould)f(also)g(exp)r(ect)h(that,)i(when)e(there)f(is)h (a)g(gap)f(in)h(the)g(lamination)g(of)g(minimizers,)456 948 y(one)41 b(can)g(also)f(construct)h(man)n(y)f(other)h(critical)g(p) r(oin)n(ts,)k(whic)n(h)c(ho)n(w)n(ev)n(er)e(could)i(fail)h(to)456 1048 y(b)r(e)33 b(Birkho\013)g(or)f(plane-lik)n(e.)53 b(This)33 b(is)h(what)f(happ)r(ens)g(in)h(dynamical)e(systems,)j(when)e (the)456 1147 y(existence)27 b(of)g(a)h(barrier)d(allo)n(ws)i(to)g (construct)g(a)g(sym)n(b)r(olic)g(dynamics.)1538 1388 y Fv(A)n(ckno)n(wledgements)555 1537 y Fu(RdlL)37 b(has)g(b)r(een)g (supp)r(orted)g(b)n(y)f(NSF)i(gran)n(ts.)63 b(EV)36 b(has)g(b)r(een)i (supp)r(orted)e(b)n(y)h(MIUR)456 1637 y(pro)5 b(ject)29 b(\\V)-7 b(ariational)29 b(Metho)r(ds)i(and)f(Nonlinear)g(Di\013eren)n (tial)g(Equations")f(and)i(INdAM.)456 1737 y(P)n(art)40 b(of)h(this)h(pap)r(er)f(w)n(as)g(written)h(while)g(the)g(authors)e(w)n (ere)h(visiting)g(the)h(Univ)n(ersitat)456 1836 y(P)n(olit)n(\022)-39 b(ecnica)35 b(de)k(Catalun)n(y)n(a)d(in)i(Barcelona)e(and)i(the)h (Institut)g(Henri)f(P)n(oincar)n(\023)-39 b(e)35 b(in)j(P)n(aris)456 1936 y(supp)r(orted)27 b(b)n(y)g(GDRE)h(Gre\014-Me\014.)956 2176 y Fv(Appendix)k Fu(A.)42 b Fv(Recalling)31 b(some)h(technical)f (resul)-6 b(ts)555 2326 y Fu(In)22 b(this)g(App)r(endix,)i(w)n(e)d (collect)h(some)f(tec)n(hnical)g(results)g(that)h(are)f(used)g(in)h (the)g(main)g(text)456 2425 y(and)28 b(w)n(e)g(giv)n(e)f(references)g (to)i(the)f(literature)g(or)f(indicate)i(the)f(small)g(mo)r (di\014cations)g(needed)456 2525 y(to)f(get)g(them.)456 2724 y(A.1.)41 b FA(Results)27 b(from)g(the)h(theory)h(of)f(elliptic)f (equations.)41 b Fu(W)-7 b(e)25 b(no)n(w)f(state)g(some)g(stan-)456 2824 y(dard)d(elliptic)h(PDE)f(results)h(in)g(a)f(form)g(whic)n(h)h(is) g(con)n(v)n(enien)n(t)f(for)g(our)g(applications.)34 b(First)22 b(of)456 2923 y(all,)i(w)n(e)f(recall)f(the)i(follo)n(wing)f (elliptic)h(strong)e(maxim)n(um)h(principle)h(for)f(w)n(eak)f (subsolutions,)456 3023 y(for)27 b(the)h(pro)r(of)f(of)g(whic)n(h)h(w)n (e)f(refer)g(to)g Fq(x)h Fu(8.7)f(of)g([GT83)o(]:)456 3152 y FA(Lemma)e(A.1.)36 b Fn(L)l(et)26 b Fu(\012)g Fn(b)l(e)g(a)h(b)l(ounde)l(d)f(c)l(onne)l(cte)l(d)g(domain)h(in)g FA(R)2541 3122 y Fp(d)2605 3152 y Fn(and)g(let)f Fr(a)2920 3164 y Fp(ij)2978 3152 y Fn(,)h Fr(c)c Fq(2)h Fr(L)3225 3122 y Fm(1)3294 3152 y Fu(\(\012\))p Fn(,)456 3252 y(for)38 b Fu(1)g Fq(\024)f Fr(i;)14 b(j)43 b Fq(\024)37 b Fr(d)p Fn(.)64 b(Assume)37 b(that)h(the)g(c)l(o)l(e\016cients)g Fr(a)2272 3264 y Fp(ij)2369 3252 y Fn(satisfy)h(the)f(uniform)g(el)t (lipticity)456 3352 y(c)l(ondition)30 b(given)h(in)f Fg(\(H1\))g Fn(and)h(that)e Fr(c)23 b Fq(\024)g Fu(0)p Fn(.)38 b(L)l(et)29 b Fr( )e Fq(2)c Fr(W)2343 3321 y Fs(1)p Fp(;)p Fs(2)2433 3352 y Fu(\(\012\))30 b Fn(satisfy)h(we)l(akly) 1316 3536 y Fr(@)1360 3548 y Fp(i)1387 3444 y Fo(\020)1437 3536 y Fr(a)1481 3548 y Fp(ij)1539 3536 y Fu(\()p Fr(x)p Fu(\))14 b Fr(@)1708 3548 y Fp(j)1744 3536 y Fr( )s Fu(\()p Fr(x)p Fu(\))1912 3444 y Fo(\021)1981 3536 y Fu(+)k Fr(c)p Fu(\()p Fr(x)p Fu(\))c Fr( )s Fu(\()p Fr(x)p Fu(\))25 b Fq(\025)e Fu(0)14 b Fr(;)456 3715 y Fn(for)30 b(any)g Fr(x)24 b Fq(2)f Fu(\012)p Fn(.)39 b(Assume)29 b(that,)h(for)h(some)f (b)l(al)t(l)h Fr(B)c Fq(\032)22 b Fu(\012)30 b Fn(we)g(have)1604 3875 y Fu(sup)1640 3944 y Fp(B)1743 3875 y Fr( )c Fu(=)d(sup)1950 3944 y Fs(\012)2050 3875 y Fr( )j Fq(\025)c Fu(0)14 b Fr(:)456 4078 y Fn(Then)30 b Fr( )j Fn(is)d(c)l(onstant)f(in)h Fu(\012)p Fn(.)555 4208 y Fu(W)-7 b(e)37 b(no)n(w)e(deduce)h(some)g (useful)g(consequences)f(from)h(Lemma)g(A.1.)62 b(First)36 b(of)g(all)g(\(see)456 4308 y(Corollary)g(A.2)j(here)g(b)r(elo)n(w\),)i (in)f(the)f(case)f(of)h(negativ)n(e)f(subsolutions,)j(w)n(e)e(remo)n(v) n(e)e(the)456 4407 y(assumption)24 b(on)h(the)g(sign)f(of)h Fr(c)g Fu(from)g(Lemma)g(A.1.)35 b(Then,)26 b(w)n(e)f(use)g(this)g(to)g (obtain)f(a)h(strong)456 4507 y(comparison)g(principle)j(\(see)g (Corollary)d(A.3\).)456 4636 y FA(Corollary)45 b(A.2.)i Fn(L)l(et)40 b Fu(\012)f Fn(b)l(e)h(a)h(b)l(ounde)l(d)f(c)l(onne)l(cte) l(d)f(domain)j(in)d FA(R)2765 4606 y Fp(d)2844 4636 y Fn(and)h(let)g Fr(a)3186 4648 y Fp(ij)3244 4636 y Fn(,)j Fr(c)e Fq(2)456 4736 y Fr(L)513 4706 y Fm(1)582 4736 y Fu(\(\012\))p Fn(,)k(for)d Fu(1)h Fq(\024)g Fr(i;)14 b(j)47 b Fq(\024)c Fr(d)p Fn(.)73 b(Assume)40 b(that)h(the)g(c)l(o)l (e\016cients)h Fr(a)2639 4748 y Fp(ij)2738 4736 y Fn(satisfy)g(the)f (uniform)456 4836 y(el)t(lipticity)32 b(c)l(ondition)e(given)h(in)f Fg(\(H1\))p Fn(.)39 b(L)l(et)30 b Fr(\036)23 b Fq(2)h Fr(W)2154 4805 y Fs(1)p Fp(;)p Fs(2)2244 4836 y Fu(\(\012\))30 b Fn(satisfy)h(we)l(akly)456 5020 y Fu(\(A.1\))676 b Fr(@)1367 5032 y Fp(i)1395 4928 y Fo(\020)1445 5020 y Fr(a)1489 5032 y Fp(ij)1547 5020 y Fu(\()p Fr(x)p Fu(\))14 b Fr(@)1716 5032 y Fp(j)1752 5020 y Fr(\036)p Fu(\()p Fr(x)p Fu(\))1912 4928 y Fo(\021)1981 5020 y Fu(+)k Fr(c)p Fu(\()p Fr(x)p Fu(\))c Fr(\036)p Fu(\()p Fr(x)p Fu(\))26 b Fq(\025)c Fu(0)14 b Fr(;)456 5199 y Fn(for)30 b(any)g Fr(x)24 b Fq(2)f Fu(\012)p Fn(.)39 b(If)30 b Fr(\036)24 b Fq(\024)e Fu(0)29 b Fn(in)h Fu(\012)p Fn(,)g(then)g(either)h Fr(\036)23 b(<)g Fu(0)29 b Fn(in)h Fu(\012)f Fn(or)i Fr(\036)23 b Fu(=)g(0)29 b Fn(in)h Fu(\012)p Fn(.)p eop %%Page: 35 35 35 34 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(35)456 450 y Fn(Pr)l(o)l(of.)43 b Fu(Assume)32 b(that)g(there)g(is)g(a)g(p)r(oin)n (t)38 b(\026)-48 b Fr(x)31 b Fq(2)g Fu(\012)h(suc)n(h)g(that)h Fr(\036)p Fu(\()5 b(\026)-47 b Fr(x)q Fu(\))31 b(=)f(0.)50 b(Let)32 b Fr(B)j Fq(\032)30 b Fu(\012)i(b)r(e)h(a)456 550 y(small)27 b(ball)g(cen)n(tered)g(at)33 b(\026)-47 b Fr(x)p Fu(.)37 b(By)27 b(construction,)1612 690 y(sup)1648 760 y Fp(B)1751 690 y Fr(\036)c Fu(=)g(sup)1950 760 y Fs(\012)2050 690 y Fr(\036)g Fu(=)g(0)14 b Fr(:)456 883 y Fu(Let)26 b Fr(c)p Fu(\()p Fr(x)p Fu(\))e(=)e Fr(c)897 853 y Fs(+)952 883 y Fu(\()p Fr(x)p Fu(\))16 b Fq(\000)f Fr(c)1195 853 y Fm(\000)1251 883 y Fu(\()p Fr(x)p Fu(\),)27 b(where)f Fr(c)1687 853 y Fs(+)1768 883 y Fu(and)f Fr(c)1963 853 y Fm(\000)2045 883 y Fu(are)g(the)h(p)r(ositiv)n(e)g(and)f(negativ) n(e)g(part)h(of)f Fr(c)p Fu(,)456 983 y(resp)r(ectiv)n(ely)-7 b(.)36 b(Hence,)27 b Fr(\036)i Fu(satisfy)e(w)n(eakly)1018 1144 y Fr(@)1062 1156 y Fp(i)1090 1051 y Fo(\020)1140 1144 y Fr(a)1184 1156 y Fp(ij)1242 1144 y Fu(\()p Fr(x)p Fu(\))14 b Fr(@)1411 1156 y Fp(j)1447 1144 y Fr(\036)p Fu(\()p Fr(x)p Fu(\))1607 1051 y Fo(\021)1676 1144 y Fq(\000)19 b Fr(c)1796 1109 y Fm(\000)1851 1144 y Fu(\()p Fr(x)p Fu(\))14 b Fr(\036)p Fu(\()p Fr(x)p Fu(\))26 b Fq(\025)d(\000)p Fr(c)2351 1109 y Fs(+)2405 1144 y Fu(\()p Fr(x)p Fu(\))14 b Fr(\036)p Fu(\()p Fr(x)p Fu(\))26 b Fq(\025)c Fu(0)14 b Fr(:)456 1304 y Fu(Th)n(us,)39 b(from)d(Lemma)h (A.1,)i Fr(\036)f Fu(is)e(constan)n(t)h(in)g(\012)g(and)g(so,)h(b)n(y)f (construction,)h(constan)n(tly)456 1404 y(equal)27 b(to)g(zero.)2432 b Fj(\003)456 1562 y FA(Corollary)28 b(A.3.)37 b Fn(L)l(et)26 b Fu(\012)g Fn(b)l(e)g(a)h(b)l(ounde)l(d)f(c)l(onne)l(cte)l(d)g(domain) h(in)g FA(R)2629 1532 y Fp(d)2693 1562 y Fn(and)g(let)f Fr(a)3008 1574 y Fp(ij)3089 1562 y Fq(2)e Fr(L)3225 1532 y Fm(1)3294 1562 y Fu(\(\012\))p Fn(,)456 1661 y(for)34 b Fu(1)c Fq(\024)g Fr(i;)14 b(j)35 b Fq(\024)30 b Fr(d)p Fn(,)36 b(satisfy)f(the)f(uniform)g(el)t(lipticity)i(c)l(ondition)f (given)g(in)f Fg(\(H1\))p Fn(.)52 b(L)l(et)33 b Fr(f)39 b Fq(2)456 1761 y Fr(W)546 1731 y Fs(1)p Fp(;)p Fm(1)669 1761 y Fu(\()p FA(R)773 1731 y Fp(d)p Fs(+1)895 1761 y Fu(\))p Fn(.)g(L)l(et)30 b Fr(u;)d(v)f Fq(2)e Fr(W)1468 1731 y Fs(1)p Fp(;)p Fs(2)1558 1761 y Fu(\(\012\))30 b Fn(satisfy)h(we)l(akly)707 1926 y Fr(@)751 1938 y Fp(i)779 1834 y Fo(\020)828 1926 y Fr(a)872 1938 y Fp(ij)931 1926 y Fu(\()p Fr(x)p Fu(\))14 b Fr(@)1100 1938 y Fp(j)1136 1926 y Fr(u)p Fu(\()p Fr(x)p Fu(\))1295 1834 y Fo(\021)1363 1926 y Fu(+)k Fr(f)9 b Fu(\()p Fr(x;)14 b(u)p Fu(\()p Fr(x)p Fu(\)\))24 b Fq(\025)f Fu(0)g Fq(\025)f Fr(@)2111 1938 y Fp(i)2139 1834 y Fo(\020)2189 1926 y Fr(a)2233 1938 y Fp(ij)2291 1926 y Fu(\()p Fr(x)p Fu(\))14 b Fr(@)2460 1938 y Fp(j)2496 1926 y Fr(v)s Fu(\()p Fr(x)p Fu(\))2650 1834 y Fo(\021)2719 1926 y Fu(+)k Fr(f)9 b Fu(\()p Fr(x;)14 b(v)s Fu(\()p Fr(x)p Fu(\)\))g Fr(;)456 2087 y Fn(for)30 b(any)g Fr(x)24 b Fq(2)f Fu(\012)p Fn(.)39 b(If)30 b Fr(u)23 b Fq(\024)f Fr(v)33 b Fn(in)d Fu(\012)p Fn(,)g(then)g(either)g Fr(u)23 b(<)g(v)32 b Fn(in)e Fu(\012)g Fn(or)g Fr(u)23 b Fu(=)f Fr(v)33 b Fn(in)d Fu(\012)p Fn(.)456 2245 y(Pr)l(o)l(of.)43 b Fu(The)27 b(function)i Fr(\036)23 b Fu(:=)g Fr(u)18 b Fq(\000)g Fr(v)31 b Fu(satis\014es)26 b(w)n(eakly)i(\(A.1\))g(with) 1227 2444 y Fr(c)p Fu(\()p Fr(x)p Fu(\))c(:=)1509 2331 y Fo(Z)1592 2352 y Fs(1)1555 2520 y(0)1643 2444 y Fr(f)1693 2410 y Fm(0)1716 2444 y Fu(\()p Fr(x;)14 b(tu)p Fu(\()p Fr(x)p Fu(\))20 b(+)e(\(1)g Fq(\000)g Fr(t)p Fu(\))p Fr(v)s Fu(\()p Fr(x)p Fu(\)\))c Fr(dt)g(:)456 2627 y Fu(Hence,)27 b(the)h(claim)g(follo)n(ws)e(from)i(Corollary)d(A.2.)1304 b Fj(\003)456 2785 y Fu(A.2.)41 b FA(Results)36 b(from)g(the)h(theory)h (of)g(parab)s(olic)f(equations.)k Fu(W)-7 b(e)33 b(consider)e(the)i (op-)456 2885 y(erator)e Fr(L)h Fu(de\014ned)h(in)h(\(5.14\))o(.)53 b(As)33 b(a)f(matter)h(of)f(fact,)j(up)e(to)g(a)2556 2816 y Fo(p)p 2639 2816 283 4 v 69 x Fu(det)q(\()p Fr(a)2831 2897 y Fp(ij)2889 2885 y Fu(\){factor,)g Fr(L)g Fu(can)456 2991 y(b)r(e)26 b(seen)f(as)g(the)h(Laplacian)e(op)r(erator)g(acting)h (on)g(the)h(Riemannian)g(manifold)f FA(R)3071 2961 y Fp(d)3135 2991 y Fu(endo)n(w)n(ed)456 3090 y(with)32 b(a)f(suitable)g(metric)h Fr(g)1341 3060 y Fp(ij)1429 3090 y Fu(=)d(det\()p Fr(a)1714 3102 y Fp(ij)1773 3090 y Fu(\))14 b Fr(a)1863 3102 y Fp(ij)1953 3090 y Fu(\(see,)33 b(e.g.,)e([T)-7 b(a)n(y96)n(],)33 b(form)n(ula)e(\(4.4\))g(on)g(page) 456 3190 y(137\))555 3290 y(The)g(theory)f(of)h(semigroups)e(with)j (monotone)e(generators)e(\(see,)k(for)e(instance,)g([Sho97)o(])456 3389 y(and)40 b([T)-7 b(a)n(y96)o(]\))41 b(sho)n(ws)f(that)h(there)g (is)g(an)g(op)r(erator)e Fr(e)2253 3359 y Fp(tL)2369 3389 y Fu(for)h(p)r(ositiv)n(e)h Fr(t)g Fu(de\014ned)g(b)n(y)g(the)456 3489 y(condition)d(that)g Fr(e)1060 3459 y Fp(tL)1135 3489 y Fr(V)1183 3501 y Fs(0)1259 3489 y Fu(to)g(is)g(unique)h (function)g Fr(V)18 b Fu(\()p Fr(x;)c(t)p Fu(\))40 b(de\014ned)e(for)g Fr(x)j Fq(2)h FA(K)3060 3459 y Fp(d)3060 3509 y(!)3146 3489 y Fu(and)c Fr(t)j Fq(2)456 3613 y Fu([0)p Fr(;)14 b Fu(+)p Fq(1)p Fu(\),)43 b(so)c(that)i(the)g(map)f Fr(t)k Fq(7!)h Fr(V)19 b Fu(\()p Fr(t;)14 b Fq(\001)p Fu(\))41 b(b)r(elongs)e(to)h Fr(C)2417 3583 y Fs(1)2455 3521 y Fo(\020)2505 3613 y Fu(\(0)p Fr(;)14 b Fu(+)p Fq(1)p Fu(\))p Fr(;)g(L)2890 3583 y Fs(2)2926 3613 y Fu(\()p FA(K)3033 3583 y Fp(d)3033 3634 y(!)3081 3613 y Fu(\))3113 3521 y Fo(\021)3203 3613 y Fu(and)41 b Fr(V)456 3733 y Fu(solv)n(es)26 b(the)i(linear)f(equation)1007 3856 y Fr(@)5 b(V)p 1007 3893 116 4 v 1026 3969 a(@)g(t)1133 3912 y Fu(\()p Fr(x;)14 b(t)p Fu(\))84 b(=)e Fr(LV)19 b Fu(\()p Fr(x;)14 b(t)p Fu(\))g Fr(;)180 b Fq(8)p Fu(\()p Fr(x;)14 b(t)p Fu(\))23 b Fq(2)h FA(K)2463 3878 y Fp(d)2463 3933 y(!)2529 3912 y Fq(\002)18 b Fu(\(0)p Fr(;)c Fu(+)p Fq(1)p Fu(\))1054 4077 y Fr(V)19 b Fu(\()p Fr(x;)14 b Fu(0\))84 b(=)e Fr(V)1590 4089 y Fs(0)1628 4077 y Fu(\()p Fr(x)p Fu(\))14 b Fr(;)181 b Fq(8)p Fr(x)22 b Fq(2)i FA(K)2227 4043 y Fp(d)2227 4098 y(!)2289 4077 y Fr(:)456 4217 y Fu(Of)g(course,)g(if)h Fr(a)962 4229 y Fp(ij)1043 4217 y Fu(=)e Fr(\016)1168 4229 y Fp(ij)1226 4217 y Fu(,)j(i.e.,)f(if)g Fr(L)f Fu(is)h(the)f(standard)g(Laplacian)f(op)r(erator,)h(and)g(the)h (domain)456 4320 y(is)32 b(the)i(whole)f FA(R)1006 4290 y Fp(d)1044 4320 y Fu(,)h(then)g Fr(e)1335 4290 y Fp(tL)1442 4320 y Fu(is)f(simply)g(the)h(con)n(v)n(olution)d(with)j(a)e(Gaussian)g (k)n(ernel)h(\(see,)456 4420 y(e.g.,)27 b([Ev)-5 b(a98)n(]\).)555 4519 y(Explicit)35 b(b)r(ounds)g(on)g(the)g(linear)f(op)r(erator)f (norm)h(of)h Fr(e)2389 4489 y Fp(tL)2499 4519 y Fu(in)g(di\013eren)n(t) g(spaces)f(can)g(b)r(e)456 4619 y(found,)21 b(for)f(instance,)h(on)e (pages)g(273{275)e(of)i([T)-7 b(a)n(y97)o(].)34 b(In)20 b(particular,)g(denoting)g(b)n(y)f Fq(L)p Fu(\()p Fr(X)r(;)14 b(Y)19 b Fu(\))456 4719 y(the)k(space)g(of)g(the)g(linear)g(op)r (erators)e(from)i Fr(X)29 b Fu(to)23 b Fr(Y)42 b Fu(\(endo)n(w)n(ed)23 b(with)g(the)h(standard)e(op)r(erator)456 4818 y(norm\),)27 b(w)n(e)g(will)h(use)f(here)h(b)r(elo)n(w)f(that)1221 4976 y Fq(k)p Fr(e)1302 4941 y Fp(tL)1376 4976 y Fq(k)1418 4992 y Fm(L)p Fs(\()o Fp(W)1560 4976 y Fk(r)o(;)p Fi(2)1638 4992 y Fs(\()p Fl(K)1722 4976 y Fk(d)1722 5009 y(!)1765 4992 y Fs(\))p Fp(;W)1882 4976 y Fk(s;)p Fi(2)1961 4992 y Fs(\()p Fl(K)2045 4976 y Fk(d)2045 5009 y(!)2088 4992 y Fs(\)\))2167 4976 y Fq(\024)c Fr(C)6 b(t)2360 4916 y Fk(r)q Fh(\000)p Fk(s)p 2360 4929 103 3 v 2397 4961 a Fi(2)2490 4976 y Fr(;)-2057 b Fu(\(A.2\))456 5116 y(for)27 b(an)n(y)g(0)c Fr(<)g(t)g Fq(\024)g Fu(1,)28 b Fr(s)23 b Fq(\025)g Fr(r)j Fq(\025)d Fu(0)28 b(and)f(some)h(constan)n(t)f Fr(C)6 b Fu(,)28 b(p)r(ossibly)f(dep)r(ending)i(on)e Fr(r)k Fu(and)c Fr(s)p Fu(.)456 5216 y(Moreo)n(v)n(er,)22 b(in)i(Prop)r(osition)f(1.1)g(on)h(page)g(273)f(of)h([T)-7 b(a)n(y97)n(],)25 b(it)g(is)f(sho)n(wn)g(that,)h(for)f(a)g(suitable)p eop %%Page: 36 36 36 35 bop 456 251 a Fs(36)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fr(T)35 b(>)25 b Fu(0,)j(for)g(an)n(y)g Fr(v)g Fq(2)d(C)5 b Fu(,)29 b(there)f(exists)g(a)g Fr(V)19 b Fu(\()p Fr(x;)14 b(t)p Fu(\))30 b(so)e(that)h(the)g(map)f Fr(t)d Fq(7!)g Fr(V)18 b Fu(\()p Fr(t;)c Fq(\001)p Fu(\))30 b(b)r(elongs)e(to)456 575 y Fr(C)521 482 y Fo(\020)570 575 y Fu([0)p Fr(;)14 b(T)e Fu(])p Fr(;)i(C)858 545 y Fs(1)894 575 y Fu(\()p FA(K)1001 545 y Fp(d)1001 595 y(!)1050 575 y Fu(\))1082 482 y Fo(\021)1159 575 y Fu(and)28 b Fr(V)46 b Fu(solv)n(es)26 b(the)i(semilinear)f(equation)707 769 y Fr(@)751 781 y Fp(t)780 769 y Fr(V)42 b Fu(=)23 b Fr(@)1002 781 y Fp(i)1029 769 y Fu(\()p Fr(a)1105 781 y Fp(ij)1164 769 y Fr(@)1208 781 y Fp(j)1243 769 y Fr(V)c Fu(\))g Fq(\000)f Fr(f)9 b Fu(\()p Fr(x;)14 b(u)1658 735 y Fs(\(0\))1765 769 y Fu(+)k Fr(V)h Fu(\))g(+)f Fr(f)9 b Fu(\()p Fr(x;)14 b(u)2263 735 y Fs(\(0\))2352 769 y Fu(\))g Fr(;)97 b Fq(8)p Fu(\()p Fr(x;)14 b(t)p Fu(\))23 b Fq(2)g FA(K)2919 735 y Fp(d)2919 790 y(!)2986 769 y Fq(\002)18 b Fu(\(0)p Fr(;)c Fu(+)p Fq(1)p Fu(\))707 901 y Fr(V)19 b Fu(\()p Fr(x;)14 b Fu(0\))23 b(=)g Fr(v)s Fu(\()p Fr(x)p Fu(\))14 b Fr(;)98 b Fq(8)p Fr(x)23 b Fq(2)g FA(K)1634 867 y Fp(d)1634 922 y(!)1696 901 y Fr(:)-1263 b Fu(\(A.3\))456 1055 y(W)-7 b(e)22 b(will)h(denote)g(b)n(y)f(\010)1178 1025 y Fp(t)1207 1055 y Fu(\()p Fr(v)s Fu(\))h(suc)n(h)f(solution)g(and)h(w)n(e)f(will)g (sometimes)g(refer)g(to)h(it)f(as)g(the)h(\\heat)456 1155 y(\015o)n(w".)555 1255 y(The)29 b(\015o)n(w)e(\010)960 1224 y Fp(t)1018 1255 y Fu(can)h(b)r(e)g(constructed)g(b)n(y)g(\014xed) g(p)r(oin)n(t)h(argumen)n(ts)e(as)g(the)i(solution)f(of)g(the)456 1354 y(follo)n(wing)e(equation:)456 1562 y(\(A.4\))208 b(\010)915 1527 y Fp(t)945 1562 y Fu(\()p Fr(v)s Fu(\))37 b(=)g Fr(e)1230 1527 y Fp(tL)1304 1562 y Fr(v)22 b Fq(\000)1449 1449 y Fo(Z)1532 1469 y Fp(t)1495 1637 y Fs(0)1575 1562 y Fr(e)1614 1527 y Fs(\()p Fp(t)p Fm(\000)p Fp(s)p Fs(\))p Fp(L)1837 1469 y Fo(h)1877 1562 y Fr(f)1940 1469 y Fo(\020)1990 1562 y Fr(x;)14 b(u)2122 1527 y Fs(\(0\))2229 1562 y Fu(+)k(\010)2372 1527 y Fp(s)2408 1562 y Fu(\()p Fr(v)s Fu(\))2515 1469 y Fo(\021)2583 1562 y Fq(\000)g Fr(f)2730 1469 y Fo(\020)2779 1562 y Fr(x;)c(u)2911 1527 y Fs(\(0\))3000 1469 y Fo(\021i)3117 1562 y Fr(ds)g(;)456 1762 y Fu(for)22 b(details,)i(see)e(again)g([T)-7 b(a)n(y97)n(])23 b(pages)f(272{274.)32 b(F)-7 b(or)22 b(further)h(prop)r(erties)f(on)g(the)h(heat)g(\015o)n(w) 456 1862 y(in)k(a)h(more)e(general)h(setting)g(see,)g(e.g.,)h([Ang90b)o (])g(and)f([Hen81)o(].)555 1962 y(In)h(particular,)e(it)i(is)g(p)r (ossible)f(to)h(sho)n(w:)456 2089 y FA(Lemma)35 b(A.4.)43 b Fn(Ther)l(e)35 b(is)g(one)f(and)h(only)g(one)f(solution)h(of)g (\(A.3\))g(in)f Fr(L)2843 2058 y Fs(2)2880 2089 y Fu(\()p FA(K)2987 2058 y Fp(d)2987 2109 y(!)3035 2089 y Fu(\))p Fn(.)52 b(F)-6 b(urther-)456 2188 y(mor)l(e,)32 b(this)f(solution)h(is) f(in)g Fr(W)1451 2158 y Fp(k)q(;)p Fs(2)1545 2188 y Fu(\()p FA(K)1652 2158 y Fp(d)1652 2209 y(!)1700 2188 y Fu(\))h Fn(for)g Fr(t)25 b(>)g(t)2073 2200 y Fs(0)2110 2188 y Fn(.)43 b(The)32 b(map)g(that)f(applies)j Fr(v)g Fn(to)d Fu(\010)3217 2158 y Fp(t)3246 2188 y Fu(\()p Fr(v)s Fu(\))h Fn(is)456 2288 y(twic)l(e)25 b(di\013er)l(entiable)i(in)e Fr(t)g Fn(for)h Fr(t)d(>)f Fu(0)j Fn(and)g(c)l(ontinuous)g(in)g Fr(v)j Fn(with)e(r)l(esp)l(e)l(ct)e(to)h(the)h Fr(W)3134 2258 y Fp(k)q(;)p Fs(2)3227 2288 y Fu(\()p FA(K)3334 2258 y Fp(d)3334 2308 y(!)3382 2288 y Fu(\))p Fn(-)456 2387 y(top)l(olo)l(gy.)456 2565 y(Pr)l(o)l(of.)43 b Fu(On)33 b(pages)f(272{274)e(of)j([T)-7 b(a)n(y97)n(],)35 b(the)f(reader)e(ma)n (y)g(\014nd)i(the)g(details)f(of)g(the)g(\014xed)456 2665 y(p)r(oin)n(t)f(argumen)n(t)f(in)i(in)f Fr(C)6 b Fu(\([0)p Fr(;)14 b(T)e Fu(])p Fr(;)i(L)1633 2635 y Fs(2)1669 2665 y Fu(\))33 b(whic)n(h)f(solv)n(es)f(\(A.3\).)51 b(The)32 b(fact)h(that)f(the)h(solution)456 2764 y(is)40 b(in)g Fr(W)751 2734 y Fp(k)q(;)p Fs(2)845 2764 y Fu(\()p FA(K)952 2734 y Fp(d)952 2785 y(!)1000 2764 y Fu(\))g(w)n(as)f (established)h(in)g(Lemma)g(5.9.)74 b(F)-7 b(or)39 b(the)i(con)n(tin)n (uit)n(y)e(of)h(the)h(map)456 2864 y Fr(v)h Fq(7!)c Fu(\010)719 2834 y Fp(t)748 2864 y Fu(\()p Fr(v)s Fu(\),)j(recall)36 b(Prop)r(osition)f(5.12.)64 b(The)37 b(only)f(thing)h(that)h(remains)e (to)h(b)r(e)g(done)g(is)456 2964 y(to)g(pro)n(v)n(e)e(the)j(di\013eren) n(tiabilit)n(y)f(with)h(resp)r(ect)f(to)g Fr(t)g Fu(for)g Fr(t)i(>)g Fu(0.)66 b(F)-7 b(or)36 b(this)i(it)g(su\016ces)f(to)456 3063 y(consider)26 b(\(A.4\))i(and)g(note)f(that)h(the)g(RHS)g(is)g (indeed)g(di\013eren)n(tiable)f(for)g Fr(t)c(>)g Fu(0.)555 3165 y(W)-7 b(e)26 b(note)e(that)h(for)g Fr(t)e(>)g Fu(0,)i(w)n(e)f(ha) n(v)n(e)1740 3133 y Fp(d)p 1728 3147 60 4 v 1728 3194 a(dt)1797 3165 y Fr(e)1836 3135 y Fp(tL)1911 3165 y Fr(v)i Fu(=)d Fr(Le)2161 3135 y Fp(tL)2235 3165 y Fr(v)s Fu(,)i(and)g(the)h(b) r(ounds)f(on)f(its)h(op)r(erator)456 3265 y(norm)33 b(follo)n(w)g(from) g(\(A.2\).)56 b(This)34 b(con)n(trols)e(the)i(\014rst)f(term)h(in)g (\(A.4\).)56 b(Similarly)-7 b(,)35 b(w)n(e)e(can)456 3365 y(see)26 b(that)h(the)f(second)g(term)h(in)g(\(A.4\))g(can)f(also) f(b)r(e)i(di\013eren)n(tiated,)g(pro)n(vided)e(that)i(\010)3225 3335 y Fp(t)3254 3365 y Fu(\()p Fr(v)s Fu(\))h(is)456 3464 y(con)n(tin)n(uous.)35 b(In)28 b(this)g(case,)f(w)n(e)g(ha)n(v)n (e)1129 3615 y Fr(d)p 1114 3652 74 4 v 1114 3729 a(dt)1211 3559 y Fo(Z)1294 3579 y Fp(t)1257 3747 y Fs(0)1337 3672 y Fr(e)1376 3637 y Fs(\()p Fp(t)p Fm(\000)p Fp(s)p Fs(\))p Fp(L)1600 3579 y Fo(h)1639 3672 y Fr(f)1702 3579 y Fo(\020)1752 3672 y Fr(x;)14 b(u)1884 3637 y Fs(\(0\))1991 3672 y Fu(+)k(\010)2134 3637 y Fp(s)2170 3672 y Fu(\()p Fr(v)s Fu(\))2277 3579 y Fo(\021)2346 3672 y Fq(\000)g Fr(f)2492 3579 y Fo(\020)2542 3672 y Fr(x;)c(u)2674 3637 y Fs(\(0\))2763 3579 y Fo(\021)o(i)2879 3672 y Fr(ds)957 3876 y Fu(=)82 b Fr(f)1168 3784 y Fo(\020)1217 3876 y Fr(x;)14 b(u)1349 3842 y Fs(\(0\))1457 3876 y Fu(+)k(\010)1600 3842 y Fp(t)1629 3876 y Fu(\()p Fr(v)s Fu(\))1736 3784 y Fo(\021)1805 3876 y Fq(\000)g Fr(f)1951 3784 y Fo(\020)2001 3876 y Fr(x;)c(u)2133 3842 y Fs(\(0\))2222 3784 y Fo(\021)456 3876 y Fu(\(A.5\))1104 4091 y(+)1183 3978 y Fo(Z)1266 3999 y Fp(t)1229 4167 y Fs(0)1309 4091 y Fr(e)1348 4057 y Fs(\()p Fp(t)p Fm(\000)p Fp(s)p Fs(\))p Fp(L)1557 4091 y Fr(L)1628 3999 y Fo(h)1667 4091 y Fr(f)1730 3999 y Fo(\020)1780 4091 y Fr(x;)g(u)1912 4057 y Fs(\(0\))2019 4091 y Fu(+)k(\010)2162 4057 y Fp(s)2198 4091 y Fu(\()p Fr(v)s Fu(\))2305 3999 y Fo(\021)2374 4091 y Fq(\000)g Fr(f)2520 3999 y Fo(\020)2570 4091 y Fr(x;)c(u)2702 4057 y Fs(\(0\))2791 3999 y Fo(\021)o(i)2907 4091 y Fr(ds)g(:)456 4292 y Fu(Giv)n(en)27 b(the)h(b)r(ounds)g(in)g(\(A.2\))g(w)n(e)f (obtain:)1250 4464 y Fq(k)p Fr(e)1331 4429 y Fp(tL)1405 4464 y Fr(L)p Fq(k)1504 4480 y Fm(L)p Fs(\()o Fp(W)1646 4464 y Fk(r)o(;)p Fi(2)1725 4480 y Fs(\()p Fl(K)1809 4464 y Fk(d)1809 4497 y(!)1851 4480 y Fs(\))p Fp(;W)1968 4464 y Fk(s;)p Fi(2)2048 4480 y Fs(\()p Fl(K)2132 4464 y Fk(d)2132 4497 y(!)2174 4480 y Fs(\)\))2253 4464 y Fq(\024)c Fr(C)6 b(t)2446 4404 y Fk(r)q Fh(\000)p Fk(s)p Fh(\000)p Fi(2)p 2446 4417 176 3 v 2520 4449 a(2)456 4618 y Fu(for)27 b(an)n(y)f(0)d Fr(<)g(t)g Fq(\024)f Fu(1,)28 b Fr(s)23 b Fq(\025)f Fr(r)k Fq(\025)d Fu(0)k(and)g(some)g (constan)n(t)g Fr(C)6 b Fu(,)28 b(p)r(ossibly)f(dep)r(ending)h(on)g Fr(r)i Fu(and)e Fr(s)p Fu(.)555 4717 y(The)e(last)f(term)g(in)h (\(A.5\))g(is)f(therefore)g(a)g(con)n(v)n(ergen)n(t)e(in)n(tegral)h(in) i Fr(W)2780 4687 y Fp(k)q(;)p Fs(2)2874 4717 y Fu(\()p FA(K)2981 4687 y Fp(d)2981 4738 y(!)3029 4717 y Fu(\))g(if)g(w)n(e)f (ha)n(v)n(e)456 4817 y(that)i(\010)695 4787 y Fp(t)725 4817 y Fu(\()p Fr(v)s Fu(\))h(in)g Fr(W)1047 4787 y Fp(k)q Fs(+1)p Fp(;)p Fs(2)1225 4817 y Fu(\()p FA(K)1332 4787 y Fp(d)1332 4838 y(!)1380 4817 y Fu(\).)555 4917 y(The)23 b(second)g(deriv)-5 b(ativ)n(e)22 b(can)h(b)r(e)g(established)g(in)g (the)h(same)e(w)n(a)n(y)-7 b(.)34 b(W)-7 b(e)24 b(just)f(need)g(to)g(c) n(hec)n(k)456 5016 y(that)29 b(the)g(form)n(ulas)f(for)g(the)h(deriv)-5 b(ativ)n(e)28 b(with)i(resp)r(ect)e(to)h Fr(t)g Fu(w)n(e)f(ha)n(v)n(e)g (just)h(established)g(can)456 5116 y(b)r(e)f(di\013eren)n(tiated)g (again)f(with)h(resp)r(ect)g(to)g Fr(t)p Fu(.)38 b(The)28 b(pro)r(cedure)f(is)h(the)g(same)g(as)f(b)r(efore)h(and)456 5216 y(w)n(e)f(th)n(us)g(lea)n(v)n(e)g(the)h(details)f(to)h(the)g (reader.)1512 b Fj(\003)p eop %%Page: 37 37 37 36 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(37)555 450 y Fu(F)-7 b(ollo)n(wing)24 b(is)i(the)f(v)n(ersion)f(of)i(the)f(parab)r (olic)f(maxim)n(um)i(principle)f(whic)n(h)g(is)h(con)n(v)n(enien)n(t) 456 550 y(for)41 b(our)f(applications.)78 b(Though)41 b(its)g(pro)r(of)g(is)h(quite)f(standard)g(and)g(it)h(is)g(inspired)f (b)n(y)456 649 y(the)e(classical)f(results)h(\(see,)j(e.g.,)f([Ev)-5 b(a98)o(]\),)42 b(w)n(e)d(giv)n(e)f(it)i(in)f(full)h(detail)f(for)g (the)h(sak)n(e)e(of)456 749 y(completeness)27 b(\(and)g(similarly)g (for)g(other)g(standard)g(pro)r(ofs)g(here)g(b)r(elo)n(w\).)456 869 y FA(Prop)s(osition)g(A.5.)39 b Fn(L)l(et)27 b Fr(T)34 b(>)23 b Fu(0)p Fn(,)28 b Fr(c)23 b Fq(2)h Fr(L)1784 839 y Fm(1)1853 869 y Fu(\()p FA(R)1957 839 y Fp(d)2010 869 y Fq(\002)14 b Fu(\(0)p Fr(;)g(T)e Fu(\)\))27 b Fn(and)i(assume)e (that)h Fr(w)r Fu(\()p Fr(x;)14 b(t)p Fu(\))29 b Fn(solves)1272 1009 y Fr(@)1316 1021 y Fp(t)1345 1009 y Fr(w)r Fu(\()p Fr(x;)14 b(t)p Fu(\))20 b(+)e Fr(c)p Fu(\()p Fr(x;)c(t)p Fu(\))g Fr(w)r Fu(\()p Fr(x;)g(t)p Fu(\))25 b(=)e(\()p Fr(Lw)r Fu(\)\()p Fr(x;)14 b(t)p Fu(\))456 1149 y Fn(for)40 b(any)f Fr(t)i Fq(2)f Fu(\(0)p Fr(;)14 b(T)e Fu(\))39 b Fn(and)g Fr(x)i Fq(2)g FA(R)1601 1119 y Fp(d)1639 1149 y Fn(.)68 b(Supp)l(ose)39 b Fr(w)r Fu(\()p Fr(x;)14 b Fu(0\))42 b Fq(\025)e Fu(0)e Fn(for)i(any)g Fr(x)h Fq(2)g FA(R)3102 1119 y Fp(d)3140 1149 y Fn(.)67 b(Then,)456 1249 y Fr(w)r Fu(\()p Fr(x;)14 b(t)p Fu(\))24 b Fq(\025)f Fu(0)29 b Fn(for)i(any)f Fr(x)23 b Fq(2)h FA(R)1391 1219 y Fp(d)1459 1249 y Fn(and)30 b(any)g Fr(t)23 b Fq(2)h Fu([0)p Fr(;)14 b(T)e Fu(\))p Fn(.)456 1406 y(Pr)l(o)l(of.)43 b Fu(Let)26 b Fr(M)32 b Fq(\025)23 b Fu(1)15 b(+)h(\(1)p Fr(=T)c Fu(\))25 b(so)h(that)h Fq(j)p Fr(c)p Fu(\()p Fr(x;)14 b(t)p Fu(\))p Fq(j)24 b(\024)e Fr(M)36 b Fu(for)25 b(an)n(y)h Fr(x)e Fq(2)f FA(R)2706 1376 y Fp(d)2771 1406 y Fu(and)j(an)n(y)g Fr(t)d Fq(2)g Fu(\(0)p Fr(;)14 b(T)e Fu(\).)456 1506 y(W)-7 b(e)31 b(pro)n(v)n(e)e(the)j(claim)e(for)h Fr(t)d Fq(2)h Fu(\(0)p Fr(;)f Fu(1)p Fr(=)p Fu(\(2)p Fr(M)9 b Fu(\)],)31 b(the)g(general)e(case)i(follo)n(wing)e(b)n(y)i (\015o)n(wing)f(the)456 1606 y(solution)d(iterativ)n(ely)-7 b(.)555 1705 y(Assume,)21 b(b)n(y)e(con)n(tradiction,)g(that)g Fr(w)r Fu(\()5 b(\026)-47 b Fr(x)r(;)1858 1690 y Fu(\026)1857 1705 y Fr(t)p Fu(\))23 b Fr(<)g Fu(0,)d(for)e(some)24 b(\026)-47 b Fr(x)23 b Fq(2)h FA(R)2654 1675 y Fp(d)2711 1705 y Fu(and)2865 1690 y(\026)2864 1705 y Fr(t)f Fq(2)g Fu(\(0)p Fr(;)28 b Fu(1)p Fr(=)p Fu(\(2)p Fr(M)9 b Fu(\)].)456 1805 y(Let)27 b(us)h(de\014ne)1356 1974 y Fr(")83 b Fu(:=)g(min)1801 1857 y Fo(\032)1918 1918 y Fu(1)p 1873 1955 132 4 v 1873 2031 a(4)p Fr(M)2014 1974 y(;)28 b Fq(j)p Fr(w)r Fu(\()5 b(\026)-47 b Fr(x)q(;)2268 1959 y Fu(\026)2266 1974 y Fr(t)p Fu(\))p Fq(j)2351 1857 y Fo(\033)2640 1974 y Fu(and)1126 2158 y Fr(W)12 b Fu(\()p Fr(x;)i(t)p Fu(\))84 b(:=)f Fr(w)r Fu(\()p Fr(x;)14 b(t)p Fu(\))20 b(+)e Fr("t)g Fu(+)g Fr(")2200 2124 y Fs(2)2251 2158 y Fr(:)456 2298 y Fu(Then,)29 b Fr(W)12 b Fu(\()p Fr(x;)i Fu(0\))25 b Fr(>)f Fu(0)k(for)h(an)n(y)e Fr(x)f Fq(2)f FA(R)1672 2268 y Fp(d)1739 2298 y Fu(and)j Fr(W)12 b Fu(\()5 b(\026)-47 b Fr(x)q(;)2109 2283 y Fu(\026)2108 2298 y Fr(t)p Fu(\))25 b Fr(<)f Fu(0,)29 b(due)g(to)f(our)g(c)n(hoice)g(of)g Fr(")p Fu(.)40 b(Then,)456 2398 y(there)e(exists)h Fr(x)967 2368 y Fp(?)1047 2398 y Fq(2)j FA(R)1216 2368 y Fp(d)1294 2398 y Fu(and)c Fr(t)1496 2368 y Fp(?)1576 2398 y Fq(2)k Fu([0)p Fr(;)1776 2382 y Fu(\026)1775 2398 y Fr(t)p Fu(])d(so)f(that)h Fr(W)12 b Fu(\()p Fr(x;)i(t)p Fu(\))43 b Fr(>)e Fu(0)e(for)f(an)n(y)g Fr(x)k Fq(2)g FA(R)3233 2368 y Fp(d)3311 2398 y Fu(and)456 2497 y Fr(t)23 b Fq(2)g Fu([0)p Fr(;)k(t)732 2467 y Fp(?)771 2497 y Fu(\),)h(and)f Fr(W)12 b Fu(\()p Fr(x)1184 2467 y Fp(?)1223 2497 y Fr(;)i(t)1290 2467 y Fp(?)1328 2497 y Fu(\))23 b(=)g(0.)36 b(In)28 b(particular,)1492 2638 y Fr(W)12 b Fu(\()p Fr(x;)i(t)1728 2603 y Fp(?)1767 2638 y Fu(\))24 b Fq(\025)e Fu(0)h(=)f Fr(W)12 b Fu(\()p Fr(x)2231 2603 y Fp(?)2270 2638 y Fr(;)i(t)2337 2603 y Fp(?)2375 2638 y Fu(\))456 2778 y(for)27 b(an)n(y)f Fr(x)e Fq(2)f FA(R)960 2748 y Fp(d)999 2778 y Fu(,)k(and)h(so)456 2918 y(\(A.6\))456 b Fr(@)1147 2930 y Fp(x)1189 2918 y Fr(W)12 b Fu(\()p Fr(x)1358 2884 y Fp(?)1397 2918 y Fr(;)i(t)1464 2884 y Fp(?)1502 2918 y Fu(\))23 b(=)g(0)165 b(and)h Fr(D)2223 2884 y Fs(2)2221 2938 y Fp(x)2263 2918 y Fr(W)12 b Fu(\()p Fr(x)2432 2884 y Fp(?)2471 2918 y Fr(;)i(t)2538 2884 y Fp(?)2576 2918 y Fu(\))23 b Fq(\025)g Fu(0)14 b Fr(:)456 3058 y Fu(Analogously)-7 b(,)1492 3163 y Fr(W)12 b Fu(\()p Fr(x)1661 3129 y Fp(?)1700 3163 y Fr(;)i(t)p Fu(\))24 b Fq(\025)e Fu(0)h(=)f Fr(W)12 b Fu(\()p Fr(x)2231 3129 y Fp(?)2270 3163 y Fr(;)i(t)2337 3129 y Fp(?)2375 3163 y Fu(\))456 3286 y(for)27 b(an)n(y)f Fr(t)e Fq(2)f Fu([0)p Fr(;)14 b(t)1003 3256 y Fp(?)1041 3286 y Fu(\))28 b(and)f(so)456 3426 y(\(A.7\))999 b Fr(@)1690 3438 y Fp(t)1720 3426 y Fr(W)12 b Fu(\()p Fr(x)1889 3392 y Fp(?)1927 3426 y Fr(;)i(t)1994 3392 y Fp(?)2032 3426 y Fu(\))24 b Fq(\024)e Fu(0)14 b Fr(:)456 3567 y Fu(Thence,)27 b(if)h Fr(z)888 3536 y Fp(?)949 3567 y Fu(:=)23 b(\()p Fr(x)1139 3536 y Fp(?)1177 3567 y Fr(;)14 b(t)1244 3536 y Fp(?)1282 3567 y Fu(\),)29 b(com)n(bining)f(\(A.6\))g(with)g(\(A.7\),)g(w)n(e)f (deduce)h(that)456 3732 y(\(A.8\))1540 3639 y Fo(\020)1590 3732 y Fu(\()p Fr(L)18 b Fq(\000)g Fr(@)1824 3744 y Fp(t)1854 3732 y Fu(\))p Fr(W)1976 3639 y Fo(\021)2025 3732 y Fu(\()p Fr(z)2100 3697 y Fp(?)2138 3732 y Fu(\))23 b Fq(\025)g Fu(0)14 b Fr(:)456 3892 y Fu(On)27 b(the)h(other)f(hand,)h(b)n(y)f(a)g (direct)h(computation,)857 3960 y Fo(\020)906 4053 y Fu(\()p Fr(L)19 b Fq(\000)f Fr(@)1141 4065 y Fp(t)1170 4053 y Fu(\))p Fr(W)1292 3960 y Fo(\021)1342 4053 y Fu(\()p Fr(z)1417 4018 y Fp(?)1455 4053 y Fu(\))83 b(=)1717 3960 y Fo(\020)1767 4053 y Fu(\()p Fr(L)18 b Fq(\000)g Fr(@)2001 4065 y Fp(t)2031 4053 y Fu(\))p Fr(w)2124 3960 y Fo(\021)2174 4053 y Fu(\()p Fr(z)2249 4018 y Fp(?)2287 4053 y Fu(\))h Fq(\000)f Fr(")k Fu(=)1570 4202 y(=)82 b Fr(c)p Fu(\()p Fr(z)1828 4168 y Fp(?)1866 4202 y Fu(\))14 b Fr(w)r Fu(\()p Fr(z)2048 4168 y Fp(?)2087 4202 y Fu(\))k Fq(\000)g Fr(")23 b Fu(=)1570 4328 y(=)82 b Fr(c)p Fu(\()p Fr(z)1828 4293 y Fp(?)1866 4328 y Fu(\))14 b Fr(W)e Fu(\()p Fr(z)2077 4293 y Fp(?)2115 4328 y Fu(\))19 b Fq(\000)f Fr(c)p Fu(\()p Fr(z)2360 4293 y Fp(?)2397 4328 y Fu(\))c(\()p Fr("t)2544 4293 y Fp(?)2601 4328 y Fu(+)k Fr(")2723 4293 y Fs(2)2760 4328 y Fu(\))h Fq(\000)f Fr(")k Fu(=)1570 4453 y(=)82 b Fq(\000)p Fr(c)p Fu(\()p Fr(z)1893 4419 y Fp(?)1931 4453 y Fu(\))14 b(\()p Fr("t)2078 4419 y Fp(?)2134 4453 y Fu(+)k Fr(")2256 4419 y Fs(2)2293 4453 y Fu(\))h Fq(\000)f Fr(")23 b Fq(\024)1570 4628 y(\024)82 b Fr(M)9 b(")1873 4511 y Fo(\022)1989 4572 y Fu(1)p 1945 4609 V 1945 4685 a(2)p Fr(M)2104 4628 y Fu(+)18 b Fr(")2226 4511 y Fo(\023)2305 4628 y Fq(\000)g Fr(")37 b(<)f Fu(0)14 b Fr(;)456 4818 y Fu(due)27 b(to)h(our)f(c)n(hoice)f(of)i Fr(")p Fu(,)f(in)h(con)n (tradiction)f(with)h(\(A.8\).)1082 b Fj(\003)555 4975 y Fu(A)29 b(consequence)f(of)h(the)h(ab)r(o)n(v)n(e)d(maxim)n(um)i (principle)g(is)f(the)i(follo)n(wing)e(parab)r(olic)f(com-)456 5075 y(parison)f(principle:)456 5195 y FA(Prop)s(osition)k(A.6.)40 b Fn(L)l(et)30 b Fr(v)1370 5207 y Fs(0)1430 5195 y Fq(\025)23 b Fr(w)1577 5207 y Fs(0)1637 5195 y Fq(2)h(C)5 b Fn(.)38 b(Then,)31 b Fu(\010)2130 5165 y Fp(t)2159 5195 y Fu(\()p Fr(v)2231 5207 y Fs(0)2269 5195 y Fu(\))23 b Fq(\025)g Fu(\010)2472 5165 y Fp(t)2501 5195 y Fu(\()p Fr(w)2592 5207 y Fs(0)2630 5195 y Fu(\))p Fn(,)31 b(for)f(any)g Fr(t)23 b Fq(\025)g Fu(0)p Fn(.)p eop %%Page: 38 38 38 37 bop 456 251 a Fs(38)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fn(Pr)l(o)l(of.)43 b Fu(Denote)28 b Fr(\020)6 b Fu(\()p Fr(x;)14 b(t)p Fu(\))24 b(:=)f(\010)1416 420 y Fp(t)1445 450 y Fu(\()p Fr(v)1517 462 y Fs(0)1555 450 y Fu(\))18 b Fq(\000)g Fu(\010)1748 420 y Fp(t)1778 450 y Fu(\()p Fr(w)1869 462 y Fs(0)1907 450 y Fu(\))27 b(and)736 645 y Fr(c)p Fu(\()p Fr(x;)14 b(t)p Fu(\))24 b(:=)1084 532 y Fo(Z)1167 553 y Fs(1)1131 721 y(0)1219 645 y Fr(@)1263 657 y Fp(\026)1307 645 y Fr(f)1357 553 y Fo(\020)1406 645 y Fr(x;)14 b(u)1538 611 y Fs(\(0\))1627 645 y Fu(\()p Fr(x)p Fu(\))20 b(+)e Fr(\033)s Fu(\010)1951 611 y Fp(t)1980 645 y Fu(\()p Fr(v)2052 657 y Fs(0)2090 645 y Fu(\)\()p Fr(x)p Fu(\))i(+)e(\(1)g Fq(\000)g Fr(\033)s Fu(\))c(\010)2667 611 y Fp(t)2697 645 y Fu(\()p Fr(w)2788 657 y Fs(0)2826 645 y Fu(\)\()p Fr(x)p Fu(\))2969 553 y Fo(\021)3034 645 y Fr(d\033)j(:)456 823 y Fu(Then,)948 961 y Fr(@)992 973 y Fp(t)1022 961 y Fr(\020)89 b Fu(=)83 b Fr(L\020)24 b Fu(+)18 b Fr(f)9 b Fu(\()p Fr(x;)14 b(u)1709 927 y Fs(\(0\))1817 961 y Fu(+)k(\010)1960 927 y Fp(t)1989 961 y Fu(\()p Fr(w)2080 973 y Fs(0)2118 961 y Fu(\)\))h Fq(\000)f Fr(f)9 b Fu(\()p Fr(x;)14 b(u)2498 927 y Fs(\(0\))2605 961 y Fu(+)k(\010)2748 927 y Fp(t)2777 961 y Fu(\()p Fr(v)2849 973 y Fs(0)2887 961 y Fu(\)\))1147 1086 y(=)83 b Fr(L\020)24 b Fq(\000)18 b Fr(c\020)i(:)456 1221 y Fu(Therefore,)26 b(b)n(y)h(Prop)r(osition)f (A.5,)i Fr(\020)h Fq(\025)23 b Fu(0.)1567 b Fj(\003)456 1373 y Fu(A.3.)41 b FA(Summary)34 b(of)i(some)d(results)i(in)g Fu([V)-7 b(al04)o(])p FA(.)41 b Fu(W)-7 b(e)32 b(no)n(w)e(recall)g(t)n (w)n(o)g(results,)i(namely)456 1472 y(Theorems)20 b(A.7{A.8)h(here)f(b) r(elo)n(w,)j(for)e(the)g(pro)r(of)g(of)h(whic)n(h)f(w)n(e)g(refer)g(to) g([V)-7 b(al04)o(],)23 b(concerning)456 1572 y(the)34 b(existence)h(and)f(the)h(geometric)e(prop)r(erties)h(of)g(lo)r(cal)g (minimizers)g(for)g(the)h(functional)456 1671 y Fq(E)500 1683 y Fp(!)547 1671 y Fu(.)i(These)28 b(results)f(are)f(used)i(during) f(the)h(pro)r(ofs)f(of)g(the)h(main)g(theorems)f(of)g(this)h(pap)r(er.) 555 1771 y(First,)g(w)n(e)f(recall)g(a)g(densit)n(y)g(estimate)h(\(see) f(also)g([CC95)o(]\):)456 1889 y FA(Theorem)i(A.7.)39 b Fn(L)l(et)28 b Fu(\012)23 b Fq(\032)g FA(R)1458 1859 y Fp(d)1525 1889 y Fn(b)l(e)29 b(a)g(b)l(ounde)l(d)g(domain.)40 b(Fix)29 b Fr(\022)2529 1901 y Fs(0)2589 1889 y Fq(2)23 b Fu(\(0)p Fr(;)14 b Fu(1\))28 b Fn(and)i Fr(r)3078 1901 y Fs(1)3138 1889 y Fr(>)23 b Fu(0)p Fn(.)38 b(L)l(et)456 1988 y Fr(u)22 b Fq(2)i Fr(W)695 1958 y Fs(1)p Fp(;)p Fs(2)785 1988 y Fu(\(\012\))p Fn(.)39 b(Assume)29 b(that)g Fr(u)h Fn(is)g(a)g(lo)l(c)l(al)h(minimizer)f(in)g Fu(\012)p Fn(,)g(i.e.,)i(that)1273 2124 y Fq(E)1317 2136 y Fs(\012)1368 2124 y Fu(\()p Fr(u)19 b Fu(+)f Fr(\036)p Fu(\))23 b Fq(\025)g(E)1786 2136 y Fs(\012)1837 2124 y Fu(\()p Fr(u)p Fu(\))184 b Fq(8)p Fr(\036)23 b Fq(2)g Fr(C)2395 2090 y Fm(1)2389 2145 y Fs(0)2466 2124 y Fu(\(\012\))14 b Fr(:)456 2260 y Fn(Assume)39 b(also)i(that)f Fq(j)p Fr(u)p Fu(\()5 b(\026)-47 b Fr(x)p Fu(\))p Fq(j)43 b Fr(<)e(\022)1529 2272 y Fs(0)1566 2260 y Fn(.)70 b(Then,)44 b(ther)l(e)c(exist)g Fr(\024;)27 b(r)2472 2272 y Fs(0)2552 2260 y Fr(>)41 b Fu(0)p Fn(,)i(dep)l(ending)f(only)e(on)456 2359 y(universal)30 b(c)l(onstants)f(and)h(on)g Fr(\022)1495 2371 y Fs(0)1562 2359 y Fn(and)g Fr(r)1760 2371 y Fs(1)1798 2359 y Fn(,)g(and)g Fr(x)2061 2371 y Fs(1)2099 2359 y Fr(;)e(x)2197 2371 y Fs(2)2257 2359 y Fq(2)c FA(R)2408 2329 y Fp(d)2446 2359 y Fn(,)30 b(such)g(that)643 2495 y Fr(B)706 2507 y Fp(\024r)781 2495 y Fu(\()p Fr(x)860 2507 y Fs(1)898 2495 y Fu(\))24 b Fq(\022)e(f)p Fr(u)g Fq(\025)h Fr(\022)1280 2507 y Fs(0)1317 2495 y Fq(g)18 b(\\)h Fr(B)1514 2507 y Fp(r)1551 2495 y Fu(\()5 b(\026)-47 b Fr(x)p Fu(\))170 b(and)g Fr(B)2199 2507 y Fp(\024r)2274 2495 y Fu(\()p Fr(x)2353 2507 y Fs(2)2391 2495 y Fu(\))24 b Fq(\022)e(f)p Fr(u)h Fq(\024)f(\000)p Fr(\022)2838 2507 y Fs(0)2875 2495 y Fq(g)c(\\)h Fr(B)3072 2507 y Fp(r)3108 2495 y Fu(\()5 b(\026)-47 b Fr(x)q Fu(\))14 b Fr(;)456 2631 y Fn(pr)l(ovide)l(d)31 b(that)f Fr(r)c Fq(\025)d Fr(r)1141 2643 y Fs(0)1208 2631 y Fn(and)30 b Fr(B)1432 2643 y Fs(2)p Fp(r)r Fs(+)p Fp(r)1580 2651 y Fi(1)1616 2631 y Fu(\()5 b(\026)-47 b Fr(x)q Fu(\))23 b Fq(\022)g Fu(\012)p Fn(.)555 2748 y Fu(No)n(w,)f(w)n(e)f(recall)g(the)h(existence)f(of)g(a) g(plane-lik)n(e,)h(lo)r(cal)e(minimizer)i(for)f Fq(E)2857 2760 y Fp(!)2904 2748 y Fu(,)i(with)f(b)r(ounded)456 2848 y(energy)-7 b(,)26 b(satisfying)h(the)h(Birkho\013)f(prop)r(ert)n (y:)456 2966 y FA(Theorem)j(A.8.)40 b Fn(Fix)30 b Fr(\022)c Fq(2)d Fu(\(0)p Fr(;)14 b Fu(1\))p Fn(.)38 b(Then,)31 b(ther)l(e)f(exists)f Fr(u)23 b Fq(2)g Fr(C)2504 2936 y Fs(2)2542 2966 y Fu(\()p FA(K)2649 2936 y Fp(d)2649 2986 y(!)2697 2966 y Fu(\))30 b Fn(such)g(that)661 3083 y Fq(\017)41 b(E)788 3095 y Fs(\012)840 3083 y Fu(\()p Fr(u)22 b Fu(+)g Fr(\036)p Fu(\))33 b Fq(\025)g(E)1285 3095 y Fs(\012)1336 3083 y Fu(\()p Fr(u)p Fu(\))p Fn(,)k(for)f(al)t(l)g (smo)l(oth)g(functions)f Fr(\036)h Fn(supp)l(orte)l(d)f(in)g(any)h(b)l (ounde)l(d)744 3183 y(domain)31 b Fu(\012)p Fn(;)661 3283 y Fq(\017)41 b(E)788 3295 y Fp(!)836 3283 y Fu(\()p Fr(u)p Fu(\))23 b Fr(<)g Fu(+)p Fq(1)p Fn(,)30 b Fq(j)p Fr(u)p Fq(j)22 b(\024)h Fu(1)29 b Fn(and)i Fr(u)e Fn(satis\014es)h(the) g(Birkho\013)h(pr)l(op)l(erty;)661 3382 y Fq(\017)41 b Fn(Ther)l(e)29 b(exists)f Fr(M)j(>)23 b Fu(0)p Fn(,)28 b(dep)l(ending)i(only)e(on)g Fr(\022)i Fn(and)f(on)f(universal)g (quantities,)h(such)744 3482 y(that)i Fr(u)p Fu(\()p Fr(x)p Fu(\))26 b Fq(2)g Fu([)p Fr(\022)r(;)14 b Fu(1])31 b Fn(pr)l(ovide)l(d)i(that)e Fr(!)22 b Fq(\001)d Fr(x)26 b Fq(\025)f Fr(M)d Fq(j)p Fr(!)s Fq(j)31 b Fn(and)h Fr(u)p Fu(\()p Fr(x)p Fu(\))26 b Fq(2)f Fu([)p Fq(\000)p Fu(1)p Fr(;)14 b Fq(\000)p Fr(\022)r Fu(])30 b Fn(pr)l(ovide)l(d)744 3582 y(that)g Fr(!)21 b Fq(\001)e Fr(x)k Fq(\024)g(\000)p Fr(M)f Fq(j)p Fr(!)s Fq(j)p Fn(.)1708 3754 y Fv(References)456 3887 y Fz([ALD83])93 b(S.)24 b(Aubry)g(and)h(P)-6 b(.)24 b(Y.)g(Le)g(Daeron,)g Ft(The)j(discr)l(ete)f(Fr)l(enkel-Kontor)l(ova)h (mo)l(del)h(and)f(its)f(exten-)810 3970 y(sions.)g(I.)g(Exact)g(r)l (esults)g(for)g(the)f(gr)l(ound-states)p Fz(,)f(Ph)n(ys.)g(D)f Fa(8)g Fz(\(1983\),)i(no.)f(3,)f(381{422.)456 4053 y([Ang90a])82 b(Sigurd)29 b(B.)f(Angenen)n(t,)j Ft(Monotone)g(r)l(e)l(curr)l(enc)l(e) g(r)l(elations,)h(their)e(Birkho\013)g(orbits)h(and)g(top)l(o-)810 4136 y(lo)l(gic)l(al)c(entr)l(opy)p Fz(,)d(Ergo)r(dic)g(Theory)g (Dynam.)e(Systems)h Fa(10)g Fz(\(1990\),)j(no.)d(1,)h(15{41.)456 4219 y([Ang90b])78 b(Sigurd)28 b(B.)g(Angenen)n(t,)i Ft(Nonline)l(ar)h(analytic)f(semi\015ows)p Fz(,)f(Pro)r(c.)f(Ro)n(y)-6 b(.)28 b(So)r(c.)g(Edin)n(burgh)g(Sect.)810 4302 y(A)c Fa(115)e Fz(\(1990\),)j(no.)f(1-2,)f(91{107.)456 4385 y([Ban87])120 b(V.)17 b(Bangert,)i Ft(The)h(existenc)l(e)f(of)h(gaps)h (in)f(minimal)g(foliations)p Fz(,)g(Aequationes)f(Math.)e Fa(34)g Fz(\(1987\),)810 4468 y(no.)24 b(2-3,)f(153{166.)456 4551 y([Ban89])120 b(V.)25 b(Bangert,)g Ft(On)i(minimal)h(laminations)g (of)f(the)g(torus)p Fz(,)e(Ann.)g(Inst.)g(H.)f(P)n(oincar)n(\023)-33 b(e)26 b(Anal.)e(Non)810 4634 y(Lin)n(\023)-33 b(eaire)24 b Fa(6)f Fz(\(1989\),)i(no.)f(2,)f(95{138.)456 4717 y([Ban90])120 b(V.)18 b(Bangert,)i Ft(L)l(aminations)i(of)f Fz(3)p Ft(-tori)g(by)f(le)l(ast)h(ar)l(e)l(a)i(surfac)l(es)p Fz(,)d(Analysis,)e(et)h(cetera,)i(Academic)810 4800 y(Press,)i(Boston,) h(MA,)f(1990,)h(pp.)f(85{114.)456 4883 y([Bes05])135 b(Ugo)35 b(Bessi,)i Ft(Many)f(plane)h(like)f(solutions)h(of)f(n)g (dimensional)i(el)t(liptic)e(pr)l(oblems)p Fz(,)i(preprin)n(t)810 4967 y(\(2005\).)456 5050 y([Bol95])139 b(Sergey)21 b(Bolotin,)g Ft(Homo)l(clinic)i(tr)l(aje)l(ctories)g(of)g(time)f(dep)l(endent)h (Hamiltonian)g(systems)p Fz(,)e(V)-6 b(ari-)810 5133 y(ational)23 b(and)g(lo)r(cal)f(metho)r(ds)g(in)g(the)h(study)g(of)f (Hamiltonian)g(systems)f(\(T)-6 b(rieste,)23 b(1994\),)g(W)-6 b(orld)810 5216 y(Sci.)24 b(Publishing,)e(Riv)n(er)i(Edge,)g(NJ,)f (1995,)h(pp.)g(1{16.)p eop %%Page: 39 39 39 38 bop 1031 251 a Fs(MUL)-5 b(TIPLICITY)29 b(F)n(OR)h(INTERF)-7 b(A)n(CES)29 b(IN)g(PERIODIC)g(MEDIA)508 b(39)456 450 y Fz([Bro65])131 b(F)-6 b(elix)31 b(E.)e(Bro)n(wder,)j Ft(In\014nite)g(dimensional)h(manifolds)h(and)f(non-line)l(ar)f(el)t (liptic)g(eigenvalue)810 533 y(pr)l(oblems)p Fz(,)25 b(Ann.)e(of)h(Math.)f(\(2\))i Fa(82)e Fz(\(1965\),)i(459{477.)456 616 y([CC95])142 b(Luis)23 b(A.)f(Ca\013arelli)g(and)h(An)n(tonio)h (C\023)-35 b(ordoba,)23 b Ft(Uniform)j(c)l(onver)l(genc)l(e)f(of)g(a)g (singular)h(p)l(erturb)l(a-)810 699 y(tion)g(pr)l(oblem)p Fz(,)e(Comm.)d(Pure)j(Appl.)f(Math.)h Fa(48)f Fz(\(1995\),)i(no.)e(1,)h (1{12.)456 782 y([CdlL98])90 b(A.)25 b(Candel)i(and)f(R.)f(de)h(la)g (Lla)n(v)n(e,)g Ft(On)h(the)h(Aubry-Mather)f(the)l(ory)h(in)f (statistic)l(al)h(me)l(chanics)p Fz(,)810 865 y(Comm.)21 b(Math.)j(Ph)n(ys.)f Fa(192)g Fz(\(1998\),)i(no.)f(3,)f(649{669.)456 948 y([CdlL01])90 b(Luis)33 b(A.)f(Ca\013arelli)h(and)g(Rafael)g(de)h (la)f(Lla)n(v)n(e,)i Ft(Planelike)g(minimizers)g(in)f(p)l(erio)l(dic)i (me)l(dia)p Fz(,)810 1031 y(Comm.)21 b(Pure)j(Appl.)f(Math.)h Fa(54)e Fz(\(2001\),)k(no.)d(12,)h(1403{1441.)456 1114 y([CdlL05])90 b(Luis)25 b(A.)g(Ca\013arelli)g(and)h(Rafael)f(de)h(la)f (Lla)n(v)n(e,)h Ft(Interfac)l(es)h(of)h(gr)l(ound)g(states)f(in)g (Ising)h(mo)l(dels)810 1197 y(with)e(p)l(erio)l(dic)h(c)l(o)l (e\016cients)p Fz(,)c(J.)g(Stat.)i(Ph)n(ys.)e Fa(118)g Fz(\(2005\),)i(no.)f(3-4,)f(687{719.)456 1285 y([CZES90])63 b(Vittorio)18 b(Coti)h(Zelati,)g(Iv)l(ar)f(Ek)n(eland,)i(and)2004 1268 y(\023)1997 1285 y(Eric)e(S)n(\023)-33 b(er)n(\023)g(e,)19 b Ft(A)h(variational)i(appr)l(o)l(ach)h(to)e(homo)l(clinic)810 1368 y(orbits)26 b(in)f(Hamiltonian)i(systems)p Fz(,)c(Math.)h(Ann.)f Fa(288)g Fz(\(1990\),)i(no.)e(1,)h(133{160.)456 1451 y([DG79])135 b(Ennio)31 b(De)g(Giorgi,)g Ft(Conver)l(genc)l(e)i(pr)l (oblems)g(for)f(functionals)h(and)g(op)l(er)l(ators)p Fz(,)h(Pro)r(ceedings)810 1534 y(of)24 b(the)i(In)n(ternational)g (Meeting)f(on)g(Recen)n(t)h(Metho)r(ds)f(in)f(Nonlinear)g(Analysis)g (\(Rome,)g(1978\))810 1617 y(\(Bologna\),)h(Pitagora,)f(1979,)g(pp.)g (131{188.)456 1700 y([Ev)l(a98])128 b(La)n(wrence)26 b(C.)e(Ev)l(ans,)g Ft(Partial)k(di\013er)l(ential)e(e)l(quations)p Fz(,)f(Graduate)h(Studies)f(in)g(Mathematics,)810 1783 y(v)n(ol.)f(19,)f(American)g(Mathematical)h(So)r(ciet)n(y)-6 b(,)24 b(Pro)n(vidence,)h(RI,)e(1998.)456 1866 y([GG82])134 b(Mariano)23 b(Giaquin)n(ta)h(and)f(Enrico)g(Giusti,)g Ft(On)h(the)h(r)l(e)l(gularity)g(of)h(the)e(minima)i(of)f(variational) 810 1949 y(inte)l(gr)l(als)p Fz(,)f(Acta)h(Math.)e Fa(148)g Fz(\(1982\),)i(31{46.)456 2037 y([GL50])145 b(V.)17 b(L.)h(Ginzburg)g (and)h(L.)e(Landau,)j Ft(On)g(the)g(the)l(ory)h(of)f(sup)l(er)l(c)l (onductivity)p Fz(,)g(Zh.)2983 2020 y(\022)2976 2037 y(Eksp)r(er.)e(T)-6 b(eoret.)810 2120 y(Fiz.)23 b Fa(20)g Fz(\(1950\),)i(1064{1082.)456 2203 y([GP58])141 b(V.)28 b(L.)g(Ginzburg)h(and)f(L.)g(P)-6 b(.)28 b(Pitaevski)-8 b(\025)-27 b(\020,)29 b Ft(On)h(the)f(the)l(ory)i(of)f(sup)l (er\015uidity)p Fz(,)f(So)n(viet)h(Ph)n(ysics.)810 2287 y(JETP)24 b Fa(34)i(\(7\))d Fz(\(1958\),)i(858{861)h(\(1240{1245)2135 2270 y(\024)2131 2287 y(Z.)d(Eksp)r(er.)h(T)-6 b(eoret.)24 b(Fiz.\).)456 2370 y([GT83])138 b(Da)n(vid)34 b(Gilbarg)f(and)i(Neil)e (S.)g(T)-6 b(rudinger,)36 b Ft(El)t(liptic)f(p)l(artial)h(di\013er)l (ential)f(e)l(quations)h(of)f(se)l(c-)810 2453 y(ond)21 b(or)l(der)p Fz(,)e(second)f(ed.,)g(Grundlehren)f(der)g(Mathematisc)n (hen)h(Wissensc)n(haften)h([F)-6 b(undamen)n(tal)810 2536 y(Principles)23 b(of)g(Mathematical)h(Sciences],)g(v)n(ol.)g(224,) g(Springer-V)-6 b(erlag,)22 b(Berlin,)g(1983.)456 2619 y([Hed32])121 b(Gusta)n(v)28 b(A.)e(Hedlund,)h Ft(Ge)l(o)l(desics)i(on) g(a)f(two-dimensional)i(R)n(iemannian)f(manifold)h(with)e(p)l(e-)810 2702 y(rio)l(dic)f(c)l(o)l(e\016cients)p Fz(,)c(Ann.)g(of)g(Math.)h Fa(33)f Fz(\(1932\),)i(719{739.)456 2785 y([Hen81])121 b(Daniel)29 b(Henry)-6 b(,)31 b Ft(Ge)l(ometric)g(the)l(ory)g(of)g (semiline)l(ar)h(p)l(ar)l(ab)l(olic)h(e)l(quations)p Fz(,)e(Lecture)g(Notes)f(in)810 2868 y(Mathematics,)24 b(v)n(ol.)f(840,)h(Springer-V)-6 b(erlag,)22 b(Berlin,)h(1981.)456 2951 y([KdlLR97])34 b(Hans)20 b(Ko)r(c)n(h,)h(Rafael)f(de)g(la)g(Lla)n (v)n(e,)h(and)f(Charles)g(Radin,)g Ft(A)n(ubry-Mather)h(the)l(ory)i (for)f(functions)810 3034 y(on)k(lattic)l(es)p Fz(,)d(Discrete)h(Con)n (tin.)g(Dynam.)f(Systems)g Fa(3)g Fz(\(1997\),)i(no.)f(1,)f(135{151.) 456 3117 y([KS00])150 b(Da)n(vid)27 b(Kinderlehrer)f(and)i(Guido)f (Stampacc)n(hia,)h Ft(A)n(n)h(intr)l(o)l(duction)h(to)e(variational)i (ine)l(qual-)810 3200 y(ities)36 b(and)h(their)f(applic)l(ations)p Fz(,)k(Classics)34 b(in)h(Applied)g(Mathematics,)j(v)n(ol.)c(31,)39 b(So)r(ciet)n(y)d(for)810 3283 y(Industrial)26 b(and)g(Applied)g (Mathematics)f(\(SIAM\),)h(Philadelphia,)g(P)-6 b(A,)25 b(2000,)h(Reprin)n(t)g(of)f(the)810 3366 y(1980)g(original.)456 3449 y([LS34])161 b(L.)30 b(Ljusternik)f(and)h(L.)g(Sc)n(hnirelmann,)g Ft(M)n(\023)-35 b(etho)l(des)33 b(top)l(olo)l(giques)g(dan)f(les)g(pr)l (obl)n(\022)-35 b(emes)33 b(varia-)810 3532 y(tionnels)p Fz(,)24 b(Hermann,)e(P)n(aris,)g(1934.)456 3616 y([Mat86])117 b(John)18 b(Mather,)g Ft(A)h(criterion)g(for)h(the)f(nonexistenc)l(e)g (of)h(invariant)f(cir)l(cles)p Fz(,)f(Inst.)g(Hautes)3238 3599 y(\023)3231 3616 y(Etudes)810 3699 y(Sci.)24 b(Publ.)f(Math.)g (\(1986\),)j(no.)d(63,)h(153{204.)456 3782 y([MF94])133 b(John)17 b(N.)e(Mather)h(and)h(Gio)n(v)l(anni)f(F)-6 b(orni,)17 b Ft(A)l(ction)h(minimizing)h(orbits)f(in)g(Hamiltonian)i (systems)p Fz(,)810 3865 y(T)-6 b(ransition)29 b(to)g(c)n(haos)g(in)f (classical)g(and)h(quan)n(tum)g(mec)n(hanics)f(\(Mon)n(tecatini)i(T)-6 b(erme,)28 b(1991\),)810 3948 y(Lecture)d(Notes)f(in)g(Math.,)f(v)n (ol.)g(1589,)i(Springer,)e(Berlin,)f(1994,)i(pp.)g(92{186.)456 4031 y([Mo)r(d87])103 b(Luciano)26 b(Mo)r(dica,)g Ft(The)h(gr)l(adient) h(the)l(ory)f(of)g(phase)i(tr)l(ansitions)f(and)g(the)f(minimal)h (interfac)l(e)810 4114 y(criterion)p Fz(,)23 b(Arc)n(h.)g(Rational)h (Mec)n(h.)g(Anal.)f Fa(98)g Fz(\(1987\),)i(no.)f(2,)f(123{142.)456 4197 y([Mos86])116 b(J)r(\177)-37 b(urgen)31 b(Moser,)f Ft(Minimal)i(solutions)g(of)g(variational)g(pr)l(oblems)h(on)e(a)h (torus)p Fz(,)f(Ann.)f(Inst.)g(H.)810 4280 y(P)n(oincar)n(\023)-33 b(e)24 b(Anal.)g(Non)f(Lin)n(\023)-33 b(eaire)24 b Fa(3)f Fz(\(1986\),)i(no.)f(3,)f(229{272.)456 4363 y([PV05a])108 b(Arshak)25 b(P)n(etrosy)n(an)h(and)g(Enrico)e(V)-6 b(aldino)r(ci,)25 b Ft(Density)h(estimates)g(for)i(a)f(de)l(gener)l(ate/singular)810 4446 y(phase-tr)l(ansition)g(mo)l(del)p Fz(,)e(SIAM)f(J.)f(Math.)h (Anal.)f Fa(36)g Fz(\(2005\),)i(no.)e(4,)h(1057{1079.)456 4529 y([PV05b])104 b(Arshak)24 b(P)n(etrosy)n(an)h(and)f(Enrico)g(V)-6 b(aldino)r(ci,)23 b Ft(Ge)l(ometric)j(pr)l(op)l(erties)h(of)f(Bernoul)t (li-typ)l(e)g(min-)810 4612 y(imizers)p Fz(,)e(In)n(terfaces)g(F)-6 b(ree)24 b(Bound.)g Fa(7)g Fz(\(2005\),)h(55{78.)456 4695 y([Ro)n(w79])108 b(J.)24 b(S.)g(Ro)n(wlinson,)g Ft(T)-5 b(r)l(anslation)28 b(of)e(J.)g(D.)g(van)g(der)h(Waals')g(\\The) f(thermo)l(dynamic)i(the)l(ory)e(of)810 4778 y(c)l(apil)t(larity)c (under)g(the)f(hyp)l(othesis)i(of)e(a)h(c)l(ontinuous)h(variation)f(of) f(density")p Fz(,)e(J.)g(Statist.)g(Ph)n(ys.)810 4861 y Fa(20)k Fz(\(1979\),)i(no.)f(2,)f(197{244.)456 4944 y([RS03])153 b(P)n(aul)25 b(H.)f(Rabino)n(witz)i(and)g(Ed)f (Stredulinsky)-6 b(,)25 b Ft(Mixe)l(d)i(states)f(for)h(an)h(Al)t (len-Cahn)f(typ)l(e)f(e)l(qua-)810 5027 y(tion)p Fz(,)d(Comm.)e(Pure)j (Appl.)f(Math.)h Fa(56)f Fz(\(2003\),)i(no.)e(8,)h(1078{1134.)456 5110 y([RS04])153 b(P)-6 b(.)19 b(H.)g(Rabino)n(witz)h(and)f(E.)g (Stredulinsky)-6 b(,)20 b Ft(On)i(some)g(r)l(esults)g(of)f(Moser)h(and) g(of)g(Bangert)p Fz(,)e(Ann.)810 5193 y(Inst.)k(H.)f(P)n(oincar)n(\023) -33 b(e)24 b(Anal.)g(Non)f(Lin)n(\023)-33 b(eaire)24 b Fa(21)f Fz(\(2004\),)i(no.)f(5,)f(673{688.)p eop %%Page: 40 40 40 39 bop 456 251 a Fs(40)631 b(RAF)-7 b(AEL)22 b(DE)g(LA)h(LLA)-7 b(VE)22 b(AND)g(ENRICO)g(V)-7 b(ALDINOCI)456 450 y Fz([Sc)n(h64])137 b(Jacob)27 b(T.)e(Sc)n(h)n(w)n(artz,)i Ft(Gener)l(alizing)h(the)f (Lusternik-Schnir)l(elman)h(the)l(ory)g(of)f(critic)l(al)h(p)l(oints)p Fz(,)810 533 y(Comm.)21 b(Pure)j(Appl.)f(Math.)h Fa(17)e Fz(\(1964\),)k(307{315.)456 616 y([Sho97])131 b(R.)22 b(E.)g(Sho)n(w)n(alter,)h Ft(Monotone)i(op)l(er)l(ators)h(in)e(Banach)i (sp)l(ac)l(e)f(and)g(nonline)l(ar)h(p)l(artial)g(di\013er)l(en-)810 699 y(tial)g(e)l(quations)p Fz(,)e(Mathematical)f(Surv)n(eys)h(and)h (Monographs,)e(v)n(ol.)g(49,)h(American)e(Mathemat-)810 782 y(ical)i(So)r(ciet)n(y)-6 b(,)24 b(Pro)n(vidence,)h(RI,)e(1997.)456 865 y([Szu89])135 b(Andrzej)29 b(Szulkin,)g Ft(Critic)l(al)i(p)l(oint)g (the)l(ory)f(of)h(Ljusternik-Schnir)l(elmann)f(typ)l(e)g(and)h(applic)l (a-)810 948 y(tions)26 b(to)f(p)l(artial)i(di\013er)l(ential)f(e)l (quations)p Fz(,)e(Minimax)e(results)h(of)g(Lusternik-Sc)n(hnirelman)g (t)n(yp)r(e)810 1031 y(and)36 b(applications)g(\(Mon)n(treal,)j(PQ,)34 b(1986\),)39 b(S)n(\023)-33 b(em.)35 b(Math.)g(Sup.,)j(v)n(ol.)d(107,)j (Presses)d(Univ.)810 1114 y(Mon)n(tr)n(\023)-33 b(eal,)24 b(Mon)n(treal,)f(QC,)g(1989,)h(pp.)g(35{96.)456 1197 y([T)-6 b(a)n(y96])129 b(Mic)n(hael)29 b(E.)g(T)-6 b(a)n(ylor,)30 b Ft(Partial)h(di\013er)l(ential)g(e)l(quations.)g(I)p Fz(,)e(Applied)g(Mathematical)g(Sciences,)810 1280 y(v)n(ol.)24 b(115,)g(Springer-V)-6 b(erlag,)22 b(New)i(Y)-6 b(ork,)23 b(1996,)h(Basic)g(theory)-6 b(.)456 1363 y([T)g(a)n(y97])129 b(Mic)n(hael)23 b(E.)g(T)-6 b(a)n(ylor,)22 b Ft(Partial)k(di\013er)l (ential)f(e)l(quations.)h(III)p Fz(,)d(Applied)g(Mathematical)g (Sciences,)810 1446 y(v)n(ol.)j(117,)g(Springer-V)-6 b(erlag,)25 b(New)h(Y)-6 b(ork,)25 b(1997,)i(Nonlinear)f(equations,)h (Corrected)f(reprin)n(t)g(of)810 1529 y(the)f(1996)f(original.)456 1612 y([T)-6 b(or04])136 b(Monica)25 b(T)-6 b(orres,)24 b Ft(Plane-like)i(minimal)h(surfac)l(es)h(in)e(p)l(erio)l(dic)h(me)l (dia)h(with)f(exclusions)p Fz(,)d(SIAM)810 1695 y(J.)g(Math.)f(Anal.)g Fa(36)g Fz(\(2004\),)i(no.)f(2,)f(523{551.)456 1778 y([V)-6 b(al04])142 b(Enrico)27 b(V)-6 b(aldino)r(ci,)28 b Ft(Plane-like)g (minimizers)h(in)g(p)l(erio)l(dic)h(me)l(dia:)40 b(jet)28 b(\015ows)h(and)h(Ginzbur)l(g--)810 1861 y(Landau-typ)l(e)c (functionals)p Fz(,)e(J.)g(Reine)g(Angew.)g(Math.)f Fa(574)g Fz(\(2004\),)i(147{185.)456 1944 y([Zie89])150 b(William)24 b(P)-6 b(.)25 b(Ziemer,)f Ft(We)l(akly)k(di\013er)l(entiable)f (functions)p Fz(,)f(Graduate)h(T)-6 b(exts)26 b(in)g(Mathematics,)810 2028 y(v)n(ol.)g(120,)h(Springer-V)-6 b(erlag,)26 b(New)h(Y)-6 b(ork,)26 b(1989,)i(Sob)r(olev)f(spaces)g(and)g(functions)g(of)f(b)r (ounded)810 2111 y(v)l(ariation.)555 2266 y Fy(Dep)l(ar)l(tment)d(of)g (Ma)l(thema)l(tics,)g(University)h(of)g(Texas)g(a)l(t)f(A)n(ustin,,)h (A)n(ustin,)h(TX)f(78712,)f(USA)555 2349 y Ft(E-mail)j(addr)l(ess)5 b Fz(:)33 b Fx(llave@math.utexas.edu)555 2490 y Fy(Dip)l(ar)l(timento) 24 b(di)h(Ma)l(tema)l(tica,)f(Universit)1892 2484 y(\022)1890 2490 y(a)g(di)i(R)n(oma)e(Tor)h(Ver)o(ga)l(t)l(a,,)e(R)n(oma,)i (I-00133,)e(It)l(al)l(y)555 2573 y Ft(E-mail)j(addr)l(ess)5 b Fz(:)33 b Fx(valdinoci@mat.uniroma2)q(.it)p eop %%Trailer end userdict /end-hook 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