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Fr(F)13 b Fu(\()p Fr(x)p Fu(\))29 b(has)f(sev)m(eral)g(lo)s (cal)50 1824 y(minima.)38 b(W)-8 b(e)32 b(alw)m(a)m(ys)f(assume)f(that) h Fr(X)1447 1838 y Fp(\017)1513 1824 y Fu(is)e(killed)f(on)i(\012)2041 1791 y Fp(c)2108 1824 y Fu(if)f(it)h(exists.)159 2042 y(F)-8 b(or)42 b(a)g(general)g(in)m(tro)s(duction)d(to)k(the)f(topic)f (and)g(its)g(history)f(w)m(e)j(refer)e(to)h(the)g(in)m(tro)s(duction)e (of)50 2187 y([BEGK3].)h(In)29 b(that)h(pap)s(er)f(w)m(e)h(ha)m(v)m(e)g (studied)e(the)i(so-called)f(metastable)h(exit)g(times)f(from)g (attractors)50 2333 y(of)h(lo)s(cal)g(minima)e(of)i Fr(F)44 b Fu(and)29 b(w)m(e)i(ha)m(v)m(e)h(giv)m(en)e(a)h(precise)e(asymptotic) i(estimate)g(for)f(the)g(mean)h(v)-5 b(alue)30 b(of)50 2478 y(these)35 b(times.)52 b(These)34 b(estimates)h(w)m(ere)g(in)e (turn)g(based)h(on)g(precise)g(estimates)h(of)g(certain)f Fo(Newtonian)50 2624 y(c)-5 b(ap)g(acities)32 b Fu(of)e(sets)h(con)m (taining)f(small)f(balls)g(cen)m(tered)i(at)g(the)g(lo)s(cations)e(of)i (the)g(minima)d(of)i Fr(F)13 b Fu(.)159 2842 y(In)20 b(the)h(presen)m(t)g(pap)s(er)e(w)m(e)i(turn)f(to)i(the)f(in)m(v)m (estigation)f(of)h(the)g(lo)m(w-lying)e(sp)s(ectrum)h(of)h(the)g (generators)50 2987 y(of)30 b(the)h(pro)s(cess)f(de\014ned)f(b)m(y)h (\(1.1\),)j(i.e.)1418 3243 y Fr(L)1480 3257 y Fp(\017)1540 3243 y Fs(\021)25 b(\000)p Fr(\017)p Fu(\001)20 b(+)g Fs(r)p Fr(F)13 b Fu(\()p Fr(x)p Fu(\))20 b Fs(\001)h(r)1181 b Fu(\(1)p Fr(:)p Fu(2\))50 3498 y(of)31 b(these)h(pro)s(cesses,)g (with)e(Diric)m(hlet)g(b)s(oundary)f(conditions)h(on)h(\012)2431 3465 y Fp(c)2499 3498 y Fu(\(if)g(\012)26 b Fs(6)p Fu(=)h Fq(R)2868 3465 y Fp(d)2917 3498 y Fu(\))k(of)h(these)g(pro)s(cesses.)50 3644 y(It)j(is)g(w)m(ell-kno)m(wn)f(that)j(the)e(sp)s(ectrum)g(of)g (suc)m(h)h(op)s(erators)f(has)h(precisely)e(one)h(exp)s(onen)m(tially)f (small)50 3789 y(eigen)m(v)-5 b(alue)37 b(for)g(eac)m(h)h(lo)s(cal)f (minim)m(um)d(of)j(the)h(function)e Fr(F)13 b Fu(,)39 b(and)d(more)i(or)f(less)f(rough)h(estimates)h(of)50 3935 y(their)31 b(precise)g(v)-5 b(alues)32 b(are)g(kno)m(wn)f ([FW,Ma,Mi].)48 b(W)-8 b(en)m(tzell)33 b([W2])g(and)f(F)-8 b(reidlin)29 b(and)j(W)-8 b(en)m(tzell)33 b([FW])50 4080 y(obtain)44 b(estimate)h(for)f(the)g(exp)s(onen)m(tial)f(rate,)49 b(i.e.)82 b(they)45 b(iden)m(tify)d(lim)2667 4094 y Fp(\017)p Fj(#)p Fv(0)2790 4080 y Fr(\017)2827 4047 y Fj(\000)p Fv(1)2940 4080 y Fu(ln)14 b Fr(\025)3084 4094 y Fp(i)3114 4080 y Fu(\()p Fr(\017)p Fu(\))45 b(using)e(large)50 4225 y(deviation)25 b(metho)s(ds.)38 b(Sharp)s(er)24 b(estimates,)k(with)c(m)m(ultiplicativ)m(e)g(errors)i(of)g(order)f Fr(\017)2996 4192 y Fj(\006)p Fp(k)r(d)3161 4225 y Fu(w)m(ere)h (obtained)50 4371 y(for)i(principal)c(eigen)m(v)-5 b(alues)28 b(b)m(y)g(Holley)-8 b(,)29 b(Kusuok)-5 b(a,)28 b(and)f(Stro)s(ok)h ([HKS])g(using)f(a)h(v)-5 b(ariational)27 b(principle;)50 4516 y(these)j(metho)s(ds)f(w)m(ere)h(extended)f(to)h(the)g(full)d(set) j(of)g(exp)s(onen)m(tially)e(small)g(eigen)m(v)-5 b(alues)29 b(b)m(y)h(Miclo)f([Mi])50 4662 y(\(see)i(also)f([Ma]\).)159 4880 y(Our)38 b(purp)s(ose)f(here)i(is)f(to)h(get)h Fo(sharp)h Fu(estimates,)h(i.e.)66 b(w)m(e)39 b(seek)h(upp)s(er)d(and)h(lo)m(w)m (er)h(b)s(ounds)e(with)50 5025 y(m)m(ultiplicativ)m(e)c(errors)i(that)i (tend)e(to)h(one)g(as)g Fr(\017)f Fu(tends)g(to)i(zero.)57 b(Suc)m(h)35 b(estimates)h(are)g(kno)m(wn)f(in)g(the)50 5171 y(one-dimensional)23 b(case)j(\(see)g(e.g.)40 b([BuMa1,BuMa2])28 b(and)d(references)g(therein\),)h(whereas)f(in)f(the)h(m)m(ulti-)50 5316 y(dimensional)e(case)k(only)d(heuristic)g(results)h(based)g(on)h (formal)f(p)s(o)m(w)m(er)h(series)f(expansions)f(of)i(WKB)h(t)m(yp)s(e) 50 5462 y(exist.)62 b(\(see)39 b(e.g.)64 b([Kolo])38 b(for)f(an)h(analysis)e(of)i(the)g(situation\).)62 b(While)36 b(the)i(metho)s(ds)f(in)m(tro)s(duced)f(in)p eop %%Page: 3 3 3 2 bop 3671 295 a Fm(3)50 513 y Fu(the)32 b(third)f(pap)s(er)g(on)i (quan)m(tum)f(mec)m(hanical)g(tunneling)e(b)m(y)i(Hel\013er)g(and)g (Sj\177)-45 b(ostrand)32 b([HS3])h(should)d(in)50 658 y(principle)23 b(allo)m(w)i(to)i(justify)e(suc)m(h)h(expansions,)g (their)f(implemen)m(tation)g(seems)h(rather)g(tedious)f(and)h(has)50 804 y(not)k(b)s(een)g(carried)g(out)g(to)h(our)f(kno)m(wledge.)159 1022 y(Here)g(w)m(e)g(will)d(resort)j(to)h(a)f(di\013eren)m(t)f (approac)m(h)h(that)h(com)m(bines)e(ideas)g(already)g(suggested)i(in)d ([W1])50 1167 y(with)43 b(p)s(oten)m(tial)g(theoretic)i(ideas.)82 b(In)43 b(fact,)49 b(this)43 b(approac)m(h)h(w)m(as)h(dev)m(elop)s(ed)e (in)g([BEGK2])j(in)d(the)50 1313 y(setting)35 b(of)g(discrete)f(Mark)m (o)m(v)j(c)m(hains,)e(where)f(indeed)g(man)m(y)h(tec)m(hnical)f (problems)f(w)m(e)j(will)c(b)s(e)i(facing)50 1458 y(here)c(disapp)s (ear,)f(and)g(that)i(ma)m(y)g(serv)m(e)g(as)g(a)g(nice)f(in)m(tro)s (duction.)159 1676 y(W)-8 b(e)29 b(will)d(from)i(no)m(w)g(on)g(assume)g (that)h Fr(F)41 b Fu(is)28 b(at)h(least)f(three)h(times)f(con)m(tin)m (uously)f(di\013eren)m(tiable)f(and)50 1822 y(has)j(a)h(\014nite)e(set) i(of)f(lo)s(cal)g(minima,)f(whic)m(h)g(w)m(e)i(denote)f(b)m(y)h Fs(M)25 b Fu(=)g Fs(f)p Fr(x)2463 1836 y Fv(1)2504 1822 y Fr(;)15 b(:)g(:)g(:)i(;)e(x)2758 1836 y Fp(n)2808 1822 y Fs(g)p Fu(.)40 b(W)-8 b(e)31 b(will)c(also)i(assume)50 1967 y(that)c Fr(F)38 b Fu(has)24 b(exp)s(onen)m(tially)f(tigh)m(t)i (lev)m(el)g(sets,)h(i.e.)39 b(that)2008 1894 y Fi(R)2051 2000 y Fp(y)r Fv(:)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fj(\025)p Fp(a)2379 1967 y Fu(exp)o(\()p Fs(\000)p Fr(F)j Fu(\()p Fr(z)t Fu(\))p Fr(=\017)p Fu(\))p Fr(dz)32 b(<)25 b(C)7 b(e)3258 1934 y Fj(\000)p Fp(a=\017)3426 1967 y Fu(,)26 b(where)50 2113 y Fr(C)32 b Fu(=)25 b Fr(C)7 b Fu(\()p Fr(a)p Fu(\))25 b Fr(<)g Fs(1)i Fu(is)e(indep)s(enden)m(t)g (of)i Fr(\017)p Fu(.)39 b(Our)26 b(main)g(in)m(terests)g(are)i(the)e (distribution)e(of)j(stopping)e(times)1366 2297 y Fr(\034)1406 2311 y Fp(A)1491 2297 y Fs(\021)g Fu(inf)20 b Fs(f)q Fr(t)25 b(>)g Fu(0)p Fs(j)p Fr(X)7 b Fu(\()p Fr(t)p Fu(\))27 b Fs(2)e Fr(A)p Fs(g)1130 b Fu(\(1)p Fr(:)p Fu(3\))50 2480 y(for)32 b(the)h(pro)s(cess)f(starting)g(in)f(one)i(minim)m(um,)e (sa)m(y)i Fr(x)c Fs(2)f(M)p Fu(,)34 b(of)e Fr(F)13 b Fu(,)34 b(when)d Fr(A)e Fu(=)g Fr(B)2959 2494 y Fp(\032)3001 2480 y Fu(\()p Fr(y)s Fu(\))k(is)e(a)i(small)e(ball)50 2626 y(of)38 b(radius)f Fr(\032)i Fu(around)e(another)i(minim)m(um,)f Fr(y)j Fs(2)e(M)p Fu(.)65 b(It)39 b(will)d(actually)i(b)s(ecome)h (apparen)m(t)f(that)i(the)50 2771 y(precise)f(c)m(hoice)h(of)g(the)f (hitting)g(set)h(is)e(often)i(not)g(imp)s(ortan)m(t,)h(and)e(that)h (the)g(problem)e(is)g(virtually)50 2917 y(equiv)-5 b(alen)m(t)27 b(to)i(considering)d(the)i(escap)s(e)g(from)g(a)g(suitably)e(c)m(hosen) i(neigh)m(b)s(orho)s(o)s(d)e(of)i Fr(x)p Fu(,)g(pro)m(vided)f(this)50 3062 y(neigh)m(b)s(orho)s(o)s(d)h(con)m(tains)i(the)h(relev)-5 b(an)m(t)31 b Fo(sadd)5 b(le)33 b(p)-5 b(oints)32 b Fu(connecting)f Fr(x)f Fu(and)g Fr(y)s Fu(.)159 3280 y(Let)e(us)e(no)m(w)h(state)h(the) g(main)e(results)g(of)h(this)f(pap)s(er.)38 b(One)27 b(k)m(ey)h(notion)f(w)m(e)g(will)e(need)h(is)h(that)g(of)h(the)50 3426 y(saddle)i(b)s(et)m(w)m(een)h(t)m(w)m(o)h(sets)g Fr(A;)15 b(B)31 b Fs(\032)26 b Fq(R)1395 3393 y Fp(d)1444 3426 y Fu(.)42 b(W)-8 b(e)32 b(sa)m(y)f(that)h Fr(z)2069 3393 y Fj(\003)2137 3426 y Fu(=)26 b Fr(z)2280 3393 y Fj(\003)2322 3426 y Fu(\()p Fr(A;)15 b(B)5 b Fu(\))31 b(is)f(a)h(saddle)f(p)s(oin)m(t)g(b)s(et)m(w)m(een)i Fr(A)50 3571 y Fu(and)e Fr(B)5 b Fu(,)30 b(if)1013 3717 y Fr(F)13 b Fu(\()p Fr(z)1165 3679 y Fj(\003)1207 3717 y Fu(\()p Fr(A;)i(B)5 b Fu(\)\))27 b(=)261 b(inf)1617 3780 y Fp(!)r Fv(:)p Fp(!)r Fv(\(0\))p Fj(2)p Fp(A;!)r Fv(\(1\))p Fj(2)p Fp(B)2252 3717 y Fu(sup)2215 3797 y Fp(t)p Fj(2)p Fv([0)p Fp(;)p Fv(1])2442 3717 y Fr(F)13 b Fu(\()p Fr(!)s Fu(\()p Fr(t)p Fu(\)\))777 b(\(1)p Fr(:)p Fu(4\))50 3932 y(where)30 b(the)h(in\014m)m(um)e(is)h(o)m(v)m(er)i(all) e(con)m(tin)m(uous)g(paths)g(going)h(from)g Fr(A)g Fu(to)g Fr(B)5 b Fu(.)42 b(Note)32 b(that)f Fr(z)3217 3899 y Fj(\003)3260 3932 y Fu(\()p Fr(A;)15 b(B)5 b Fu(\))31 b(ma)m(y)50 4077 y(not)f(b)s(e)g(uniquely)e(de\014ned;)h(w)m(e)i(call)f Fs(Z)7 b Fu(\()p Fr(A;)15 b(B)5 b Fu(\))31 b(the)g(set)f(of)h(all)e(p)s (ossible)f(solutions.)159 4296 y(Giv)m(en)j(t)m(w)m(o)i(disjoin)m(t)d (closed)h(sets)h Fr(A;)15 b(D)s Fu(,)32 b(w)m(e)g(will)d(denote)j(b)m (y)f Fr(h)2393 4310 y Fp(A;D)2536 4296 y Fu(\()p Fr(x)p Fu(\))h(the)f(equilibrium)c(p)s(oten)m(tial,)50 4441 y(b)m(y)22 b Fr(e)210 4455 y Fp(A;D)353 4441 y Fu(\()p Fr(dy)s Fu(\))h(the)g(equilibrium)18 b(measure,)24 b(and)e(b)m(y)g(cap) 1977 4455 y Fp(A)2036 4441 y Fu(\()p Fr(D)s Fu(\))h(the)g(Newtonian)f (capacit)m(y)i(corresp)s(onding)50 4586 y(to)38 b(the)g(Diric)m(hlet)f (problem)f(with)h(b)s(oundary)f(conditions)g(one)i(on)g Fr(A)g Fu(and)f(zero)i(on)e Fr(D)s Fu(.)63 b(The)37 b(precise)50 4732 y(de\014nitions)29 b(of)j(these)g(classical)e(quan)m(tities)h (\(see)i(e.g.)45 b([BluGet,Do)s(o,Szni]\))33 b(are)f(recalled)f(in)f (Section)h(2)50 4877 y(of)37 b([BEGK3].)50 5096 y Fh(Theorem)d(1.1:)49 b Fo(Assume)31 b(that)g Fr(F)44 b Fo(has)31 b Fr(n)f Fo(lo)-5 b(c)g(al)33 b(minima,)e Fr(x)2243 5110 y Fv(1)2284 5096 y Fr(;)15 b(:)g(:)g(:)i(;)e(x)2538 5110 y Fp(n)2618 5096 y Fo(and)31 b(that)h(for)f(some)g Fr(\022)d(>)d Fu(0)30 b Fo(the)50 5241 y(minima)j Fr(x)436 5255 y Fp(i)499 5241 y Fo(of)g Fr(F)46 b Fo(c)-5 b(an)33 b(b)-5 b(e)32 b(lab)-5 b(ele)g(d)34 b(in)f(such)g(a)g(way)g(that,)h(with)f Fs(M)2377 5255 y Fp(k)2448 5241 y Fs(\021)25 b(f)p Fr(x)2641 5255 y Fv(1)2682 5241 y Fr(;)15 b(:)g(:)g(:)h(;)f(x)2935 5255 y Fp(k)2980 5241 y Fs(g)33 b Fo(and)h Fs(M)3344 5255 y Fv(0)3410 5241 y Fs(\021)25 b Fu(\012)3572 5208 y Fp(c)3609 5241 y Fo(,)585 5425 y Fr(F)13 b Fu(\()p Fr(z)737 5387 y Fj(\003)780 5425 y Fu(\()p Fr(x)867 5439 y Fp(k)912 5425 y Fr(;)i Fs(M)1061 5439 y Fp(k)r Fj(\000)p Fv(1)1199 5425 y Fu(\)\))21 b Fs(\000)f Fr(F)13 b Fu(\()p Fr(x)1539 5439 y Fp(k)1584 5425 y Fu(\))26 b Fs(\024)f Fu(min)1755 5484 y Fp(i)d Fu(0)34 b Fo(such)h(that)h(the)e Fr(n)g Fo(exp)-5 b(onential)5 b(ly)37 b(smal)5 b(l)35 b(eigenvalues)f Fr(\025)2743 963 y Fv(1)2813 949 y Fr(<)28 b(\025)2965 963 y Fv(2)3034 949 y Fr(<)h Fs(\001)15 b(\001)g(\001)29 b Fr(<)g(\025)3421 963 y Fp(n)3504 949 y Fo(of)35 b Fr(L)3675 963 y Fp(\017)50 1095 y Fo(satisfy:)190 1302 y Fr(\025)243 1316 y Fp(k)313 1302 y Fu(=)420 1240 y(cap)572 1254 y Fp(B)627 1264 y Fk(k)671 1240 y Fu(\()p Fs(S)761 1254 y Fp(k)r Fj(\000)p Fv(1)899 1240 y Fu(\))p 420 1281 515 4 v 563 1364 a Fs(k)p Fr(h)660 1378 y Fp(k)706 1364 y Fs(k)751 1333 y Fv(2)751 1388 y(2)946 1302 y Fu(\(1)21 b(+)e Fr(O)s Fu(\()p Fr(e)1286 1264 y Fj(\000)p Fp(\016)r(=\017)1451 1302 y Fu(\)\))313 1545 y(=)575 1483 y(1)p 420 1524 356 4 v 420 1607 a Fq(E)475 1621 y Fp(x)516 1631 y Fk(k)566 1607 y Fr(\034)606 1621 y Fj(S)651 1631 y Fk(k)q Fg(\000)p Fl(1)786 1545 y Fu(\(1)i(+)f Fr(O)s Fu(\()p Fr(e)1127 1507 y Fj(\000)p Fp(\016)r(=\017)1292 1545 y Fu(\)\))313 1845 y(=)420 1784 y Fs(j)p Fr(\025)498 1751 y Fj(\003)498 1806 y Fv(1)540 1784 y Fu(\()p Fr(z)621 1751 y Fj(\003)664 1784 y Fu(\()p Fr(x)751 1798 y Fp(k)796 1784 y Fr(;)15 b Fs(M)945 1798 y Fp(k)r Fj(\000)p Fv(1)1083 1784 y Fu(\)\))p Fs(j)p 420 1824 759 4 v 749 1907 a Fu(2)p Fr(\031)1190 1682 y Fi(s)p 1281 1682 1086 4 v 1548 1784 a Fu(det\()p Fs(r)1785 1757 y Fv(2)1826 1784 y Fr(F)e Fu(\()p Fr(x)1984 1798 y Fp(k)2029 1784 y Fu(\)\))p 1292 1824 1064 4 v 1292 1907 a Fs(j)i Fu(det)q(\()p Fs(r)1570 1881 y Fv(2)1610 1907 y Fr(F)e Fu(\()p Fr(z)1762 1881 y Fj(\003)1805 1907 y Fu(\()p Fr(x)1892 1921 y Fp(k)1937 1907 y Fr(;)i Fs(M)2086 1921 y Fp(k)r Fj(\000)p Fv(1)2224 1907 y Fu(\)\)\))p Fs(j)2367 1845 y Fr(e)2409 1808 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(z)2609 1780 y Fg(\003)2646 1808 y Fv(\()p Fp(x)2715 1818 y Fk(k)2755 1808 y Fp(;)p Fj(M)2862 1818 y Fk(k)q Fg(\000)p Fl(1)2982 1808 y Fv(\)\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)3220 1818 y Fk(k)3261 1808 y Fv(\)])p Fp(=\017)494 2092 y Fs(\002)585 1991 y Fi(\020)639 2092 y Fu(1)21 b(+)f Fr(O)883 1991 y Fi(\020)937 2092 y Fr(\017)974 2054 y Fv(1)p Fp(=)p Fv(2)1089 2092 y Fs(j)15 b Fu(ln)f Fr(\017)p Fs(j)1282 1991 y Fi(\021\021)3523 1684 y Fu(\(1)p Fr(:)p Fu(6\))50 2278 y Fo(wher)-5 b(e)34 b Fr(\025)360 2245 y Fj(\003)360 2301 y Fv(1)402 2278 y Fu(\()p Fr(z)483 2245 y Fj(\003)525 2278 y Fu(\))g Fo(denotes)g(the)g(unique)e(ne)-5 b(gative)34 b(eigenvalue)e(of)i(the)f(Hessian)h(of)f Fr(F)46 b Fo(at)34 b(the)g(sadd)5 b(le)34 b(p)-5 b(oint)50 2424 y Fr(z)96 2391 y Fj(\003)138 2424 y Fo(.)42 b(Note)32 b(that)i(if)e Fu(\012)25 b(=)g Fq(R)s Fo(,)38 b(then)33 b Fu(cap)1367 2438 y Fp(B)1422 2447 y Fl(1)1462 2424 y Fu(\()p Fs(M)1606 2438 y Fv(0)1647 2424 y Fu(\))26 b(=)f(cap)1955 2438 y Fp(B)2010 2447 y Fl(1)2050 2424 y Fu(\()p Fs(;)p Fu(\))i(=)e(0)p Fo(.)50 2642 y Ft(Remark:)40 b Fu(The)20 b(theorem)g(can)h(b)s(e)f(seen)g(as)h(con)m(taining)e (three)i(results:)34 b(First,)22 b(an)e(asymptotically)g(sharp)50 2788 y(iden)m(ti\014cation)35 b(of)h(the)g(exp)s(onen)m(tially)f(small) f(eigen)m(v)-5 b(alues)36 b(with)f(the)h(in)m(v)m(erse)g(mean)g(exit)h (times)e(from)50 2933 y(lo)s(cal)h(minima;)i(this)e(is)g(a)i(general)f (feature)g(of)g(metastable)h(systems)f(\(see)h(e.g.)61 b([D1,D2,D3,GS,GM])50 3078 y(for)39 b(earlier)f(results\).)66 b(Second,)42 b(it)c(relates)i(these)g(eigen)m(v)-5 b(alues)39 b(precisely)e(to)j(Newtonian)f(capacities;)50 3224 y(this)32 b(is)f(the)j(k)m(ey)f(di\013erence)f(from)h(our)f(results)g(to)h(e.g.) 50 b(the)33 b(approac)m(h)g(of)g(Kolok)m(oltso)m(v)h(and)e(Mak)-5 b(aro)m(v)50 3369 y([KoMa1,KoM2,Kol],)39 b(since)c(it)f(allo)m(ws)h (thirdly)d(to)k(get)g(an)f(explicit)f(expression)g(for)h(the)g(eigen)m (v)-5 b(alues)50 3515 y(in)29 b(terms)h(of)h(the)f(p)s(oten)m(tial)g Fr(F)13 b Fu(.)50 3733 y Ft(Remark:)42 b Fu(Conditions)21 b(\(1.5\))k(state)g(that)f Fo(\\al)5 b(l)26 b(val)5 b(leys)27 b(of)g Fr(F)39 b Fo(have)27 b(di\013er)-5 b(ent)26 b(depth")p Fu(,)h(whic)m(h)22 b(is)g(in)g(some)50 3878 y(sense)f(the)h(generic)f (situation.)37 b(In)20 b(this)h(case)h(a)g(n)m(um)m(b)s(er)e(of)h (simpli\014cations)d(tak)m(e)23 b(place,)h(in)c(particular)f(w)m(e)50 4024 y(do)30 b(not)g(ha)m(v)m(e)h(to)g(deal)e(with)g(degenerate)j (eigen)m(v)-5 b(alues.)40 b(These)30 b(conditions)e(are)j(completely)e (analogous)50 4169 y(to)39 b(the)g(conditions)e(imp)s(osed)g(in)g ([BEGK2].)66 b(Our)38 b(general)g(approac)m(h)h(do)s(es,)i(ho)m(w)m(ev) m(er,)h(in)37 b(principle)50 4315 y(also)29 b(allo)m(w)h(to)g(treat)h (degenerate)g(situations.)39 b(W)-8 b(e)31 b(p)s(ostp)s(one)e(the)h (treatmen)m(t)h(of)f(suc)m(h)g(cases)g(to)h(future)50 4460 y(w)m(ork.)159 4678 y(In)k(the)g(course)h(of)f(the)h(pro)s(of)f (of)g(Theorem)h(1.1)g(w)m(e)g(will)d(also)i(obtain)g(rather)g(detailed) g(con)m(trol)h(on)50 4824 y(the)30 b(eigenfunctions)f(of)i Fr(L)966 4838 y Fp(\017)1031 4824 y Fu(corresp)s(onding)d(to)j(its)f (small)f(eigen)m(v)-5 b(alues.)50 5042 y Fh(Theorem)54 b(1.2:)152 b Fo(Under)45 b(the)h(assumptions)h(of)e(The)-5 b(or)g(em)47 b(1.1,)i(if)44 b Fr(\036)2725 5056 y Fp(k)2815 5042 y Fo(denote)i(the)f(normalize)-5 b(d)50 5188 y(eigenfunction)32 b(c)-5 b(orr)g(esp)g(onding)36 b(to)d(the)g(eigenvalue)f Fr(\025)1938 5202 y Fp(k)1983 5188 y Fo(,)g(then)h(ther)-5 b(e)34 b(exists)f Fr(\016)c(>)c Fu(0)33 b Fo(s.t.)881 5402 y Fr(\036)935 5416 y Fp(k)980 5402 y Fu(\()p Fr(y)s Fu(\))26 b(=)1237 5333 y Fr(h)1289 5349 y Fp(B)1344 5358 y Fk(\017)1377 5349 y Fv(\()p Fp(x)1446 5359 y Fk(k)1486 5349 y Fv(\))p Fp(;)p Fj(S)1581 5359 y Fk(k)q Fg(\000)p Fl(1)1706 5333 y Fu(\()p Fr(y)s Fu(\))p 1231 5381 600 4 v 1231 5465 a Fs(k)p Fr(h)1328 5481 y Fp(B)1383 5490 y Fk(\017)1416 5481 y Fv(\()p Fp(x)1485 5491 y Fk(k)1525 5481 y Fv(\))p Fp(;)p Fj(S)1620 5491 y Fk(k)q Fg(\000)p Fl(1)1745 5465 y Fs(k)1790 5479 y Fv(2)1842 5402 y Fu(\(1)21 b(+)f Fr(O)s Fu(\()p Fr(e)2183 5365 y Fj(\000)p Fp(\016)r(=\017)2347 5402 y Fu(\)\))h(+)f Fr(O)s Fu(\()p Fr(e)2678 5365 y Fj(\000)p Fp(\016)r(=\017)2843 5402 y Fu(\))645 b(\(1)p Fr(:)p Fu(7\))p eop %%Page: 5 5 5 4 bop 3671 295 a Fm(5)50 513 y Fo(wher)-5 b(e)34 b Fr(h)359 529 y Fp(B)414 538 y Fk(\017)446 529 y Fv(\()p Fp(x)515 539 y Fk(k)556 529 y Fv(\))p Fp(;)p Fj(S)651 539 y Fk(k)q Fg(\000)p Fl(1)775 513 y Fu(\()p Fr(y)s Fu(\))26 b(=)f Fq(P)1070 527 y Fp(y)1129 439 y Fi(\002)1167 513 y Fr(\034)1207 529 y Fp(B)1262 538 y Fk(\017)1294 529 y Fv(\()p Fp(x)1363 539 y Fk(k)1403 529 y Fv(\))1462 513 y Fr(<)f(\034)1597 527 y Fj(S)1642 537 y Fk(k)q Fg(\000)p Fl(1)1766 439 y Fi(\003)50 731 y Ft(Remark:)47 b Fu(W)-8 b(e)33 b(giv)m(e)e(ev)m(en)h(more)f(precise)g(expressions)e(for)i(the)g (eigenfunctions)f(in)g(the)h(course)g(of)h(the)50 876 y(pro)s(ofs)26 b(later)h(on.)39 b(Note)29 b(that)e(there)g(is)f (considerable)g(in)m(terest)h(in)e(the)i(kno)m(wledge)g(of)h (eigenfunctions)d(in)50 1022 y(the)e(con)m(text)j(of)d(n)m(umerical)f (sc)m(hemes)i(designed)e(to)i(reco)m(v)m(er)h(metastable)f(sets)g(from) f(the)g(computation)h(of)50 1167 y(eigenfunctions.)38 b(See)28 b(in)f(particular)f(references)i([S,SFHD,HMS].)g(Let)h(us)e (emphasise)f(that,)j(using)e(the)50 1313 y(b)s(ounds)d(on)h (equilibrium)d(p)s(oten)m(tials)j(obtained)g(in)g(Corollary)f(4.8)j(of) 34 b([BEGK3],)28 b(Theorem)d(1.2)j(implies)50 1458 y(that)35 b(the)g(eigenfunction)f(corresp)s(onding)e(to)k(a)f(Minim)m(um)e Fr(x)2230 1472 y Fp(i)2295 1458 y Fu(is)h(exp)s(onen)m(tially)f(close)i (to)g(a)g(constan)m(t)50 1604 y(\()p Fs(\030)28 b Fr(e)226 1571 y Fp(F)10 b Fv(\()p Fp(x)351 1580 y Fk(i)380 1571 y Fv(\))p Fp(=\017)481 1604 y Fu(\))32 b(in)e(the)j(connected)f(comp)s (onen)m(t)g(of)g(the)g(lev)m(el)g(set)g Fs(f)p Fr(y)f Fu(:)d Fr(F)13 b Fu(\()p Fr(y)s Fu(\))29 b Fr(<)e(F)13 b Fu(\()p Fr(z)2962 1571 y Fj(\003)3005 1604 y Fu(\()p Fr(x)3092 1618 y Fp(i)3122 1604 y Fr(;)i Fs(M)3271 1618 y Fp(i)p Fj(\000)p Fv(1)3395 1604 y Fu(\)\))p Fs(g)33 b Fu(that)50 1749 y(con)m(tains)38 b Fr(x)466 1763 y Fp(i)533 1749 y Fu(\(i.e.)63 b(in)36 b(the)i(v)-5 b(alley)37 b(b)s(elo)m(w)f(the)i(saddle)f(p)s(oin)m(t)f(that)j(connects)f Fr(x)2880 1763 y Fp(i)2948 1749 y Fu(to)g(the)g(set)g(that)g(lies)50 1895 y(b)s(elo)m(w)e Fr(x)366 1909 y Fp(i)396 1895 y Fu(\),)i(while)d(it)h(drops)g(exp)s(onen)m(tially)e(in)i(the)g(other)h (connected)h(comp)s(onen)m(ts)e(of)h(the)g(lev)m(el)f(set)50 2040 y(of)h(this)e(saddle;)k(b)s(elo)m(w)d(the)h(lev)m(el)g(of)g Fr(x)1459 2054 y Fp(i)1525 2040 y Fu(it)g(is)e(exp)s(onen)m(tially)g (small)h(in)f(absolute)h(terms.)60 b(Note)38 b(that)50 2185 y(this)33 b(implies)f(that)j(the)f(zeros)h(of)g Fr(\036)f Fu(are)h(generally)f(not)h(in)e(the)h(neigh)m(b)s(orho)s(o)s (d)e(of)j(the)f(saddle)g(p)s(oin)m(ts,)50 2331 y(but)g(m)m(uc)m(h)g (closer)g(to)i(the)e(minima)f(in)g Fs(M)1552 2345 y Fp(i)p Fj(\000)p Fv(1)1675 2331 y Fu(.)53 b(This)33 b(fact)i(w)m(as)g(also)f (observ)m(ed)h(in)e([HMS].)i(W)-8 b(e)36 b(w)m(ould)50 2476 y(lik)m(e)d(to)i(stress)f(that)h(the)f(fact)h(that)g(the)f (eigenfunctions)f(drop)g(sharply)g(at)h(the)h(saddle)e(p)s(oin)m(ts)g (mak)m(es)50 2622 y(them)c(v)m(ery)g(go)s(o)s(d)g(indicators)e(of)i (the)g(actual)g(v)-5 b(alley)28 b(structure)h(of)g(the)g(p)s(oten)m (tial)f Fr(F)13 b Fu(,)29 b(i.e.)40 b(they)29 b(b)s(ecome)50 2767 y(excellen)m(t)34 b(appro)m(ximations)e(to)j(the)f(indicator)e (functions)h(of)h(the)g(metastable)g(sets)g(corresp)s(onding)e(to)50 2913 y(the)e(metastable)h(exit)g(time)f(1)p Fr(=\025)1196 2927 y Fp(i)1227 2913 y Fu(.)159 3131 y(Finally)-8 b(,)26 b(it)h(is)f(almost)h(a)h(corollary)e(from)h(the)g(results)f(obtained)g (ab)s(o)m(v)m(e)j(that)e(metastable)h(exit)f(times)50 3276 y(are)40 b(asymptotically)f(exp)s(onen)m(tially)f(distributed,)i (when)f(appropriate)g(non-degeneracy)h(conditions)50 3422 y(are)31 b(met.)50 3640 y Fh(Theorem)43 b(1.3:)63 b Fo(Assume)37 b(that)h(the)f(Hessian)g(of)h Fr(F)50 b Fo(is)37 b(non-de)-5 b(gener)g(ate)38 b(at)g(al)5 b(l)37 b(lo)-5 b(c)g(al)39 b(minima)f(and)50 3785 y(sadd)5 b(le)34 b(p)-5 b(oints.)43 b(L)-5 b(et)33 b Fr(x)842 3799 y Fp(i)905 3785 y Fo(b)-5 b(e)32 b(a)h(minimum)g(of)g Fr(F)46 b Fo(and)33 b(let)g Fr(D)i Fo(b)-5 b(e)33 b(any)g(close)-5 b(d)34 b(subset)f(of)f Fq(R)3118 3752 y Fp(d)3199 3785 y Fo(such)h(that:)10 4004 y(\(i\))47 b(If)36 b Fs(M)367 4018 y Fp(i)430 4004 y Fs(\021)d(f)p Fr(y)624 4018 y Fv(1)665 4004 y Fr(;)15 b(:)g(:)g(:)h(;)f(y)911 4018 y Fp(k)956 4004 y Fs(g)33 b(\032)f(M)37 b Fo(enumer)-5 b(ates)38 b(al)5 b(l)37 b(those)h(minima)f(of)g Fr(F)50 b Fo(such)36 b(that)i Fr(F)13 b Fu(\()p Fr(y)3248 4018 y Fp(j)3287 4004 y Fu(\))33 b Fs(\024)f Fr(F)13 b Fu(\()p Fr(x)3616 4018 y Fp(i)3647 4004 y Fu(\))p Fo(,)159 4149 y(then)33 b Fs([)422 4116 y Fp(k)422 4173 y(j)t Fv(=1)552 4149 y Fr(B)621 4163 y Fp(\017)656 4149 y Fu(\()p Fr(y)736 4163 y Fp(j)774 4149 y Fu(\))26 b Fs(\032)f Fr(D)s Fo(,)32 b(and)-18 4367 y(\(ii\))62 b Fu(dist)14 b(\()q Fs(S)7 b Fu(\()p Fr(x)521 4381 y Fp(i)551 4367 y Fr(;)15 b Fs(M)700 4381 y Fp(i)731 4367 y Fu(\))p Fr(;)g(D)s Fu(\))26 b Fs(\025)f Fr(\016)k(>)c Fu(0)33 b Fo(for)g(some)g Fr(\016)k Fo(indep)-5 b(endent)34 b(of)f Fr(\017)p Fo(.)159 4585 y(Assume)i(further)h(that)g(the)f(c)-5 b(onditions)37 b(of)f(The)-5 b(or)g(em)37 b(1.1)e(ar)-5 b(e)36 b(satis\014e)-5 b(d.)51 b(Then,)36 b(ther)-5 b(e)36 b(exist)f Fr(\016)f(>)29 b Fu(0)50 4731 y Fo(indep)-5 b(endent)34 b(of)f Fr(\017)f Fo(and)i(of)f Fr(t)p Fo(,)f(such)h(that)h(for)f(al)5 b(l)33 b Fr(t)25 b(>)g Fu(0)p Fo(,)552 4996 y Fq(P)607 5010 y Fp(x)648 5020 y Fk(k)708 4996 y Fu([)p Fr(\034)773 5010 y Fp(D)864 4996 y Fr(>)g(t)p Fq(E)1048 5010 y Fp(x)1089 5020 y Fk(k)1139 4996 y Fr(\034)1179 5010 y Fp(D)1245 4996 y Fu(])g(=)1391 4895 y Fi(\020)1446 4996 y Fu(1)20 b(+)g Fr(O)1689 4895 y Fi(\020)1743 4996 y Fr(e)1785 4958 y Fj(\000)p Fp(\016)r(=\017)1950 4895 y Fi(\021\021)2074 4996 y Fr(e)2116 4958 y Fp(t)p Fv(\(1+)p Fp(O)r Fv(\()p Fp(e)p Fj(\000)p Fp(\016)r(=\017)r Fv(\)\))1290 5188 y Fu(+)1381 5102 y Fi(X)1381 5297 y Fp(j)t(>k)1528 5188 y Fr(O)s Fu(\()p Fr(e)1677 5151 y Fj(\000)p Fp(\016)r(=\017)1841 5188 y Fu(\))p Fr(e)1918 5151 y Fj(\000)p Fp(t\025)2045 5160 y Fk(j)2080 5151 y Ff(E)2117 5160 y Fk(x)2153 5176 y(k)2203 5151 y Fp(\034)2237 5160 y Fk(D)2318 5188 y Fu(+)g Fr(O)s Fu(\(1\))p Fr(e)2638 5151 y Fj(\000)p Fp(tO)r Fv(\()p Fp(e)2842 5123 y Fk(d)p Fg(\000)p Fl(1)2962 5151 y Fv(\))p Ff(E)3027 5160 y Fk(x)3063 5176 y(k)3113 5151 y Fp(\034)3147 5160 y Fk(D)3523 5128 y Fu(\(1)p Fr(:)p Fu(8\))p eop %%Page: 6 6 6 5 bop 50 295 a Fm(6)159 513 y Fu(The)44 b(results)g(of)h(this)f(pap)s (er)g(together)i(with)e(those)h(of)52 b([BEGK3])46 b(sho)m(w)f(that)h (the)f(metho)s(ds)f(to)50 658 y(analyse)28 b(metastable)h(b)s(eha)m (viour)e(in)g(discrete)h(Mark)m(o)m(v)i(c)m(hains)d(in)m(tro)s(duced)g (in)g([BEGK1,BEGK2])j(can)50 804 y(b)s(e)24 b(naturally)f(extended)i (to)g(the)g(treatmen)m(t)i(of)e(con)m(tin)m(uous)f(di\013usion)e(pro)s (cesses.)39 b(In)24 b(particular)f(w)m(e)i(see)50 949 y(that)i(the)g(metastable)g(b)s(eha)m(viour)e(of)i(con)m(tin)m(uous)f (and)g(discrete)g(di\013usions)d(is)j(virtually)e(iden)m(tical,)i(and) 50 1095 y(that)36 b(all)f(results)g(for)g(the)h(discrete)g(c)m(hains)f (treated)i(in)d([BEGK1])j(carry)f(o)m(v)m(er)h(to)g(the)f(corresp)s (onding)50 1240 y(di\013usion)26 b(appro)m(ximations.)39 b(In)29 b(fact,)i(our)d(results)g(in)g(the)i(di\013usion)c(case)k(are)g (sharp)s(er,)e(since)h(w)m(e)g(w)m(ere)50 1385 y(able)41 b(to)h(iden)m(tify)e(the)h(constan)m(ts)i(in)d(the)h(prefactors)h(of)f (exp)s(onen)m(tially)f(small)g(or)h(large)h(terms)f(\(w)m(e)50 1531 y(exp)s(ect,)f(ho)m(w)m(ev)m(er,)i(that)c(with)f(some)h(extra)h(w) m(ork)f(this)e(impro)m(v)m(emen)m(t)j(can)f(also)g(b)s(e)f(carried)g(o) m(v)m(er)i(to)50 1676 y(the)28 b(discrete)f(c)m(hains,)h(at)g(least)g (under)e(certain)h(conditions\).)39 b(There)27 b(are)h(a)g(n)m(um)m(b)s (er)e(of)i(generalizations)50 1822 y(of)43 b(these)h(results)d(that)j (can)f(b)s(e)g(in)m(v)m(estigated:)67 b(First,)46 b(one)d(can)g (naturally)f(di\013usion)e(pro)s(cesses)j(on)50 1967 y(more)26 b(general)g(Riemannian)e(manifolds.)37 b(Second,)27 b(one)f(can)h(consider)e(extensions)g(to)i(lo)s(cally)e(in\014nitely)50 2113 y(divisible)i(pro)s(cesses)32 b(with)e(mixed)g(di\013usion)f(and)i (jump)f(comp)s(onen)m(ts.)44 b(Suc)m(h)30 b(extensions)h(will)e (require)50 2258 y(some)h(extra)g(w)m(ork,)f(but)g(in)f(principle)e (our)j(approac)m(h)g(app)s(ears)g(applicable,)f(and)g(qualitativ)m(ely) g(similar)50 2404 y(results)37 b(should)g(b)s(e)h(obtainable.)64 b(Another)39 b(p)s(oten)m(tially)e(in)m(teresting)h(generalization)g (concerns)h(non-)50 2549 y(rev)m(ersible)31 b(di\013usion)g(pro)s (cesses.)48 b(Here)34 b(the)f(main)f(di\016cult)m(y)f(is)h(the)h (determination)f(of)h(the)h(in)m(v)-5 b(arian)m(t)50 2695 y(measure,)40 b(whic)m(h)d(our)g(metho)s(ds)g(do)h(not)h(address)e (at)h(all.)63 b(Ho)m(w)m(ev)m(er,)42 b(it)c(is)f(to)i(b)s(e)e(exp)s (ected)h(that)h(at)50 2840 y(least)29 b(in)e(uniquely)f(ergo)s(dic)j (situations,)f(some)h(of)g(our)f(results)g(can)h(still)d(b)s(e)j (carried)e(o)m(v)m(er.)42 b(W)-8 b(e)30 b(hop)s(e)e(to)50 2985 y(address)h(these)i(issues)e(in)g(future)h(publications.)159 3204 y(The)38 b(remainder)g(of)h(this)f(pap)s(er)g(is)f(organized)i(as) h(follo)m(ws:)57 b(In)38 b(Section)g(2)i(w)m(e)f(pro)m(v)m(e)h(an)f(a)g (priori)50 3349 y(estimate)h(on)f(the)g(sp)s(ectrum)f(of)i(the)f (generator)i(when)d(Diric)m(hlet)g(conditions)g(are)h(applied)e(to)j (small)50 3495 y(neigh)m(b)s(orho)s(o)s(ds)29 b(of)i(all)f(the)i(lo)s (cal)e(minima)f(of)j Fr(F)13 b Fu(.)43 b(In)31 b(Section)g(3)h(w)m(e)f (then)g(sho)m(w)h(that)f(the)h(eigen)m(v)-5 b(alues)50 3640 y(of)32 b(the)h(full)d(generator)j(are)g(asymptotically)e(close)i (to)g(those)f(of)h(the)f(capacit)m(y)i(matrix,)e(whic)m(h)f(in)g(turn) 50 3785 y(are)36 b(then)f(ev)-5 b(aluated)35 b(in)f(the)i(generic)f (situation.)54 b(In)35 b(the)h(course)f(of)h(the)f(pro)s(of)g(w)m(e)g (also)h(iden)m(tify)e(the)50 3931 y(eigen)m(v)-5 b(alues)26 b(of)g(the)g(generator)h(with)d(the)i(principle)d(eigen)m(v)-5 b(alues)26 b(of)g(appropriate)f(Diric)m(hlet)f(op)s(erators.)50 4076 y(Finally)-8 b(,)29 b(w)m(e)i(deriv)m(e)f(from)g(these)g(results)f (the)i(exp)s(onen)m(tial)e(distribution)e(of)k(the)f(mean)h(exit)f (times.)50 4295 y Ft(Ac)m(kno)m(wledgemen)m(ts:)41 b Fu(W)-8 b(e)21 b(thank)f(an)g(anon)m(ymous)h(referee)f(of)28 b([BEGK2])22 b(for)e(dra)m(wing)f(our)h(atten)m(tion)50 4440 y(to)30 b(the)f(pap)s(er)f([W2])i(b)m(y)f(W)-8 b(en)m(tzell.)41 b(W)-8 b(e)31 b(also)e(thank)f(M.)i(Ec)m(kho\013)g(for)f(participation) e(in)h(an)h(early)f(stage)50 4585 y(of)40 b(this)e(w)m(ork.)70 b(A.)40 b(Bo)m(vier)g(thanks)f(the)h(EPFL)g(and)f(V.)h(Ga)m(yrard)h (the)f(WIAS)f(for)h(hospitalit)m(y)e(and)50 4731 y(\014nancial)28 b(supp)s(ort)h(that)i(made)f(this)g(collab)s(oration)f(p)s(ossible.)50 4953 y Fw(2.)55 b(A)40 b(priori)i(sp)s(ectral)f(estimates.)159 5171 y Fu(Most)c(of)g(the)f(preparatory)g(bac)m(kground)h(and)e (necessary)i(tec)m(hnical)f(a)h(priori)d(estimates)j(w)m(ere)g(in-)50 5316 y(tro)s(duced)29 b(in)f([BEGK3])j(and)e(will)e(b)s(e)i(imp)s (orted)f(from)i(there.)40 b(In)29 b(this)g(section)h(w)m(e)g(giv)m(e)g (an)g(additional)50 5462 y(a)h(priori)e(estimate)i(on)g(the)g(sp)s (ectrum)f(of)h(certain)g(Diric)m(hlet)f(op)s(erators)h(asso)s(ciated)h (to)f Fr(L)3215 5476 y Fp(\017)3250 5462 y Fu(.)42 b(More)32 b(pre-)p eop %%Page: 7 7 7 6 bop 3671 295 a Fm(7)50 513 y Fu(cisely)-8 b(,)28 b(w)m(e)h(deriv)m(e)f(a)g(priori)e(lo)m(w)m(er)j(b)s(ounds)d(on)i (principal)d(eigen)m(v)-5 b(alues)28 b(and)g(for)g(the)g(Diric)m(hlet)f (problem)50 658 y(in)34 b(\(regular\))h(op)s(en)f(sets)h Fr(D)h Fs(\032)d Fq(R)1226 625 y Fp(d)1309 658 y Fu(with)h(closure)1850 635 y(\026)1829 658 y Fr(D)s Fu(.)55 b(W)-8 b(e)36 b(denote)g(b)m(y)e Fr(@)5 b(D)39 b Fu(the)c(b)s(oundary)e(of)3447 635 y(\026)3426 658 y Fr(D)s Fu(.)55 b(W)-8 b(e)50 804 y(denote)31 b(b)m(y)473 780 y(\026)469 804 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))26 b Fs(\021)796 780 y Fu(\026)792 804 y Fr(\025)845 818 y Fv(1)886 804 y Fu(\()p Fr(D)s Fu(\))k(the)h(principal)c(eigen)m(v)-5 b(alue)30 b(of)h(the)f(Diric)m(hlet)g(problem)1338 1022 y(\()p Fr(L)1435 1036 y Fp(\017)1490 1022 y Fs(\000)20 b Fr(\025)p Fu(\))p Fr(f)10 b Fu(\()p Fr(x)p Fu(\))25 b(=)g(0)p Fr(;)107 b(x)25 b Fs(2)g Fr(D)1669 1194 y(f)10 b Fu(\()p Fr(x)p Fu(\))25 b(=)g(0)p Fr(;)107 b(x)25 b Fs(2)g Fr(D)2385 1157 y Fp(c)3523 1108 y Fu(\(2)p Fr(:)p Fu(1\))50 1424 y(and)k(sometimes)i(use)f(the)g(notation)h Fr(L)1406 1391 y Fp(D)1467 1364 y Fk(c)1406 1447 y Fp(\017)1536 1424 y Fu(to)g(indicate)e(the)i(Diric)m(hlet)e(op)s(erator)h(corresp)s (onding)f(to)i(the)50 1570 y(problem)e(\(2.1\).)159 1788 y(The)h(follo)m(wing)f(lemma)h(is)f(a)i(classical)e(result)g(of)i (Donsk)m(er)g(and)f(V)-8 b(aradhan)30 b([D)m(V]:)50 2006 y Fh(Lemma)38 b(2.1:)51 b Fo(The)33 b(princip)-5 b(al)34 b(eigenvalue)1691 1982 y Fu(\026)1687 2006 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))f Fo(satis\014es)1434 2243 y Fu(\026)1430 2267 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))26 b Fs(\025)2018 2205 y Fu(1)p 1764 2246 555 4 v 1764 2329 a(sup)1901 2351 y Fp(x)p Fj(2)p Fp(D)2072 2329 y Fq(E)2127 2343 y Fp(x)2179 2329 y Fr(\034)2219 2343 y Fp(D)2280 2325 y Fk(c)3523 2267 y Fu(\(2)p Fr(:)p Fu(2\))159 2739 y(In)21 b(the)h(case)h(when)e(w)m(e)h(consider)f(di\013usions)e(on)i(a)h (compact)h(set,)i(Lemma)d(2.1)h(will)c(yield)h(a)i(su\016cien)m(tly)50 2885 y(go)s(o)s(d)43 b(estimate.)79 b(If)43 b Fr(D)j Fu(is)c(un)m(b)s(ounded,)i(the)f(suprem)m(um)e(on)i(the)h(righ)m(t)e (ma)m(y)i(b)s(e)e(in\014nite)f(and)i(the)50 3030 y(estimate)32 b(b)s(ecomes)g(useless.)42 b(Ho)m(w)m(ev)m(er,)34 b(it)d(is)g(easy)h (to)g(mo)s(dify)d(the)j(pro)s(of)f(of)g(Lemma)h(2.1)g(to)g(yield)e(an) 50 3176 y(impro)m(v)m(emen)m(t.)50 3394 y Fh(Lemma)36 b(2.2:)50 b Fo(L)-5 b(et)32 b Fr(\036)892 3408 y Fp(D)990 3394 y Fo(denote)g(the)g(eigenfunction)f(c)-5 b(orr)g(esp)g(onding)35 b(to)d(the)g(princip)-5 b(al)33 b(eigenvalue)e(of)50 3539 y Fr(L)112 3506 y Fp(D)112 3562 y(\017)178 3539 y Fo(.)159 3758 y(L)-5 b(et)33 b Fr(A)25 b Fs(\032)g Fr(D)36 b Fo(b)-5 b(e)32 b(any)h(c)-5 b(omp)g(act)35 b(set,)e(Then)794 4028 y Fu(\026)790 4052 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))26 b Fs(\025)1375 3991 y Fu(1)p 1124 4032 548 4 v 1124 4115 a(sup)1261 4137 y Fp(x)p Fj(2)p Fp(A)1425 4115 y Fq(E)1480 4129 y Fp(x)1532 4115 y Fr(\034)1572 4129 y Fp(D)1633 4110 y Fk(c)1697 3897 y Fi( )1769 4052 y Fu(1)21 b Fs(\000)1926 3929 y Fi(Z)1976 4135 y Fp(D)r Fj(n)p Fp(A)2149 4052 y Fr(dy)s(e)2286 4015 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2568 4052 y Fs(j)p Fr(\036)2647 4066 y Fp(D)2713 4052 y Fu(\()p Fr(y)s Fu(\))p Fs(j)2856 4015 y Fv(2)2898 3897 y Fi(!)3523 4052 y Fu(\(2)p Fr(:)p Fu(3\))50 4347 y Fo(Mor)-5 b(e)g(over,)33 b(for)h(any)f Fr(\016)c(>)c Fu(0)p Fo(,)33 b(ther)-5 b(e)33 b(exists)g(b)-5 b(ounde)g(d)34 b Fr(A)26 b Fs(\032)f Fr(D)35 b Fo(such)e(that)1302 4579 y Fu(\026)1298 4603 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))26 b Fs(\025)1883 4541 y Fu(1)p 1632 4582 V 1632 4665 a(sup)1769 4688 y Fp(x)p Fj(2)p Fp(A)1933 4665 y Fq(E)1988 4679 y Fp(x)2040 4665 y Fr(\034)2080 4679 y Fp(D)2141 4661 y Fk(c)2190 4603 y Fu(\(1)21 b Fs(\000)f Fr(\016)s Fu(\))1063 b(\(2)p Fr(:)p Fu(4\))50 5076 y Fe(Pro)s(of:)47 b Fu(Let)31 b Fr(w)r Fu(\()p Fr(x)p Fu(\))h(denote)f(the)f(solution)f(of)i(the)f (Diric)m(hlet)f(problem)1448 5294 y Fr(L)1510 5308 y Fp(\017)1545 5294 y Fr(w)r Fu(\()p Fr(x)p Fu(\))e(=)e(1)p Fr(;)106 b(x)26 b Fs(2)e Fr(D)1545 5466 y(w)r Fu(\()p Fr(x)p Fu(\))j(=)e(0)p Fr(;)106 b(x)26 b Fs(2)e Fr(D)2274 5429 y Fp(c)3523 5380 y Fu(\(2)p Fr(:)p Fu(5\))p eop %%Page: 8 8 8 7 bop 50 295 a Fm(8)50 513 y Fu(Note)44 b(that)f(\(see)h(e.g.)78 b(Eq)43 b(\(2.22\))i(of)50 b([BEGK3]\))44 b Fr(w)r Fu(\()p Fr(x)p Fu(\))j(=)e Fq(E)2299 527 y Fp(x)2351 513 y Fr(\034)2391 527 y Fp(D)2452 508 y Fk(c)2490 513 y Fu(.)78 b(Using)42 b(that)h(for)f(an)m(y)h Fr(C)53 b(>)45 b Fu(0,)50 658 y Fr(ab)25 b Fs(\024)269 622 y Fv(1)p 269 637 37 4 v 269 690 a(2)316 658 y Fu(\()p Fr(C)7 b(a)471 625 y Fv(2)532 658 y Fu(+)20 b Fr(b)662 625 y Fv(2)703 658 y Fr(=C)7 b Fu(\))30 b(with)f Fr(ab)d Fu(=)f Fr(\036)p Fu(\()p Fr(x)p Fu(\))p Fr(\036)p Fu(\()p Fr(y)s Fu(\))32 b(and)d Fr(C)j Fu(=)25 b Fr(w)r Fu(\()p Fr(y)s Fu(\))p Fr(=w)r Fu(\()p Fr(x)p Fu(\),)33 b(one)e(sho)m(ws)f(readily)f(that)629 790 y Fi(Z)680 996 y Fp(D)761 914 y Fr(dxe)902 876 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(x)p Fv(\))p Fp(=\017)1185 914 y Fr(\036)p Fu(\()p Fr(x)p Fu(\)\()p Fr(L)1458 928 y Fp(\017)1494 914 y Fr(\036)p Fu(\)\()p Fr(x)p Fu(\))27 b Fs(\025)1828 790 y Fi(Z)1878 996 y Fp(D)1959 914 y Fr(dxe)2100 876 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(x)p Fv(\))p Fp(=\017)2401 852 y Fr(\036)p Fu(\()p Fr(x)p Fu(\))p 2395 893 191 4 v 2395 976 a Fr(w)r Fu(\()p Fr(x)p Fu(\))2596 914 y(\()p Fr(L)2693 928 y Fp(\017)2728 914 y Fr(w)r Fu(\)\()p Fr(x)p Fu(\))p Fr(\036)p Fu(\()p Fr(x)p Fu(\))1732 1165 y(=)1828 1041 y Fi(Z)1878 1248 y Fp(D)1959 1165 y Fr(dxe)2100 1127 y Fj(\000)p Fp(F)g Fv(\()p Fp(x)p Fv(\))p Fp(=\017)2401 1103 y Fr(\036)p Fu(\()p Fr(x)p Fu(\))p 2395 1144 V 2395 1227 a Fr(w)r Fu(\()p Fr(x)p Fu(\))2596 1165 y Fr(\036)p Fu(\()p Fr(x)p Fu(\))1732 1410 y Fs(\025)2062 1349 y Fu(1)p 1839 1389 492 4 v 1839 1472 a(sup)1976 1495 y Fp(x)p Fj(2)p Fp(A)2140 1472 y Fr(w)r Fu(\()p Fr(x)p Fu(\))2356 1286 y Fi(Z)2407 1493 y Fp(A)2481 1410 y Fr(dxe)2622 1373 y Fj(\000)p Fp(F)g Fv(\()p Fp(x)p Fv(\))p Fp(=\017)2906 1410 y Fr(\036)2960 1373 y Fv(2)3001 1410 y Fu(\()p Fr(x)p Fu(\))3523 1164 y(\(2)p Fr(:)p Fu(6\))50 1662 y(Cho)s(osing)29 b Fr(\036)h Fu(as)h(the)f(normalized)f(eigenfunction)g(with)g(maximal)h(eigen)m(v) -5 b(alue)30 b(yields)f(\(2.3\).)159 1880 y(W)-8 b(e)31 b(no)m(w)g(claim)e(that)i(for)f(an)m(y)h Fr(\015)f(>)25 b Fu(0,)1242 1992 y Fi(Z)1348 2116 y Fr(dy)s(e)1485 2079 y Fj(\000)p Fp(\015)t(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1809 2116 y Fs(j)p Fr(\036)p Fu(\()p Fr(y)s Fu(\))p Fs(j)2031 2079 y Fv(2)2098 2116 y Fr(<)25 b(C)2259 2130 y Fp(\015)2331 2116 y Fr(<)g Fs(1)1005 b Fu(\(2)p Fr(:)p Fu(7\))50 2357 y(This)29 b(clearly)h(implies)d(\(2.4\).)44 b(The)30 b(estimate)h(\(2.7\))i(follo)m(ws)c(from)i(a)g(standard)e(Com)m(b)s (es-Thomas)i(esti-)50 2503 y(mate)22 b(for)e(the)h(ground-state)h (eigenfunction,)g Fr(\036)p Fu(.)37 b(It)21 b(is)f(con)m(v)m(enien)m(t) i(to)g(in)m(tro)s(duce)d Fr(v)s Fu(\()p Fr(y)s Fu(\))27 b Fs(\021)e Fr(e)3195 2470 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=)p Fv(2)p Fp(\017)3512 2503 y Fr(\036)p Fu(\()p Fr(y)s Fu(\),)50 2648 y(whic)m(h)29 b(is)g(the)i(corresp)s (onding)d(ground)i(state)h(eigenfunction)e(of)i(the)f(op)s(erator)1402 2879 y Fr(H)1478 2893 y Fp(\017)1538 2879 y Fs(\021)25 b Fr(e)1676 2841 y Fj(\000)p Fp(F)10 b Fv(\()p Fj(\001)p Fv(\))p Fp(=)p Fv(2)p Fp(\017)1976 2879 y Fr(L)2038 2893 y Fp(\017)2072 2879 y Fr(e)2114 2841 y Fp(F)g Fv(\()p Fj(\001)p Fv(\))p Fp(=)p Fv(2)p Fp(\017)3523 2879 y Fu(\(2)p Fr(:)p Fu(8\))50 3110 y(whic)m(h)29 b(is)g(a)i(symmetric)f(op)s(erator) g(on)h Fr(L)1475 3077 y Fv(2)1515 3110 y Fu(\()p Fq(R)1610 3077 y Fp(d)1659 3110 y Fr(;)15 b(dy)s Fu(\).)41 b(By)31 b(a)g(standard)e(computation,)972 3351 y Fr(t)p Fu(\()p Fr(\013)p Fu(\)[)p Fr(u)p Fu(])e Fs(\021)1358 3227 y Fi(Z)1464 3351 y Fr(dy)s(u)1611 3313 y Fj(\003)1653 3351 y Fu(\()p Fr(y)s Fu(\))p Fr(e)1813 3313 y Fp(i\013F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2111 3351 y Fr(H)2187 3365 y Fp(\017)2221 3351 y Fr(e)2263 3313 y Fj(\000)p Fp(i\013F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2617 3351 y Fr(u)p Fu(\()p Fr(y)s Fu(\))736 b(\(2)p Fr(:)p Fu(9\))50 3592 y(de\014nes)42 b(a)h(closed)g(sectoral)h(form)f(\(in)f(the)h (sense)g(of)g(Kato)h([Ka]\),)k(whic)m(h)41 b(is)i(analytic)f(in)g(the)h (strip)50 3737 y Fs(j=)p Fr(\013)p Fs(j)26 b Fr(<)f Fu(1)p Fr(=)p Fu(2.)42 b(The)30 b(Com)m(b)s(es-Thomas)f(estimate)j(\(see)f (e.g.)42 b([RS]\))30 b(then)g(implies)e(that)j Fr(v)j Fu(satis\014es)1199 3854 y Fi(Z)1305 3978 y Fr(dy)s(e)1442 3940 y Fv(\(1)p Fj(\000)p Fp(\015)t Fv(\))p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1858 3978 y Fs(j)p Fr(v)s Fu(\()p Fr(y)s Fu(\))p Fs(j)2073 3940 y Fv(2)2141 3978 y Fr(<)25 b(C)2302 3992 y Fp(\015)2374 3978 y Fr(<)g Fs(1)916 b Fu(\(2)p Fr(:)p Fu(10\))50 4219 y(whic)m(h)29 b(is)g(equiv)-5 b(alen)m(t)30 b(to)h(\(2.11\).)43 b(This)29 b(completes)i(the)f(pro)s(of)g(of)g(the)h(lemma.)40 b Fs(})159 4437 y Fu(W)-8 b(e)32 b(will)c(\014rst)h(establish)g(that)1251 4413 y(\026)1248 4437 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))i(is)e(at)j (most)f(p)s(olynomially)c(small)i(in)g Fr(\017)h Fu(if)g Fr(D)j Fu(do)s(es)e(not)f(con)m(tain)50 4583 y(lo)s(cal)f(minima,)g (more)i(precisely)-8 b(,)29 b(de\014ne)1284 4813 y Fs(M)1393 4827 y Fp(\017)1453 4813 y Fs(\021)c(f)p Fr(z)30 b Fs(2)25 b Fu(\012)p Fs(j)15 b Fu(dist)o(\()p Fr(z)t(;)g Fs(M)p Fu(\))27 b Fs(\024)e Fr(\017)p Fs(g)1002 b Fu(\(2)p Fr(:)p Fu(12\))50 5117 y Fh(Lemma)41 b(2.3:)103 b Fo(Assume)35 b(that)h Fr(D)25 b Fs(\\)c(M)1561 5131 y Fp(\017)1626 5117 y Fu(=)29 b Fs(;)p Fo(.)49 b(Then)35 b(ther)-5 b(e)36 b(is)f(a)h(\014nite)f(p)-5 b(ositive)35 b(c)-5 b(onstant)37 b Fr(C)42 b Fo(such)50 5262 y(that)987 5408 y Fu(sup)980 5484 y Fp(x)p Fj(2)p Fp(D)1146 5408 y Fq(E)1201 5422 y Fp(x)1253 5408 y Fr(\034)1293 5422 y Fp(D)1354 5403 y Fk(c)1418 5408 y Fs(\024)25 b Fr(C)k Fu(sup)1600 5484 y Fp(x)p Fj(2)p Fp(D)1767 5408 y Fs(jf)p Fr(y)g Fu(:)c Fr(F)13 b Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fr(F)13 b Fu(\()p Fr(x)p Fu(\))p Fs(gj)k Fr(\017)2589 5370 y Fj(\000)p Fp(d)p Fv(+1)3477 5408 y Fu(\(2)p Fr(:)p Fu(13\))p eop %%Page: 9 9 9 8 bop 3671 295 a Fm(9)50 731 y Fe(Pro)s(of:)54 b Fu(The)33 b(starting)h(p)s(oin)m(t)e(of)i(the)f(pro)s(of)g(is)g(the)g(relation)g (\(whic)m(h)g(is)f(an)h(immediate)g(consequence)50 876 y(of)k([BEGK3],)32 b(Eq.)41 b(\(2.27\)\))748 1017 y Fi(Z)799 1224 y Fp(D)880 1141 y Fr(dy)s(e)1017 1103 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1298 1141 y Fr(h)1350 1157 y Fp(B)1405 1166 y Fk(\032)1443 1157 y Fv(\()p Fp(x)p Fv(\))p Fp(;D)1623 1139 y Fk(c)1663 1141 y Fu(\()p Fr(y)s Fu(\))25 b Fs(\025)130 b Fu(inf)1902 1204 y Fp(z)s Fj(2)p Fp(@)t(B)2087 1213 y Fk(\032)2124 1204 y Fv(\()p Fp(x)p Fv(\))2237 1141 y Fq(E)2292 1155 y Fp(z)2340 1141 y Fr(\034)2380 1155 y Fp(D)2441 1136 y Fk(c)2479 1141 y Fu(cap)2631 1157 y Fp(B)2686 1166 y Fk(\032)2723 1157 y Fv(\()p Fp(x)p Fv(\))2826 1141 y Fu(\()p Fr(D)2939 1103 y Fp(c)2976 1141 y Fu(\))466 b(\(2)p Fr(:)p Fu(14\))50 1412 y(b)s(et)m(w)m(een)40 b(mean)g(time,)h(equilibrium)35 b(p)s(oten)m(tial)k(and)g(capacities.) 69 b(It)39 b(follo)m(ws)g(from)g(the)h(w)m(ell)e(kno)m(wn)50 1557 y(relation)1409 1703 y Fq(E)1463 1717 y Fp(x)1515 1703 y Fr(\034)1555 1717 y Fp(D)1616 1698 y Fk(c)1680 1703 y Fu(=)1776 1579 y Fi(Z)1826 1786 y Fp(D)1908 1703 y Fr(G)1980 1717 y Fp(D)2045 1703 y Fu(\()p Fr(x;)15 b(y)s Fu(\))p Fr(dy)1130 b Fu(\(2)p Fr(:)p Fu(15\))50 1926 y(b)s(et)m(w)m(een)39 b(mean)f(time)g(and)g(Green)g(function)f (that)i(the)g(Harnac)m(k)g(inequalit)m(y)d(of)46 b([BEGK3],)c(Lemma)50 2072 y(4.1,)32 b(carries)d(o)m(v)m(er)j(to)f Fq(E)871 2086 y Fp(z)919 2072 y Fr(\034)959 2086 y Fp(D)1020 2067 y Fk(c)1058 2072 y Fu(,)g(implying)c(that,)32 b(if)d Fr(\032)c Fu(=)g Fr(c\017)p Fu(,)31 b(then)1290 2326 y(sup)1199 2407 y Fp(z)s Fj(2)p Fp(@)t(B)1384 2416 y Fk(\032)1420 2407 y Fv(\()p Fp(x)p Fv(\))1534 2326 y Fq(E)1588 2340 y Fp(z)1636 2326 y Fr(\034)1676 2340 y Fp(D)1737 2322 y Fk(c)1801 2326 y Fs(\024)25 b Fr(C)126 b Fu(inf)1984 2389 y Fp(z)s Fj(2)p Fp(@)t(B)2169 2398 y Fk(\032)2205 2389 y Fv(\()p Fp(x)p Fv(\))2318 2326 y Fq(E)2373 2340 y Fp(z)2421 2326 y Fr(\034)2461 2340 y Fp(D)2522 2322 y Fk(c)3477 2326 y Fu(\(2)p Fr(:)p Fu(16\))50 2620 y(Com)m(bining)28 b(this)h(with)g(\(2.14\))k(giv)m(es)e(us)e(that) 1063 2911 y(sup)971 2992 y Fp(z)s Fj(2)p Fp(@)t(B)1156 3001 y Fk(\032)1193 2992 y Fv(\()p Fp(x)p Fv(\))1306 2911 y Fq(E)1361 2925 y Fp(z)1409 2911 y Fr(\034)1449 2925 y Fp(D)1510 2907 y Fk(c)1574 2911 y Fs(\024)c Fr(C)1752 2768 y Fi(R)1795 2874 y Fp(D)1876 2841 y Fr(dy)s(e)2013 2808 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2294 2841 y Fr(h)2346 2858 y Fp(B)2401 2867 y Fk(\032)2439 2858 y Fv(\()p Fp(x)p Fv(\))p Fp(;D)2619 2839 y Fk(c)2659 2841 y Fu(\()p Fr(y)s Fu(\))p 1752 2890 1026 4 v 1999 2973 a(cap)2150 2990 y Fp(B)2205 2999 y Fk(\032)2243 2990 y Fv(\()p Fp(x)p Fv(\))2345 2973 y Fu(\()p Fr(D)2458 2947 y Fp(c)2495 2973 y Fu(\))3477 2912 y(\(2)p Fr(:)p Fu(17\))50 3205 y(W)-8 b(e)31 b(no)m(w)g(distinguish)26 b(the)31 b(regions)f Fs(f)p Fr(y)e Fu(:)e Fr(F)13 b Fu(\()p Fr(y)s Fu(\))26 b Fr(>)f(F)13 b Fu(\()p Fr(x)p Fu(\))p Fs(g)31 b Fu(and)f Fs(f)p Fr(y)f Fu(:)c Fr(F)13 b Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fr(F)13 b Fu(\()p Fr(x)p Fu(\))p Fs(g)31 b Fu(in)e(the)i(in)m(tegral.)40 b(In)50 3350 y(the)29 b(former,)h(w)m(e)g(just)e(use)h(that)h Fr(h)1236 3367 y Fp(B)1291 3376 y Fk(\032)1329 3367 y Fv(\()p Fp(x)p Fv(\))p Fp(;D)1509 3348 y Fk(c)1549 3350 y Fu(\()p Fr(y)s Fu(\))c Fs(\024)f Fu(1,)30 b(while)d(in)h(the)i (latter)g(w)m(e)g(in)m(v)m(ok)m(e)g(the)f(upp)s(er)f(b)s(ound)50 3496 y(from)i(Prop)s(osition)e(4.3)k(in)d([BEGK3].)42 b(This)28 b(giv)m(es)345 3801 y(sup)253 3881 y Fp(z)s Fj(2)p Fp(@)t(B)438 3890 y Fk(\032)475 3881 y Fv(\()p Fp(x)p Fv(\))588 3801 y Fq(E)643 3815 y Fp(z)691 3801 y Fr(\034)731 3815 y Fp(D)792 3796 y Fk(c)856 3801 y Fs(\024)c Fr(C)1034 3644 y Fi(R)1077 3749 y Fp(y)r Fj(2)p Fp(D)r Fv(:)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(>F)g Fv(\()p Fp(x)p Fv(\))1629 3717 y Fr(dy)s(e)1766 3684 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)p 1034 3780 1014 4 v 1275 3863 a Fu(cap)1426 3880 y Fp(B)1481 3889 y Fk(\032)1519 3880 y Fv(\()p Fp(x)p Fv(\))1621 3863 y Fu(\()p Fr(D)1734 3837 y Fp(c)1771 3863 y Fu(\))850 4083 y(+)20 b Fr(C)1267 4021 y Fu(1)p 1024 4062 533 4 v 1024 4145 a(cap)1175 4162 y Fp(B)1230 4171 y Fk(\032)1268 4162 y Fv(\()p Fp(x)p Fv(\))1370 4145 y Fu(\()p Fr(D)1483 4119 y Fp(c)1521 4145 y Fu(\))1582 3959 y Fi(Z)1632 4166 y Fp(y)r Fj(2)p Fp(D)r Fv(:)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fj(\024)p Fp(F)g Fv(\()p Fp(x)p Fv(\))2185 4083 y Fr(dy)s(e)2322 4045 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2614 4013 y Fu(cap)2766 4029 y Fp(B)2821 4038 y Fk(\032)2858 4029 y Fv(\()p Fp(y)r Fv(\))2959 4013 y Fu(\()p Fr(B)3063 4027 y Fp(\032)3105 4013 y Fu(\()p Fr(x)p Fu(\)\))p 2614 4062 650 4 v 2674 4145 a(cap)2825 4162 y Fp(B)2880 4171 y Fk(\032)2918 4162 y Fv(\()p Fp(y)r Fv(\))3018 4145 y Fu(\()p Fr(D)3131 4119 y Fp(c)3168 4145 y Fu(\))3477 3932 y(\(2)p Fr(:)p Fu(18\))50 4368 y(Using)35 b(the)g(upp)s(er)f(and)h(lo)m(w)m(er)h(b)s(ounds)d(on)j(the) g(capacities)g(from)f(Prop)s(osition)e(4.7)k(of)43 b([BEGK3],)38 b(w)m(e)50 4513 y(get)31 b(that)756 4664 y(sup)664 4744 y Fp(z)s Fj(2)p Fp(@)t(B)849 4753 y Fk(\032)886 4744 y Fv(\()p Fp(x)p Fv(\))999 4664 y Fq(E)1054 4678 y Fp(z)1102 4664 y Fr(\034)1142 4678 y Fp(D)1203 4659 y Fk(c)1267 4664 y Fs(\024)25 b Fr(C)1435 4626 y Fj(0)1460 4664 y Fr(\017\032)1544 4626 y Fj(\000)p Fp(d)p Fv(+2)1735 4664 y Fr(e)1777 4626 y Fv(+)p Fp(F)10 b Fv(\()p Fp(x)p Fv(\))p Fp(=\017)2074 4540 y Fi(Z)2125 4746 y Fp(y)r Fj(2)p Fp(D)r Fv(:)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(>F)g Fv(\()p Fp(x)p Fv(\))2677 4664 y Fr(dy)s(e)2814 4626 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1262 4931 y Fu(+)19 b Fr(C)1424 4894 y Fj(0)1449 4931 y Fr(\017\032)1533 4894 y Fj(\000)p Fp(d)p Fv(+2)1740 4808 y Fi(Z)1790 5014 y Fp(y)r Fj(2)p Fp(D)r Fv(:)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fj(\024)p Fp(F)g Fv(\()p Fp(x)p Fv(\))2343 4931 y Fr(dy)3477 4808 y Fu(\(2)p Fr(:)p Fu(19\))50 5171 y(By)26 b(our)g(assumption)f(on)h Fr(F)13 b Fu(,)27 b(the)g(\014rst)e(in)m (tegral)h(is)g(b)s(ounded)e(b)m(y)i(a)g(constan)m(t)i(times)e(exp\()p Fs(\000)p Fr(F)13 b Fu(\()p Fr(x)p Fu(\))p Fr(=\017)p Fu(\))27 b(and)50 5316 y(the)33 b(second)g(is)f(equal)g(to)i(the)f(v)m (olume)g(of)g(the)g(lev)m(el)g(set)g Fs(f)p Fr(F)13 b Fu(\()p Fr(y)s Fu(\))31 b Fs(\024)e Fr(F)13 b Fu(\()p Fr(x)p Fu(\))p Fs(g)p Fu(.)49 b(This)32 b(implies)e(the)j(claimed)50 5462 y(b)s(ound.)p Fs(})p eop %%Page: 10 10 10 9 bop 50 295 a Fm(10)159 513 y Fu(Com)m(bining)28 b(our)i(results)f(yields)g(the)50 731 y Fh(Corollary)56 b(2.4:)86 b Fo(If)45 b Fr(D)33 b Fs(\\)c(M)1256 745 y Fp(\017)1339 731 y Fu(=)48 b Fs(;)p Fo(,)h(then)d(ther)-5 b(e)47 b(exists)f(a)f(\014nite)h(p)-5 b(ositive)46 b(c)-5 b(onstant)48 b Fr(C)55 b(<)48 b Fs(1)p Fo(,)50 876 y(indep)-5 b(endent)34 b(of)f Fr(\017)p Fo(,)f(such)h(that)1600 998 y Fu(\026)1597 1022 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))25 b Fs(\025)g Fr(C)7 b(\017)2028 984 y Fp(d)p Fj(\000)p Fv(1)3477 1022 y Fu(\(2)p Fr(:)p Fu(20\))159 1444 y(W)-8 b(e)23 b(can)g(generalize)g(the)f(b)s(ounds)e(obtained)i(so)g(far)h(to) g(sets)f Fr(D)k Fu(con)m(taining)21 b(some)i(of)g(the)f(lo)s(cal)g (minima)50 1589 y(of)30 b Fr(F)13 b Fu(.)41 b(I.e.)g(let)31 b Fs(N)38 b(\032)25 b(M)30 b Fu(b)s(e)g(nonempt)m(y)g(and)g(let)1282 1831 y Fs(N)1357 1845 y Fp(\017)1416 1831 y Fu(=)1512 1757 y Fi(\010)1565 1831 y Fr(y)f Fs(2)24 b Fq(R)1784 1794 y Fp(d)1832 1831 y Fs(j)15 b Fu(dist\()p Fr(y)s(;)g Fs(N)e Fu(\))26 b Fs(\024)f Fr(\017)2425 1757 y Fi(\011)3477 1831 y Fu(\(2)p Fr(:)p Fu(21\))50 2073 y(Assume)j(that)i Fr(D)e Fs(\033)d(N)855 2087 y Fp(\017)918 2073 y Fu(and)j(set)i Fr(A)p Fu(\()p Fr(x)p Fu(\))c(=)f Fs(f)p Fr(y)k Fu(:)c Fr(h)1767 2090 y Fp(B)1822 2099 y Fk(\017)1855 2090 y Fv(\()p Fp(x)p Fv(\))p Fp(;D)2035 2071 y Fk(c)2070 2090 y Fj(n)p Fp(B)2162 2099 y Fk(\017)2195 2090 y Fv(\()p Fp(x)p Fv(\))2297 2073 y Fu(\()p Fr(y)s Fu(\))h(=)f(max)2706 2087 y Fp(y)r Fj(2M)2899 2073 y Fr(h)2951 2090 y Fp(B)3006 2099 y Fk(\017)3039 2090 y Fv(\()p Fp(x)p Fv(\))p Fp(;D)3219 2071 y Fk(c)3284 2090 y Fp(baB)3410 2099 y Fk(\017)3443 2090 y Fv(\()p Fp(x)p Fv(\))3546 2073 y Fu(\()p Fr(y)s Fu(\))p Fs(g)50 2219 y Fu(Then)50 2437 y Fh(Lemma)38 b(2.5:)51 b Fo(Under)33 b(the)g(assumptions)h(of)f(lemma)h(2.2,)1239 2663 y Fu(1)p 1161 2704 202 4 v 1165 2766 a(\026)1161 2790 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))1399 2725 y Fs(\025)1557 2638 y Fi(X)1494 2833 y Fp(i)p Fv(:)p Fp(x)1582 2842 y Fk(i)1611 2833 y Fj(2N)1720 2842 y Fk(\017)1857 2568 y Fi(R)1900 2673 y Fp(A)p Fv(\()p Fp(x)2024 2682 y Fk(i)2053 2673 y Fv(\))2101 2641 y Fr(e)2143 2608 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2424 2641 y Fr(dy)p 1778 2704 821 4 v 1778 2787 a Fu(cap)1929 2803 y Fp(B)1984 2812 y Fk(\017)2017 2803 y Fv(\()p Fp(x)2086 2812 y Fk(i)2115 2803 y Fv(\))2148 2787 y Fu(\()p Fr(D)s Fs(n)p Fr(B)2375 2801 y Fp(\017)2410 2787 y Fu(\()p Fr(x)2497 2801 y Fp(i)2528 2787 y Fu(\)\))3477 2725 y(\(2)p Fr(:)p Fu(22\))50 3185 y Fe(Pro)s(of:)56 b Fu(The)34 b(pro)s(of)f(is)g(similar)f(to)i(that)h (of)g(the)f(preceding)f(corollary)g(com)m(bined)h(with)f(the)h (estimate)50 3330 y(on)c(mean)h(times)e(giv)m(en)i(in)e(Theorem)h(6.2)h (of)38 b([BEGK3].)k(W)-8 b(e)32 b(lea)m(v)m(e)f(the)g(details)e(to)i (the)g(reader.)p Fs(})50 3549 y Ft(Remark:)46 b Fu(The)30 b(k)m(ey)h(fact)g(w)m(e)g(need)f(to)h(extract)h(from)e(lemma)g(2.5)h (is)f(that)1109 3767 y(\026)1105 3791 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))c Fs(\025\030)66 b Fu(min)1499 3849 y Fp(i)p Fv(:)p Fp(x)1587 3858 y Fk(i)1615 3849 y Fj(2N)1748 3791 y Fr(e)1790 3753 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(z)1990 3726 y Fg(\003)2028 3753 y Fv(\()p Fp(x)2097 3762 y Fk(i)2126 3753 y Fp(;)p Fj(M)2233 3763 y Fk(k)2273 3753 y Fv(\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)2483 3762 y Fk(i)2513 3753 y Fv(\)])p Fp(=\017)p Fv(])3477 3791 y Fu(\(2)p Fr(:)p Fu(23\))50 4153 y Fw(3.)55 b(Characterization)39 b(of)i(the)g(small)f(eigen)m(v)-7 b(alues.)159 4371 y Fu(It)34 b(is)f(a)i(w)m(ell-kno)m(wn)e(fact)h(that)h(if)e Fr(F)47 b Fu(has)34 b Fr(n)g Fu(lo)s(cal)f(minima,)g(then)h Fr(L)2591 4385 y Fp(\017)2659 4371 y Fu(has)g Fr(n)g Fu(eigen)m(v)-5 b(alues)34 b(that)g(are)50 4516 y(exp)s(onen)m(tially)i (small)g(in)h Fr(\017)h Fu(and)f(that)h(the)h(next)f(largest)g(eigen)m (v)-5 b(alue)38 b(is)f(of)h(the)g(order)f(of)h(a)g(constan)m(t)50 4662 y([FW,Kolo].)79 b(It)43 b(is)f(also)h(kno)m(wn)f(\([Kolo],)47 b(Chapter)42 b(8,)47 b(Prop)s(osition)41 b(2.2\))j(that)f(the)g (eigenspace)h(of)50 4807 y(these)39 b(eigenfunctions)f(is)g(exp)s(onen) m(tially)g(close)h(in)f(the)h Fr(L)2132 4774 y Fv(2)2173 4807 y Fu(\(exp\()p Fs(\000)p Fr(F)13 b Fu(\()p Fr(y)s Fu(\)\))p Fr(dy)s Fu(\)-distance)41 b(to)f(the)f(linear)50 4953 y(h)m(ull)24 b(of)i(the)g Fr(n)g Fu(indicator)f(functions)f Fr(\037)1381 4967 y Fp(i)1437 4953 y Fu(of)i(the)h(attractors)h(of)e (the)g(minima)e Fr(x)2732 4967 y Fp(i)2788 4953 y Fu(under)h(the)h (deterministic)50 5098 y(dynamical)j(system)47 b(_)-42 b Fr(y)s Fu(\()p Fr(t)p Fu(\))26 b(=)f Fs(\000r)p Fr(F)13 b Fu(\()p Fr(y)s Fu(\()p Fr(t)p Fu(\)\).)159 5316 y(In)34 b(this)g(section)h(w)m(e)g(will)d(deriv)m(e)i(a)i(precise)e(c)m (haracterization)i(of)f(these)g(eigen)m(v)-5 b(alues)35 b(that)g(togeth-)50 5462 y(er)h(with)g(the)h(estimates)g(on)g (capacities)f(of)44 b([BEGK3])38 b(will)c(ultimately)h(yield)g(the)i (exact)h(asymptotic)p eop %%Page: 11 11 11 10 bop 3633 295 a Fm(11)50 513 y Fu(form)m(ulae)38 b(of)i(Theorem)e(1.1.)68 b(This)38 b(is)g(the)h(analogue)g(of)h (Section)e(4)i(of)46 b([BEGK2])40 b(for)f(the)g(di\013usion)50 658 y(case.)h(Our)26 b(approac)m(h)h(can)g(in)f(to)h(some)g(exten)m(t)i (b)s(e)d(seen)h(as)g(an)f(application)g(of)h(the)g(ideas)f(of)h(W)-8 b(en)m(tzell's)50 804 y(remark)j(able)36 b(pap)s(er)g(from)g(1973)j ([W2].)61 b(As)37 b(w)m(e)g(will)d(see,)40 b(the)d(application)e(of)i (these)g(ideas)g(is)e(not)i(as)50 949 y(straigh)m(tforw)m(ard)30 b(as)h(in)e(the)h(discrete)g(case,)i(but)e(in)f(principle)e(v)m(ery)j (similar.)159 1167 y(Before)23 b(w)m(e)f(turn)f(to)i(the)f(details)g (of)g(this)f(construction,)i(it)f(is)f(useful)f(to)j(explain)d(the)j (general)f(strategy)-8 b(.)159 1385 y(Let)41 b(us)g(no)m(w)g(consider)e (a)j(set)f(of)g(disjoin)m(t)f(compact)i(sets)f Fr(B)2343 1399 y Fp(i)2417 1385 y Fs(\021)h Fr(B)2599 1399 y Fp(\017)2634 1385 y Fu(\()p Fr(x)2721 1399 y Fp(i)2752 1385 y Fu(\),)i Fr(i)f Fu(=)f(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(k)s Fu(.)73 b(Let)3616 1361 y(\026)3612 1385 y Fr(\025)3665 1399 y Fp(k)50 1531 y Fu(denote)44 b(the)f(principal)d(eigen)m(v)-5 b(alue)43 b(of)h(the)f(Diric)m(hlet)f(op)s(erator)i Fr(L)2480 1545 y Fp(\017)2558 1531 y Fu(with)e(Diric)m(hlet)g(conditions)g(on)50 1676 y Fs(S)105 1690 y Fp(k)175 1676 y Fs(\021)25 b([)332 1643 y Fp(k)332 1700 y(i)p Fv(=1)454 1676 y Fr(B)523 1690 y Fp(i)583 1676 y Fu(\(and)30 b(p)s(ossibly)d(on)i(some)i(further) d(set)j(\012\).)40 b(Consider,)29 b(for)h Fr(\025)25 b(<)2832 1652 y Fu(\026)2828 1676 y Fr(\025)2881 1690 y Fp(k)2926 1676 y Fu(,)30 b(the)g(solution)e(of)j(the)50 1822 y(Diric)m(hlet)e(problem)1229 1930 y(\()p Fr(L)1326 1944 y Fp(\017)1381 1930 y Fs(\000)20 b Fr(\025)p Fu(\))p Fr(f)1615 1893 y Fp(\025)1662 1930 y Fu(\()p Fr(x)p Fu(\))26 b(=)f(0)p Fr(;)107 b(x)25 b Fs(2)g Fu(\012)p Fs(n)p Fr(@)5 b Fs(S)2465 1944 y Fp(k)1561 2103 y Fr(f)1616 2066 y Fp(\025)1662 2103 y Fu(\()p Fr(x)p Fu(\))26 b(=)f Fr(\036)p Fu(\()p Fr(x)p Fu(\))p Fr(;)108 b(x)25 b Fs(2)g Fr(@)5 b Fs(S)2486 2117 y Fp(k)3523 2014 y Fu(\(3)p Fr(:)p Fu(1\))50 2273 y(\(i.e.)43 b(w)m(e)31 b(consider)f(the)h(Diric)m(hlet)f(problems) g(in)f(the)i(exterior)g(and)g(the)g(in)m(terior)f(of)h(the)g(balls)f (sim)m(ulta-)50 2419 y(neously;)h(note)h(that)g(the)g(principal)c (eigen)m(v)-5 b(alue)32 b(of)g Fr(L)1945 2433 y Fp(\017)2011 2419 y Fu(within)d(a)j(ball)e(will)f(alw)m(a)m(ys)j(b)s(e)f(larger)g (than)3616 2395 y(\026)3612 2419 y Fr(\025)3665 2433 y Fp(k)50 2564 y Fu(and)25 b(so)h(pla)m(ys)g(no)f(r^)-45 b(ole\).)40 b(The)25 b(basic)g(idea)h(is)f(no)m(w)g(to)i(construct)f (an)g(eigenfunction)f(of)h(the)g(full)d(op)s(erator)50 2710 y Fr(L)112 2724 y Fp(\017)176 2710 y Fu(as)31 b(a)f(solution)f(of) h(the)g(problem)e(\(3.1\))k(with)d(suitably)f(c)m(hosen)j Fr(\036)p Fu(.)40 b(Indeed,)30 b(if)f Fr(\025)h Fu(is)f(an)h(eigen)m(v) -5 b(alue)30 b(of)50 2855 y Fr(L)112 2869 y Fp(\017)181 2855 y Fu(and)j(if)g(w)m(e)i(c)m(ho)s(ose)g Fr(\036)p 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Fu(is)g(not)h(in)e(the)i (sp)s(ectrum)f(of)g Fr(L)1599 3465 y Fj(S)1644 3475 y Fk(k)1599 3521 y Fp(\017)1688 3498 y Fu(,)h(this)f(implies)e(that)j Fr(f)2475 3465 y Fp(\025)2522 3498 y Fu(\()p Fr(x)p Fu(\))d(=)f Fr(\036)p Fu(\()p Fr(x)p Fu(\))k(on)g Fs(S)3158 3465 y Fp(c)3151 3524 y(k)3224 3498 y Fu(as)g(w)m(ell.)39 b(The)50 3644 y(same)30 b(argumen)m(t)f(applies)f(in)g(the)i(in)m (terior)e(of)i Fs(S)1724 3658 y Fp(k)1798 3644 y Fu(This)d(means)j (that)g Fr(\025)25 b(<)2656 3620 y Fu(\026)2652 3644 y Fr(\025)2705 3658 y Fp(k)2779 3644 y Fu(is)j(an)i(eigen)m(v)-5 b(alue)29 b(of)h Fr(L)3593 3658 y Fp(\017)3657 3644 y Fu(if)50 3789 y(and)g(only)g(if)f(w)m(e)i(can)g(\014nd)f(a)h(function)e Fr(\036)i Fu(on)f Fr(@)5 b Fs(S)1745 3803 y Fp(k)1790 3789 y Fu(,)31 b(suc)m(h)g(that)g(the)g(solution)e(of)i(the)g(Diric)m (hlet)e(problem)50 3935 y(\(3.1\))i(is)e(actually)h(an)g(eigenfunction) e(of)i Fr(L)1551 3949 y Fp(\017)1616 3935 y Fu(with)e(eigen)m(v)-5 b(alue)30 b Fr(\025)p Fu(.)41 b(In)29 b(other)h(w)m(ords,)g(an)m(y)g (eigenfunction)50 4080 y(corresp)s(onding)h(to)k(eigen)m(v)-5 b(alues)33 b(b)s(elo)m(w)g(the)h(principal)d(Diric)m(hlet)h(eigen)m(v) -5 b(alue)34 b(can)g(b)s(e)f(represen)m(ted)g(as)50 4225 y(solution)c(of)h(\(3.1\).)159 4444 y(Th)m(us)41 b(the)i(eigen)m(v)-5 b(alue)43 b(problem)e(reduces)h(to)i(\014nding)c(out)j(for)f(whic)m(h)f (v)-5 b(alues)42 b(of)h Fr(\025)g Fu(for)f(suitable)50 4589 y Fr(\036)34 b Fu(on)h(the)g(b)s(oundaries)d(of)j Fr(B)1074 4603 y Fp(i)1104 4589 y Fu(,)h(\()p Fr(L)1262 4603 y Fp(\017)1320 4589 y Fs(\000)22 b Fr(\025)p Fu(\))p Fr(f)1556 4556 y Fp(\025)1636 4589 y Fu(=)32 b(0)j(ev)m(erywhere.)54 b(In)33 b(fact,)k(\()p Fr(L)2764 4603 y Fp(\017)2822 4589 y Fs(\000)23 b Fr(\025)p Fu(\))p Fr(f)3059 4556 y Fp(\025)3141 4589 y Fu(is)33 b(in)h(general)g(a)50 4735 y(measure)25 b(concen)m(trated)h(on)f(the)g(surface)f Fr(@)5 b Fs(S)1611 4749 y Fp(k)1656 4735 y Fu(;)27 b(demanding)c(that)j (this)d(surface)i(measure)g(b)s(e)f(zero)i(yields)50 4880 y(in)j(general)h(an)h(in)m(tegral)f(equation)g(for)h Fr(\036)p Fu(\()p Fr(x)p Fu(\))g(on)f Fr(@)5 b Fs(S)1879 4894 y Fp(k)1924 4880 y Fu(,)31 b(whic)m(h)e(is)h(not)g(particularly)f (easy)i(to)g(handle.)39 b(In)50 5025 y(the)24 b(case)h(of)f(discrete)g (Mark)m(o)m(v)h(pro)s(cesses,)h(w)m(e)e(ha)m(v)m(e)h(considered)e(a)i (v)m(ery)f(similar)d(problem)i(in)f([BEGK2].)50 5171 y(There,)28 b(the)h(balls)d Fr(B)770 5185 y Fp(i)829 5171 y Fu(w)m(ere,)j(ho)m(w)m(ev)m(er,)i(simply)25 b(the)k(p)s(oin)m (ts)e Fr(x)2201 5185 y Fp(i)2231 5171 y Fu(.)40 b(The)28 b(measure)g(\()p Fr(L)2930 5185 y Fp(\017)2981 5171 y Fs(\000)16 b Fr(\025)p Fu(\))p Fr(f)3211 5138 y Fp(\025)3287 5171 y Fu(w)m(as)28 b(then)g(a)50 5316 y(simple)h(measure)h(on)h(the)g (\014nite)e(set)j Fs(M)1456 5330 y Fp(k)1501 5316 y Fu(,)f(and)f(the)h (b)s(oundary)d(condition)i(reduces)g(to)h(the)g Fr(k)j Fu(n)m(um)m(b)s(ers)50 5462 y Fr(\036)p Fu(\()p Fr(x)191 5476 y Fp(i)222 5462 y Fu(\),)39 b(and)e(the)h(in)m(tegral)f(equation)g (w)m(as)h(reduced)e(to)i(a)g(simple)d(linear)h(equation)h(for)h(the)f (unkno)m(wn)p eop %%Page: 12 12 12 11 bop 50 295 a Fm(12)50 513 y Fu(v)m(ector)42 b Fr(\036)p Fu(\()p Fr(x)474 527 y Fp(i)505 513 y Fu(\))p Fr(;)15 b(i)43 b Fu(=)f(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(k)s Fu(.)72 b(The)40 b(condition)f(for)i Fr(\025)f Fu(to)i(b)s(e)e(an)g(eigen)m(v) -5 b(alue)41 b(w)m(as)g(th)m(us)f(simply)e(that)50 658 y(a)j(certain)f(determinan)m(t)g(v)-5 b(anishes.)69 b(It)40 b(w)m(ould)f(b)s(e)h(more)h(than)f(nice)f(if)h(w)m(e)h(could)e(reduce)h (ourselv)m(es)50 804 y(to)h(a)h(similarly)37 b(simple)i(condition)h(in) f(the)i(presen)m(t)g(case.)73 b(Indeed)40 b(this)g(w)m(ould)g(b)s(e)g (so,)k(if)c(w)m(e)h(knew)50 949 y(b)s(eforehand)31 b(that)i Fr(\036)p Fu(\()p Fr(x)p Fu(\))h(is)e(constan)m(t)i(on)e(eac)m(h)i (surface)f Fr(@)5 b(B)2160 963 y Fp(i)2190 949 y Fu(.)48 b(While)31 b(this)h(cannot)h(b)s(e)f(truly)g(the)g(case,)50 1095 y(if)h Fr(\017)h Fu(is)f(small)g(w)m(e)h(ma)m(y)h(exp)s(ect)f (that)h Fr(\036)f Fu(v)-5 b(aries)34 b(little.)51 b(In)33 b(that)i(case,)h(w)m(e)f(could,)f(as)g(w)m(e)h(shall)d(see,)k(use)50 1240 y(p)s(erturbativ)m(e)g(argumen)m(ts)j(to)f(arriv)m(e)g(at)g(the)g (desired)e(conclusion.)62 b(Unfortunately)-8 b(,)39 b(to)g(obtain)e (suc)m(h)50 1385 y(con)m(trol)c(on)f(eigenfunctions)f(lo)s(oks)h (rather)g(di\016cult.)45 b(While)31 b(the)h(Harnac)m(k-)i(and)e(H\177) -45 b(older)31 b(inequalities)50 1531 y(will)f(giv)m(e)j(us)f(the)h (desired)f(con)m(trol)h(if)f(w)m(e)h(kno)m(w)g(that)g(the)g (eigenfunction)f(do)s(es)g(not)h(c)m(hange)h(sign)e(in)f(a)50 1676 y(suitable)j(neigh)m(b)s(orho)s(o)s(d)e(of)k(the)f(minim)m(um,)f (one)i(cannot)g(exclude)e(that)i(some)g(minima)d(are)i(close)h(to)50 1822 y(suc)m(h)30 b(zeros.)41 b(T)-8 b(o)31 b(deal)f(with)f(these)i (cases)g(creates)h(a)f(n)m(um)m(b)s(er)e(of)h(complications.)50 2040 y Ft(Regularit)m(y)38 b(prop)s(erties)h(of)f(p)s(ositiv)m(e)h (harmonic)f(functions.)49 b Fu(W)-8 b(e)35 b(\014rst)d(state)j(a)e (simple)f(appli-)50 2185 y(cation)d(of)g(the)g(Harnac)m(k-)h(and)f (H\177)-45 b(older)28 b(inequalities)e(\(see)k([GT])f(,)g(Corollaries)e (9.25)k(and)d(9.24\))j(that)e(w)m(e)50 2331 y(ha)m(v)m(e)i(stated)h(as) e(Lemmata)i(4.1)f(and)f(4.2)h(in)e([BEGK3].)50 2549 y Fh(Lemma)40 b(3.1:)54 b Fo(Assume)34 b(that)h Fr(x)f Fo(is)g(a)h(lo)-5 b(c)g(al)36 b(minimum)e(of)g Fr(F)13 b Fo(.)46 b(L)-5 b(et)35 b Fr(\036)f Fo(b)-5 b(e)34 b(a)g(p)-5 b(ositive)35 b(str)-5 b(ong)35 b(solution)50 2695 y(of)29 b Fu(\()p Fr(L)250 2709 y Fp(\017)298 2695 y Fs(\000)13 b Fr(\025)p Fu(\))p Fr(\036)26 b Fu(=)f(0)p Fo(,)30 b Fs(j)p Fr(\025)p Fs(j)c(\024)e Fu(1)p Fo(,)31 b(on)e(a)h(b)-5 b(al)5 b(l)30 b Fr(B)1512 2716 y Fv(4)1548 2669 y Fj(p)p 1608 2669 31 4 v 47 x Fp(\017)1643 2695 y Fu(\()p Fr(x)p Fu(\))p Fo(.)41 b(Then)29 b(ther)-5 b(e)31 b(exists)e(a)h(c)-5 b(onstant)31 b Fr(C)h(<)25 b Fs(1)k Fo(and)h Fr(\013)c(>)e Fu(0)p Fo(,)50 2840 y(b)-5 b(oth)34 b(indep)-5 b(endent)34 b(of)f Fr(\017)f Fo(such)h(that)1176 3053 y Fu(osc)1297 3069 y Fp(y)r Fj(2)p Fp(B)1440 3078 y Fk(\017)1473 3069 y Fv(\()p Fp(x)p Fv(\))1576 3053 y Fr(\036)p Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)e Fr(C)7 b(\017)1978 3015 y Fp(\013=)p Fv(2)2179 3053 y Fu(min)2118 3116 y Fp(y)r Fj(2)p Fp(B)2261 3125 y Fk(\017)2294 3116 y Fv(\()p Fp(x)p Fv(\))2407 3053 y Fr(\036)p Fu(\()p Fr(x)p Fu(\))940 b(\(3)p Fr(:)p Fu(3\))50 3484 y Fe(Pro)s(of:)50 b Fu(W)-8 b(e)32 b(can)g(use)f(Lemmata)h(4.1)h(and)e(4.2)h(stated)g(in)f([BEGK3]) h(with)e(\003)d(=)g Fr(\025)g Fu(=)g Fr(\017)p Fu(,)k Fr(\015)i Fu(=)26 b(1,)33 b Fr(c)27 b Fu(=)g Fr(\025)p Fu(,)50 3629 y(and)1070 3774 y Fr(\027)k Fu(=)25 b Fr(\017)1279 3737 y Fj(\000)p Fv(2)1501 3774 y Fu(sup)1392 3855 y Fp(y)r Fj(2)p Fp(B)1535 3872 y Fl(4)1566 3838 y Fg(p)p 1616 3838 28 3 v 34 x Fk(\017)1648 3855 y Fv(\()p Fp(x)p Fv(\))1761 3774 y Fs(kr)p Fr(F)13 b Fu(\()p Fr(y)s Fu(\))p Fs(k)2116 3737 y Fv(2)2116 3797 y Fj(1)2220 3774 y Fs(\024)25 b Fr(const:\017)2592 3737 y Fj(\000)p Fv(1)3523 3774 y Fu(\(3)p Fr(:)p Fu(4\))50 4017 y(Then,)k(with)h Fr(R)25 b Fu(=)g(2)755 3952 y Fs(p)p 832 3952 37 4 v 832 4017 a Fr(\017)p Fu(,)30 b(w)m(e)h(obtain)f(\014rst)f(from)h(Lemma)h(4.2)g (that)1342 4230 y(sup)1234 4310 y Fp(y)r Fj(2)p Fp(B)1377 4327 y Fl(2)1408 4293 y Fg(p)p 1458 4293 28 3 v 34 x Fk(\017)1490 4310 y Fv(\()p Fp(x)p Fv(\))1603 4230 y Fr(\036)p Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fr(C)144 b Fu(inf)1984 4293 y Fp(y)r Fj(2)p Fp(B)2127 4309 y Fl(2)2158 4276 y Fg(p)p 2208 4276 V 33 x Fk(\017)2240 4293 y Fv(\()p Fp(x)p Fv(\))2353 4230 y Fr(\036)p Fu(\()p Fr(y)s Fu(\))998 b(\(3)p Fr(:)p Fu(5\))50 4485 y(and)30 b(then)g(from)g(Lemma)g(4.1)i (that)820 4698 y(osc)942 4714 y Fp(y)r Fj(2)p Fp(B)1085 4723 y Fk(\017)1117 4714 y Fv(\()p Fp(x)p Fv(\))1220 4698 y Fr(\036)p Fu(\()p Fr(x)p Fu(\))26 b Fs(\024)f Fr(C)7 b(\017)1627 4660 y Fp(\013=)p Fv(2)1875 4698 y Fu(sup)1767 4778 y Fp(y)r Fj(2)p Fp(B)1910 4795 y Fl(2)1941 4761 y Fg(p)p 1990 4761 V 1990 4795 a Fk(\017)2023 4778 y Fv(\()p Fp(x)p Fv(\))2136 4698 y Fr(\036)p Fu(\()p Fr(y)s Fu(\))2323 4597 y Fi(\020)2378 4698 y Fu(1)21 b(+)2535 4628 y Fs(p)p 2611 4628 37 4 v 70 x Fr(\017)2647 4643 y Fp(d)p Fv(+1)2782 4698 y Fs(j)p Fr(\025)p Fs(j)2885 4597 y Fi(\021)3523 4698 y Fu(\(3)p Fr(:)p Fu(6\))50 4953 y(This)28 b(implies)g(the)j(lemma)e(if)h Fr(\025)g Fu(is)g(not)g(to)s(o)h(large.)p Fs(})50 5171 y Ft(Principal)52 b(eigen)m(v)-6 b(alues)51 b(revisited.)82 b Fu(W)-8 b(e)45 b(will)d(no)m(w)i(impro)m(v)m(e)g(on)g(the)g(estimates)h(on)f (principal)50 5316 y(eigenfunctions)653 5292 y(\026)650 5316 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))36 b(obtained)g(in)f(Section)h(2) h(b)m(y)f(sho)m(wing)f(that)i(in)e(the)h(case)i(when)d Fr(D)k Fu(con)m(tains)d(a)50 5462 y(lo)s(cal)29 b(minim)m(um)f(of)j Fr(F)13 b Fu(,)30 b(these)h(estimates)g(are)g(essen)m(tially)e(exact.)p eop %%Page: 13 13 13 12 bop 3633 295 a Fm(13)50 513 y Fh(Prop)s(osition)38 b(3.2:)k Fo(Assume)33 b(that)i Fr(D)g Fo(c)-5 b(ontains)35 b Fr(l)28 b Fs(\025)e Fu(1)34 b Fo(lo)-5 b(c)g(al)35 b(minima)f(of)f(the)h(function)f Fr(F)46 b Fo(and)34 b(that)50 658 y(ther)-5 b(e)33 b(is)g(a)g(single)f(minimum)i Fr(x)25 b Fs(2)g Fr(D)35 b Fo(that)f(r)-5 b(e)g(alizes)866 870 y Fr(F)13 b Fu(\()p Fr(z)1018 833 y Fj(\003)1061 870 y Fu(\()p Fr(x;)i(D)1266 833 y Fp(c)1304 870 y Fu(\)\))20 b Fs(\000)g Fr(F)13 b Fu(\()p Fr(x)p Fu(\))26 b(=)1873 813 y Fp(l)1800 870 y Fu(max)1826 928 y Fp(i)p Fv(=1)1969 870 y Fu([)p Fr(F)13 b Fu(\()p Fr(z)2146 833 y Fj(\003)2189 870 y Fu(\()p Fr(x)2276 884 y Fp(i)2307 870 y Fr(;)i(D)2425 833 y Fp(c)2462 870 y Fu(\)\))21 b Fs(\000)f Fr(F)13 b Fu(\()p Fr(x)2802 884 y Fp(i)2833 870 y Fu(\)])630 b(\(3)p Fr(:)p Fu(7\))50 1082 y Fo(We)37 b(write)h Fr(B)g Fs(\021)33 b Fr(B)736 1096 y Fp(\017)771 1082 y Fu(\()p Fr(x)p Fu(\))p Fo(.)57 b(Then)37 b(ther)-5 b(e)38 b(exists)g Fr(\013)c(>)g Fu(0)p Fr(;)15 b(C)41 b(<)33 b Fs(1)p Fr(;)15 b(\016)38 b(>)c Fu(0)p Fo(,)k(indep)-5 b(endent)39 b(of)f Fr(\017)p Fo(,)g(such)f(that)50 1228 y(princip)-5 b(al)34 b(eigenvalue)870 1204 y Fu(\026)866 1228 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))f Fo(of)g(the)g(Dirichlet)g(pr)-5 b(oblem)34 b(on)g Fr(D)h Fo(satis\014es)259 1394 y Fu(cap)411 1408 y Fp(B)473 1394 y Fu(\()p Fr(D)586 1361 y Fp(c)623 1394 y Fu(\))p 259 1435 400 4 v 277 1518 a Fs(k)p Fr(h)374 1532 y Fp(B)s(;D)515 1514 y Fk(c)554 1518 y Fs(k)599 1487 y Fv(2)599 1543 y(2)670 1456 y Fu(\(1)21 b Fs(\000)f Fr(C)7 b(\017)971 1418 y Fp(\013=)p Fv(2)1095 1456 y Fu(\)\(1)22 b Fs(\000)d Fr(e)1364 1418 y Fj(\000)p Fp(\016)r(=\017)1529 1456 y Fu(\))26 b Fs(\024)1689 1432 y Fu(\026)1686 1456 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))f Fs(\024)2019 1394 y Fu(cap)2171 1408 y Fp(B)2233 1394 y Fu(\()p Fr(D)2346 1361 y Fp(c)2384 1394 y Fu(\))p 2019 1435 V 2037 1518 a Fs(k)p Fr(h)2134 1532 y Fp(B)s(;D)2275 1514 y Fk(c)2315 1518 y Fs(k)2360 1487 y Fv(2)2360 1543 y(2)2430 1456 y Fu(\(1)c(+)f Fr(C)7 b(\017)2731 1418 y Fp(\013=)p Fv(2)2855 1456 y Fu(\)\(1)22 b(+)e Fr(e)3125 1418 y Fj(\000)p Fp(\016)r(=\017) 3289 1456 y Fu(\))199 b(\(3)p Fr(:)p Fu(8\))50 1701 y Fo(wher)-5 b(e)29 b(her)-5 b(e)28 b(and)h(henc)-5 b(eforth)29 b Fs(k)10 b(\001)g(k)1235 1715 y Fv(2)1304 1701 y Fo(denotes)29 b(the)f Fr(L)1835 1668 y Fv(2)1904 1701 y Fo(norm)h(with)f(r)-5 b(esp)g(e)g(ct)30 b(to)e(the)g(me)-5 b(asur)g(e)29 b Fr(e)3265 1668 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)3547 1701 y Fr(dy)s Fo(.)50 1919 y Fe(Pro)s(of:)47 b Fu(Set)31 b Fr(D)614 1886 y Fv(0)680 1919 y Fu(=)25 b Fr(D)s Fs(n)p Fr(B)5 b Fu(.)40 b(Then)29 b(w)m(e)i(kno)m(w)g(b)m(y)f (Lemma)g(2.5)i(that)f(there)f(exists)g Fr(\016)f(>)c Fu(0)31 b(suc)m(h)f(that)1214 2107 y(\026)1210 2131 y Fr(\025)p Fu(\()p Fr(D)1376 2093 y Fv(0)1417 2131 y Fu(\))c Fs(\025)f Fr(e)1616 2093 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(z)1816 2066 y Fg(\003)1853 2093 y Fv(\()p Fp(x;D)2005 2066 y Fk(c)2040 2093 y Fv(\)\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)p Fv(\)])p Fp(=\017)2400 2131 y Fr(e)2442 2093 y Fp(\016)r(=\017)3523 2131 y Fu(\(3)p Fr(:)p Fu(9\))50 2343 y(while)23 b(w)m(e)j(kno)m(w)g(that)840 2319 y(\026)836 2343 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))g Fr(<)1162 2319 y Fu(\026)1159 2343 y Fr(\025)p Fu(\()p Fr(D)1325 2310 y Fv(0)1366 2343 y Fu(\))f(\(and)h(exp)s(ect)1920 2319 y(\026)1917 2343 y Fr(\025)f Fs(\030)g Fr(e)2133 2310 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(z)2333 2283 y Fg(\003)2370 2310 y Fv(\()p Fp(x;D)2522 2283 y Fk(c)2557 2310 y Fv(\)\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)p Fv(\)])p Fp(=e)2921 2343 y Fu(,)27 b(i.e.)39 b(m)m(uc)m(h)26 b(smaller\).)50 2489 y(By)33 b(the)g(general)g(philosoph)m(y)e(outlined)g(ab)s(o)m(v)m (e,)k(w)m(e)e(kno)m(w)g(that)h(the)f(principal)c(eigenfunction)j(can)h (b)s(e)50 2634 y(represen)m(ted)d(as)h(the)f(solution)f(of)i(the)f (Diric)m(hlet)g(problem)e(\(b)s(oth)i(inside)e Fr(B)35 b Fu(and)30 b(outside)g Fr(B)5 b Fu(\))1215 2814 y(\()p Fr(L)1312 2828 y Fp(\017)1368 2814 y Fs(\000)19 b Fr(\025)p Fu(\))p Fr(f)1601 2777 y Fp(\025)1649 2814 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(0)p Fr(;)106 b(y)28 b Fs(2)d Fr(D)s Fs(n)p Fr(@)5 b(B)1547 2987 y(f)1602 2949 y Fp(\025)1649 2987 y Fu(\()p Fr(y)s Fu(\))26 b(=)f Fr(\036)1943 3001 y Fp(D)2009 2987 y Fu(\()p Fr(y)s Fu(\))p Fr(;)107 b(y)28 b Fs(2)c Fr(@)5 b(B)1547 3160 y(f)1602 3122 y Fp(\025)1649 3160 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(0)p Fr(;)106 b(y)28 b Fs(2)d Fr(D)2302 3122 y Fp(c)3477 2985 y Fu(\(3)p Fr(:)p Fu(10\))50 3330 y(where)32 b(the)g(b)s(oundary)f(conditions)g Fr(\036)1376 3344 y Fp(D)1474 3330 y Fu(are)i(giv)m(en)f(b)m(y)h(the)f (actual)h(principal)c(eigenfunction.)46 b(W)-8 b(e)33 b(will)50 3476 y(assume)e(that)47 b(dist)o(\()p Fr(x;)15 b(D)930 3443 y Fp(c)967 3476 y Fu(\))27 b Fs(\025)f Fr(\016)31 b(>)26 b Fu(0,)32 b(indep)s(enden)m(t)d(of)i Fr(\017)p Fu(.)43 b(Then)30 b Fr(B)2428 3497 y Fv(4)2464 3451 y Fj(p)p 2524 3451 31 4 v 46 x Fp(\017)2559 3476 y Fu(\()p Fr(x)p Fu(\))d Fs(\032)f Fr(D)s Fu(,)31 b(and)g(since)f Fr(\036)3394 3490 y Fp(D)3491 3476 y Fu(is)h(the)50 3621 y(principal)g(eigenfunction,)i(it)h(ma)m(y)h(b)s(e)e(c)m(hosen)i(p)s (ositiv)m(e)e(on)h Fr(D)s Fu(.)51 b(Therefore)34 b(Lemma)g(3.1)h (applies)d(and)50 3767 y(sho)m(ws)e(that)1025 3912 y(inf)985 3972 y Fp(y)r Fj(2)p Fp(@)t(B)1190 3912 y Fr(\036)1244 3926 y Fp(D)1310 3912 y Fu(\()p Fr(y)s Fu(\))25 b Fs(\021)g Fr(c)h Fs(\024)51 b Fu(sup)1710 3989 y Fp(y)r Fj(2)p Fp(@)t(B)1914 3912 y Fr(\036)1968 3926 y Fp(D)2034 3912 y Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fu(\(1)c(+)f Fr(C)7 b(\017)2575 3875 y Fp(\013=)p Fv(2)2699 3912 y Fu(\))p Fr(c)704 b Fu(\(3)p Fr(:)p Fu(11\))50 4142 y(W)-8 b(e)28 b(will)23 b(normalize)j(the)h(eigenfunction)e(s.t.)40 b Fr(c)25 b Fu(=)g(1,)j(Th)m(us)e(w)m(e)h(can)g(write)e Fr(f)2710 4109 y Fp(\025)2757 4142 y Fu(\()p Fr(x)p Fu(\))h(=)f Fr(h)3053 4109 y Fp(\025)3053 4167 y(B)s(;D)3194 4149 y Fk(c)3233 4142 y Fu(\()p Fr(x)p Fu(\))13 b(+)g Fr( )3514 4109 y Fp(\025)3562 4142 y Fu(\()p Fr(x)p Fu(\),)50 4287 y(where)30 b Fr(h)365 4254 y Fp(\025)365 4312 y(B)s(;D)506 4294 y Fk(c)570 4287 y Fs(\021)25 b Fr(h)718 4254 y Fp(\025)796 4287 y Fu(is)30 b(the)g Fr(\025)p Fu(-equilibrium)c(p)s(oten)m(tial)k (\(see)h([BEGK3],)h(Chapter)e(2\))h(that)g(solv)m(es)1251 4482 y(\()p Fr(L)1348 4496 y Fp(\017)1404 4482 y Fs(\000)19 b Fr(\025)p Fu(\))p Fr(h)1634 4445 y Fp(\025)1683 4482 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(0)p Fr(;)106 b(y)28 b Fs(2)d Fr(D)s Fs(n)p Fr(@)5 b(B)1583 4655 y(h)1635 4617 y Fp(\025)1683 4655 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(1)p Fr(;)106 b(y)28 b Fs(2)d Fr(@)5 b(B)1583 4828 y(h)1635 4790 y Fp(\025)1683 4828 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(0)p Fr(;)106 b(y)28 b Fs(2)d Fr(D)2336 4790 y Fp(c)3477 4653 y Fu(\(3)p Fr(:)p Fu(12\))50 5012 y(while)j Fr( )349 4979 y Fp(\025)428 5012 y Fu(solv)m(es)1133 5121 y(\()p Fr(L)1230 5135 y Fp(\017)1285 5121 y Fs(\000)20 b Fr(\025)p Fu(\))p Fr( )1526 5083 y Fp(\025)1575 5121 y Fu(\()p Fr(y)s Fu(\))25 b(=)g(0)p Fr(;)107 b(y)28 b Fs(2)d Fr(D)s Fs(n)p Fr(@)5 b(B)1464 5294 y( )1526 5256 y Fp(\025)1575 5294 y Fu(\()p Fr(y)s Fu(\))25 b(=)g Fr(\036)1868 5308 y Fp(D)1934 5294 y Fu(\()p Fr(y)s Fu(\))c Fs(\000)f Fu(1)p Fr(;)107 b(y)28 b Fs(2)d Fr(@)5 b(B)1464 5466 y( )1526 5429 y Fp(\025)1575 5466 y Fu(\()p Fr(y)s Fu(\))25 b(=)g(0)p Fr(;)107 b(y)28 b Fs(2)d Fr(D)2228 5429 y Fp(c)3477 5291 y Fu(\(3)p Fr(:)p Fu(13\))p eop %%Page: 14 14 14 13 bop 50 295 a Fm(14)159 513 y Fu(W)-8 b(e)34 b(w)m(an)m(t)g(that)f (\()p Fr(L)839 527 y Fp(\017)896 513 y Fs(\000)21 b Fr(\025)p Fu(\))p Fr(f)1131 480 y Fp(\025)1179 513 y Fu(\()p Fr(x)p Fu(\))30 b(=)f(0)k(on)f(all)g(of)h Fr(D)s Fu(.)48 b(Here)33 b(w)m(e)g(ha)m(v)m(e)h(to)g(in)m(terpret)e(\()p Fr(L)3178 527 y Fp(\017)3235 513 y Fs(\000)21 b Fr(\025)p Fu(\))p Fr(f)3470 480 y Fp(\025)3550 513 y Fu(as)33 b(a)50 658 y(surface)d(measure)g(on)h Fr(@)5 b(B)g Fu(.)40 b(I.e.,)31 b(if)f Fr(g)k Fu(is)29 b(a)i(smo)s(oth)f(test)h(function)e(that)i(v)-5 b(anishes)29 b(on)i Fr(D)3149 625 y Fp(c)3186 658 y Fu(,)171 786 y Fi(Z)222 992 y Fp(D)303 910 y Fr(dy)s(e)440 872 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)721 910 y Fr(g)s Fu(\()p Fr(y)s Fu(\)\()p Fr(L)982 924 y Fp(\017)1039 910 y Fs(\000)19 b Fr(\025)p Fu(\))p Fr(f)1272 872 y Fp(\025)1320 910 y Fu(\()p Fr(y)s Fu(\))26 b Fs(\021)1560 786 y Fi(Z)1610 992 y Fp(D)1691 910 y Fr(dy)s(e)1828 872 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2109 910 y Fr(f)2164 872 y Fp(\025)2211 910 y Fu(\()p Fr(y)s Fu(\)\()p Fr(L)2426 924 y Fp(\017)2482 910 y Fs(\000)20 b Fr(\025)p Fu(\))p Fr(f)2716 872 y Fp(\025)2763 910 y Fu(\()p Fr(y)s Fu(\))p Fr(g)s Fu(\()p Fr(y)s Fu(\))423 1153 y(=)519 1029 y Fi(Z)570 1235 y Fp(D)r Fj(n)p Fp(B)746 1153 y Fr(dy)s(e)883 1115 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1164 1153 y Fr(f)1219 1115 y Fp(\025)1266 1153 y Fu(\()p Fr(y)s Fu(\)\()p Fr(L)1481 1167 y Fp(\017)1537 1153 y Fs(\000)20 b Fr(\025)p Fu(\))p Fr(g)s Fu(\()p Fr(y)s Fu(\))i(+)1993 1029 y Fi(Z)2043 1235 y Fv(in)n(t)14 b Fp(B)2221 1153 y Fr(dy)s(e)2358 1115 y Fj(\000)p Fp(F)c Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2639 1153 y Fr(f)2694 1115 y Fp(\025)2741 1153 y Fu(\()p Fr(y)s Fu(\)\()p Fr(L)2956 1167 y Fp(\017)3012 1153 y Fs(\000)19 b Fr(\025)p Fu(\))p Fr(g)s Fu(\()p Fr(y)s Fu(\))423 1411 y(=)25 b Fr(\017)571 1288 y Fi(Z)622 1494 y Fp(@)t(B)742 1411 y Fr(e)784 1374 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1081 1338 y Fi(\000)1122 1411 y Fr(g)s Fu(\()p Fr(y)s Fu(\))p Fr(@)1334 1428 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1481 1411 y Fr(f)1536 1374 y Fp(\025)1583 1411 y Fu(\()p Fr(y)s Fu(\))21 b Fs(\000)f Fr(f)1868 1374 y Fp(\025)1915 1411 y Fu(\()p Fr(y)s Fu(\))p Fr(@)2081 1428 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2227 1411 y Fr(g)s Fu(\()p Fr(y)s Fu(\))2391 1338 y Fi(\001)2449 1411 y Fr(d\033)2548 1425 y Fp(B)2611 1411 y Fu(\()p Fr(y)s Fu(\))418 1654 y(+)g Fr(\017)561 1531 y Fi(Z)612 1737 y Fp(@)t(B)732 1654 y Fr(e)774 1617 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1070 1581 y Fi(\000)1112 1654 y Fr(g)s Fu(\()p Fr(y)s Fu(\))p Fr(@)1324 1671 y Fj(\000)p Fp(n)p Fv(\()p Fp(y)r Fv(\))1528 1654 y Fr(f)1583 1617 y Fp(\025)1630 1654 y Fu(\()p Fr(y)s Fu(\))20 b Fs(\000)g Fr(f)1914 1617 y Fp(\025)1961 1654 y Fu(\()p Fr(y)s Fu(\))p Fr(@)2127 1671 y Fj(\000)p Fp(n)p Fv(\()p Fp(y)r Fv(\))2330 1654 y Fr(g)s Fu(\()p Fr(y)s Fu(\))2494 1581 y Fi(\001)2552 1654 y Fr(d\033)2651 1668 y Fp(B)2714 1654 y Fu(\()p Fr(y)s Fu(\))423 1897 y(=)25 b Fr(\017)571 1774 y Fi(Z)622 1980 y Fp(@)t(B)742 1897 y Fr(e)784 1860 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1081 1824 y Fi(\000)1122 1897 y Fr(g)s Fu(\()p Fr(y)s Fu(\))p Fr(@)1334 1914 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1481 1897 y Fr(f)1536 1860 y Fp(\025)1583 1897 y Fu(\()p Fr(y)s Fu(\))21 b(+)f Fr(g)s Fu(\()p Fr(y)s Fu(\))p Fr(@)2025 1914 y Fj(\000)p Fp(n)p Fv(\()p Fp(y)r Fv(\))2228 1897 y Fr(f)2283 1860 y Fp(\025)2330 1897 y Fu(\()p Fr(y)s Fu(\))2448 1824 y Fi(\001)2506 1897 y Fr(d\033)2605 1911 y Fp(B)2668 1897 y Fu(\()p Fr(y)s Fu(\))3477 1406 y(\(3)p Fr(:)p Fu(14\))50 2158 y(where)32 b Fr(d\033)414 2172 y Fp(B)477 2158 y Fu(\()p Fr(y)s Fu(\))g(denotes)h(the)g(Euclidean)d (surface)i(measure)h(on)f Fr(@)5 b(B)g Fu(,)33 b(and)f Fr(@)2744 2175 y Fj(\006)p Fp(n)p Fv(\()p Fp(y)r Fv(\))2978 2158 y Fu(denote)h(the)g(normal)50 2304 y(deriv)-5 b(ativ)m(e)30 b(at)h Fr(y)d Fs(2)d Fr(@)5 b(B)35 b Fu(from)30 b(the)g(exterior)h(and) f(in)m(terior)f(of)h Fr(B)5 b Fu(,)30 b(resp)s(ectiv)m(ely)-8 b(.)41 b(Th)m(us)29 b(w)m(e)i(can)g(iden)m(tify)364 2550 y Fr(dy)s(e)501 2513 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)782 2550 y Fu(\()p Fr(L)879 2564 y Fp(\017)935 2550 y Fs(\000)20 b Fr(\025)p Fu(\))p Fr(f)1169 2513 y Fp(\025)1216 2550 y Fu(\()p Fr(y)s Fu(\))26 b(=)f Fr(\017e)1535 2513 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1816 2550 y Fu(\()p Fr(@)1899 2567 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2045 2550 y Fr(f)2100 2513 y Fp(\025)2147 2550 y Fu(\()p Fr(y)s Fu(\))20 b(+)g Fr(@)2424 2567 y Fj(\000)p Fp(n)p Fv(\()p Fp(y)r Fv(\))2627 2550 y Fr(f)2682 2513 y Fp(\025)2729 2550 y Fu(\()p Fr(y)s Fu(\)\))p Fr(d\033)2981 2564 y Fp(B)3045 2550 y Fu(\()p Fr(y)s Fu(\))314 b(\(3)p Fr(:)p Fu(15\))50 2797 y(T)-8 b(o)30 b(get)h(con)m(trol)f(on)768 2773 y(\026)764 2797 y Fr(\025)p Fu(,)g(w)m(e)g(can)g(ask)g(at)g(least) g(that)g(the)g(total)h(mass)e(of)h(this)e(measure)i(on)f Fr(@)5 b(B)35 b Fu(v)-5 b(anishes,)50 2942 y(i.e.)40 b(that)876 3088 y(0)25 b(=)1042 2964 y Fi(Z)1093 3170 y Fp(@)t(B)1213 3088 y Fr(e)1255 3050 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1536 3088 y Fu(\()p Fr(@)1619 3104 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1766 3088 y Fr(f)1821 3050 y Fp(\025)1867 3088 y Fu(\()p Fr(y)s Fu(\))21 b(+)f Fr(@)2145 3104 y Fj(\000)p Fp(n)p Fv(\()p Fp(y)r Fv(\))2348 3088 y Fr(f)2403 3050 y Fp(\025)2450 3088 y Fu(\()p Fr(y)s Fu(\)\))p Fr(d\033)2702 3102 y Fp(B)2765 3088 y Fu(\()p Fr(y)s Fu(\))594 b(\(3)p Fr(:)p Fu(16\))50 3308 y(T)-8 b(o)38 b(ev)-5 b(aluate)38 b(this)e(expression)g (it)h(will)d(b)s(e)j(con)m(v)m(enien)m(t)h(to)g(observ)m(e)g(that)g(on) f Fr(@)5 b(B)g Fu(,)39 b Fr(h)3075 3322 y Fp(B)s(;D)3216 3303 y Fk(c)3255 3308 y Fu(\()p Fr(y)s Fu(\))f(=)e(1)i(for)50 3453 y Fr(y)52 b Fs(2)d Fr(@)5 b(B)49 b Fu(\(where)c Fr(h)793 3467 y Fp(B)s(;D)934 3449 y Fk(c)1022 3453 y Fs(\021)k Fr(h)1194 3420 y Fp(\025)p Fv(=0)1194 3478 y Fp(B)s(;D)1335 3460 y Fk(c)1419 3453 y Fu(is)44 b(the)h(Newtonian)f (p)s(oten)m(tial)h(\(see)g([BEGK3],)50 b(Chapter)44 b(2\)\).)50 3599 y(Moreo)m(v)m(er,)d(on)c Fr(B)5 b Fu(,)38 b Fr(h)805 3613 y Fp(B)s(;D)946 3594 y Fk(c)985 3599 y Fu(\()p Fr(y)s Fu(\))e Fs(\021)g Fu(1,)j(so)f(that)f Fr(@)1725 3615 y Fj(\000)p Fp(n)p Fv(\()p Fp(y)r Fv(\))1928 3599 y Fr(h)1980 3613 y Fp(B)s(;D)2121 3594 y Fk(c)2160 3599 y Fu(\()p Fr(y)s Fu(\))g(v)-5 b(anishes)36 b(on)g Fr(@)5 b(B)g Fu(.)61 b(Using)36 b(these)h(facts)50 3744 y(together)h(with)e(Green's) i(second)f(iden)m(tit)m(y)f(\(see)j(Eq.)60 b(\(2.8\))39 b(in)d([BEGK3]\),)41 b(w)m(e)c(get)i(from)d(\(3.15\))k(the)50 3890 y(condition)339 4124 y(0)25 b(=)495 4000 y Fi(Z)546 4207 y Fp(@)t(B)666 4124 y Fr(e)708 4087 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)989 4124 y Fr(@)1037 4141 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1183 4124 y Fr(h)1235 4138 y Fp(B)s(;D)1376 4120 y Fk(c)1415 4124 y Fu(\()p Fr(y)s Fu(\))p Fr(f)1588 4087 y Fp(\025)1636 4124 y Fu(\()p Fr(y)s Fu(\))20 b Fs(\000)1876 4063 y Fr(\025)p 1876 4103 54 4 v 1884 4187 a(\017)1955 4000 y Fi(Z)2006 4207 y Fp(D)2087 4124 y Fr(dy)s(e)2224 4087 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2505 4124 y Fr(h)2557 4138 y Fp(B)s(;D)2698 4120 y Fk(c)2737 4124 y Fu(\()p Fr(y)s Fu(\))p Fr(f)2910 4087 y Fp(\025)2958 4124 y Fu(\()p Fr(y)s Fu(\))505 4368 y(=)601 4244 y Fi(Z)652 4451 y Fp(@)t(B)772 4368 y Fr(e)814 4331 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1095 4368 y Fr(@)1143 4384 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1289 4368 y Fr(h)1341 4382 y Fp(B)s(;D)1482 4364 y Fk(c)1521 4368 y Fu(\()p Fr(y)s Fu(\))21 b Fs(\000)1762 4307 y Fr(\025)p 1762 4347 V 1770 4430 a(\017)1841 4244 y Fi(Z)1891 4451 y Fp(D)1972 4368 y Fr(dy)s(e)2109 4331 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2391 4368 y Fr(h)2443 4382 y Fp(B)s(;D)2584 4364 y Fk(c)2623 4368 y Fu(\()p Fr(y)s Fu(\))p Fr(h)2793 4331 y Fp(\025)2793 4391 y(B)s(;D)2934 4372 y Fk(c)2973 4368 y Fu(\()p Fr(y)s Fu(\))500 4612 y(+)591 4488 y Fi(Z)642 4695 y Fp(@)t(B)762 4612 y Fr(e)804 4574 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1085 4612 y Fr(@)1133 4628 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1279 4612 y Fr(h)1331 4626 y Fp(B)s(;D)1472 4607 y Fk(c)1511 4612 y Fu(\()p Fr(y)s Fu(\))p Fr( )1691 4574 y Fp(\025)1740 4612 y Fu(\()p Fr(y)s Fu(\))21 b Fs(\000)1980 4550 y Fr(\025)p 1980 4591 V 1988 4674 a(\017)2059 4488 y Fi(Z)2110 4695 y Fp(D)2191 4612 y Fr(dy)s(e)2328 4574 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2609 4612 y Fr(h)2661 4626 y Fp(B)s(;D)2802 4607 y Fk(c)2841 4612 y Fu(\()p Fr(y)s Fu(\))p Fr( )3021 4574 y Fp(\025)3070 4612 y Fu(\()p Fr(y)s Fu(\))3477 4370 y(\(3)p Fr(:)p Fu(17\))50 4873 y(\(Note)45 b(that)f(the)g(deriv)-5 b(ativ)m(e)43 b Fr(@)1176 4889 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1365 4873 y Fu(is)g(in)f(the)h(direction)g(of)g(the)h(in)m(terior)e(of)i Fr(B)5 b Fu(\).)80 b(The)43 b(t)m(w)m(o)i(terms)50 5018 y(in)m(v)m(olving)28 b Fr( )500 4985 y Fp(\025)578 5018 y Fu(will)f(b)s(e)i(naturally)f(treated)j(as)f(error)f(terms.)40 b(In)29 b(fact,)i(since)e Fr(@)2784 5035 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2930 5018 y Fr(h)2982 5032 y Fp(B)s(;D)3123 5014 y Fk(c)3162 5018 y Fu(\()p Fr(y)s Fu(\))d Fr(>)f Fu(0,)30 b(using)50 5164 y(Lemma)g(3.1,)i(w)m(e)f(get)g(that)343 5415 y(0)25 b Fs(\024)509 5292 y Fi(Z)560 5498 y Fp(@)t(B)680 5415 y Fr(e)722 5378 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1003 5415 y Fr(@)1051 5432 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1197 5415 y Fr(h)1249 5429 y Fp(B)s(;D)1390 5411 y Fk(c)1429 5415 y Fu(\()p Fr(y)s Fu(\))p Fr( )1609 5378 y Fp(\025)1658 5415 y Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fr(C)7 b(\017)2007 5378 y Fp(\013=)p Fv(2)2146 5292 y Fi(Z)2197 5498 y Fp(@)t(B)2317 5415 y Fr(e)2359 5378 y Fj(\000)p Fp(F)j Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2640 5415 y Fr(@)2688 5432 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2834 5415 y Fr(h)2886 5429 y Fp(B)s(;D)3027 5411 y Fk(c)3066 5415 y Fu(\()p Fr(y)s Fu(\))293 b(\(3)p Fr(:)p Fu(18\))p eop %%Page: 15 15 15 14 bop 3633 295 a Fm(15)50 513 y Fu(If)30 b(w)m(e)h(de\014ne)e Fr(\016)s( )642 480 y Fp(\025)716 513 y Fu(=)c Fr( )874 480 y Fp(\025)943 513 y Fs(\000)20 b Fr( )1096 480 y Fv(0)1137 513 y Fu(,)30 b(w)m(e)h(see)g(that)g Fr(\016)s( )1775 480 y Fp(\025)1855 513 y Fu(solv)m(es)f(the)h(Diric)m(hlet)e(problem) 1110 724 y(\()p Fr(L)1207 738 y Fp(\017)1262 724 y Fs(\000)20 b Fr(\025)p Fu(\))p Fr(\016)s( )1546 687 y Fp(\025)1595 724 y Fu(\()p Fr(y)s Fu(\))26 b(=)f Fr(\025 )1950 687 y Fv(0)1991 724 y Fu(\()p Fr(y)s Fu(\))p Fr(;)107 b(y)28 b Fs(2)d Fr(D)s Fs(n)p Fr(@)5 b(B)1441 897 y(\016)s( )1546 859 y Fp(\025)1595 897 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(0)p Fr(;)107 b(y)28 b Fs(2)d Fr(@)5 b(B)1441 1070 y(\016)s( )1546 1032 y Fp(\025)1595 1070 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(0)p Fr(;)107 b(y)28 b Fs(2)d Fr(D)2249 1032 y Fp(c)3477 895 y Fu(\(3)p Fr(:)p Fu(19\))50 1277 y(and)30 b(th)m(us)1245 1422 y Fr(\016)s( )1350 1385 y Fp(\025)1399 1422 y Fu(\()p Fr(y)s Fu(\))c(=)f Fr(\025)p Fu(\()p Fr(L)1789 1385 y Fp(D)1850 1357 y Fk(c)1884 1385 y Fj([)p Fp(B)1789 1445 y(\017)2016 1422 y Fs(\000)20 b Fr(\025)p Fu(\))2195 1385 y Fj(\000)p Fv(1)2293 1422 y Fr( )2355 1385 y Fv(0)2396 1422 y Fu(\()p Fr(y)s Fu(\))963 b(\(3)p Fr(:)p Fu(20\))50 1628 y(and)30 b(so)1345 1774 y Fs(k)p Fr(\016)s( )1495 1736 y Fp(\025)1544 1774 y Fs(k)1589 1788 y Fv(2)1656 1774 y Fs(\024)1939 1712 y Fr(\025)p 1763 1753 407 4 v 1767 1815 a Fu(\026)1763 1839 y Fr(\025)p Fu(\()p Fr(D)1929 1813 y Fv(0)1970 1839 y Fu(\))20 b Fs(\000)g Fr(\025)2180 1774 y Fs(k)p Fr( )2287 1736 y Fv(0)2329 1774 y Fs(k)2374 1788 y Fv(2)3477 1774 y Fu(\(3)p Fr(:)p Fu(21\))50 1995 y(By)30 b(the)h(same)g(argumen)m(t)g(w)m(e)f(also)h(ha)m(v)m(e)g(that) 1070 2250 y Fs(k)p Fr(h)1167 2212 y Fp(\025)1167 2272 y(B)s(;D)1308 2254 y Fk(c)1367 2250 y Fs(\000)20 b Fr(h)1510 2264 y Fp(B)s(;D)1651 2245 y Fk(c)1690 2250 y Fs(k)1735 2264 y Fv(2)1802 2250 y Fs(\024)2085 2188 y Fr(\025)p 1909 2229 V 1913 2291 a Fu(\026)1909 2315 y Fr(\025)p Fu(\()p Fr(D)2075 2289 y Fv(0)2116 2315 y Fu(\))g Fs(\000)g Fr(\025)2326 2250 y Fs(k)p Fr(h)2423 2264 y Fp(B)s(;D)2564 2245 y Fk(c)2604 2250 y Fs(k)2649 2264 y Fv(2)3477 2250 y Fu(\(3)p Fr(:)p Fu(22\))50 2524 y(On)29 b(the)i(other)g(hand,)e (using)g(the)h(P)m(oisson)h(k)m(ernel)e(represen)m(tation)i(of)f Fr( )2598 2491 y Fv(0)2640 2524 y Fu(,)878 2777 y Fr( )940 2740 y Fv(0)981 2777 y Fu(\()p Fr(z)t Fu(\))c(=)f Fs(\000)p Fr(\017)1342 2654 y Fi(Z)1393 2860 y Fp(@)t(B)1498 2777 y Fu(\()p Fr(\036)1587 2791 y Fp(D)1653 2777 y Fu(\()p Fr(y)s Fu(\))c Fs(\000)f Fu(1\))p Fr(@)2011 2794 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2158 2777 y Fr(G)2230 2794 y Fp(D)r Fj(n)p Fp(B)2390 2777 y Fu(\()p Fr(x;)15 b(y)s Fu(\))p Fr(d\033)2699 2791 y Fp(B)2763 2777 y Fu(\()p Fr(y)s Fu(\))596 b(\(3)p Fr(:)p Fu(23\))50 3040 y(where)32 b Fr(G)387 3057 y Fp(D)r Fj(n)p Fp(B)548 3040 y Fu(\()p Fr(x;)15 b(y)s Fu(\))34 b(denotes)f(the)h(Green's)f(function)f(for)h (the)g(Diric)m(hlet)f(problem)f(with)h(in)g Fr(D)s Fs(n)p Fr(B)38 b Fu(\(see)50 3186 y([BEGK3],)28 b(Chapter)e(2\).)40 b(Since)26 b(the)g(normal)f(deriv)-5 b(ativ)m(e)26 b(of)h(the)f (Green's)h(function)e(is)g(negativ)m(e)j(on)e Fr(@)5 b(B)g Fu(,)50 3331 y(w)m(e)31 b(get)g(that)1334 3477 y(0)26 b Fs(\024)f Fr( )1563 3439 y Fv(0)1604 3477 y Fu(\()p Fr(z)t Fu(\))i Fs(\024)e Fr(C)7 b(\017)1952 3439 y Fp(\013=)p Fv(2)2076 3477 y Fr(h)2128 3491 y Fp(B)s(;D)2269 3472 y Fk(c)2308 3477 y Fu(\()p Fr(z)t Fu(\))1053 b(\(3)p Fr(:)p Fu(24\))50 3683 y(With)1074 3829 y Fr(\017)1126 3705 y Fi(Z)1177 3911 y Fp(@)t(B)1297 3829 y Fr(e)1339 3791 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1620 3829 y Fr(@)1668 3845 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1814 3829 y Fr(h)1866 3843 y Fp(B)s(;D)2007 3824 y Fk(c)2046 3829 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(cap)2437 3843 y Fp(B)2500 3829 y Fu(\()p Fr(D)2613 3791 y Fp(c)2650 3829 y Fu(\))792 b(\(3)p Fr(:)p Fu(25\))50 4050 y(\(3.17\))32 b(implies)c(that)701 4270 y(0)e Fs(\025)e Fu(cap)1019 4284 y Fp(B)1082 4270 y Fu(\()p Fr(D)1195 4232 y Fp(c)1232 4270 y Fu(\))c Fs(\000)g Fr(\025)p Fs(k)p Fr(h)1528 4284 y Fp(B)s(;D)1669 4265 y Fk(c)1709 4270 y Fs(k)1754 4232 y Fv(2)1754 4292 y(2)1795 4270 y Fu(\(1)h Fs(\000)f Fr(C)7 b(\017)2096 4232 y Fp(\013=)p Fv(2)2221 4270 y Fu(\)\(1)21 b Fs(\000)f Fr(\025=)p Fu(\()2585 4246 y(\026)2581 4270 y Fr(\025)q Fu(\()p Fr(D)2748 4232 y Fv(0)2789 4270 y Fu(\))g Fs(\000)g Fr(\025)p Fu(\)\))701 4442 y(0)26 b Fs(\024)e Fu(cap)1019 4456 y Fp(B)1082 4442 y Fu(\()p Fr(D)1195 4405 y Fp(c)1232 4442 y Fu(\)\(1)d(+)f Fr(C)7 b(\017)1568 4405 y Fp(\013=)p Fv(2)1693 4442 y Fu(\))20 b Fs(\000)g Fr(\025)p Fs(k)p Fr(h)1989 4456 y Fp(B)s(;D)2136 4442 y Fs(k)2181 4405 y Fv(2)2181 4465 y(2)3477 4352 y Fu(\(3)p Fr(:)p Fu(26\))50 4662 y(This)34 b(implies)g(the)i(claimed)g (b)s(ound)e(on)1502 4638 y(\026)1498 4662 y Fr(\025)p Fu(\()p Fr(D)s Fu(\).)59 b(Note)38 b(that,)g(while)c(w)m(e)j(ha)m(v)m (e)h(only)d(used)h(a)g(necessary)50 4807 y(condition)e(for)601 4783 y(\026)597 4807 y Fr(\025)p Fu(\()p Fr(D)s Fu(\),)k(the)d(fact)h (that)g(there)f(m)m(ust)g(b)s(e)f(suc)m(h)h(an)g(eigen)m(v)-5 b(alue)35 b(implies)e(that)i(it)g(actually)50 4953 y(lies)29 b(in)g(the)i(b)s(ounds)d(giv)m(en)i(b)m(y)g(\(3.26\).)43 b Fs(})50 5171 y Ft(Remark:)50 b Fu(In)31 b(the)h(case)i(when)d(sev)m (eral)h(of)g(the)h(minima)d(within)f Fr(D)35 b Fu(satisfy)d(\(3.7\))i (\(i.e.)46 b(if)31 b Fr(D)k Fu(con)m(tains)50 5316 y(sev)m(eral)j (minima)d(that)j(are)g(\\equally)e(deep"\),)k(one)e(has)f(to)h(remo)m (v)m(e)h(balls)d Fr(B)2814 5330 y Fp(\017)2849 5316 y Fu(\()p Fr(x)2936 5330 y Fp(i)2966 5316 y Fu(\))i(for)f(eac)m(h)i(of)e (these)50 5462 y(minima.)h(Then)26 b(one)i(ma)m(y)g(pro)s(ceed)g(as)g (b)s(efore.)39 b(The)27 b(only)g(di\013erence)g(is)g(that)h(no)m(w)g (there)g(app)s(ears)e(one)p eop %%Page: 16 16 16 15 bop 50 295 a Fm(16)50 513 y Fu(v)-5 b(alue)34 b Fr(c)328 527 y Fp(i)393 513 y Fu(for)g(eac)m(h)i(of)f(the)f(minima)f (that)i(is)e(y)m(et)j(to)f(b)s(e)f(determined.)52 b(One)34 b(sees)h(that)g(in)f(suc)m(h)g(a)h(case)54 634 y(\026)50 658 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))c(is)e(determined)g(b)m(y)h(a)h(v) -5 b(ariational)29 b(form)m(ula)578 930 y(\026)575 954 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))c(=)113 b(min)897 1012 y Fp(c)930 1021 y Fl(1)965 1012 y Fp(;:::)o(;c)1107 1022 y Fk(l)1132 1012 y Fj(\025)p Fv(0)1251 812 y Fi(R)1294 918 y Fp(D)1375 886 y Fr(e)1417 853 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1698 886 y Fs(kr)p Fr(h)p Fu(\()p Fr(c)1945 900 y Fv(1)1987 886 y Fr(;)15 b(:)g(:)g(:)i(;)e(c)2228 900 y Fp(l)2257 886 y Fu(\))p Fs(k)2337 853 y Fv(2)2337 908 y(2)p 1251 933 1128 4 v 1513 1016 a Fs(k)p Fr(h)p Fu(\()p Fr(c)1684 1030 y Fv(1)1726 1016 y Fr(;)g(:)g(:)g(:)i(;)e(c)1967 1030 y Fp(l)1995 1016 y Fu(\))p Fs(k)2075 985 y Fv(2)2075 1040 y(2)2389 954 y Fu(\(1)21 b(+)f Fr(O)s Fu(\()p Fr(\017)2725 916 y Fp(\013=)p Fv(2)2850 954 y Fr(;)15 b(e)2932 916 y Fj(\000)p Fp(\016)r(=\013)3114 954 y Fu(\)\))293 b(\(3)p Fr(:)p Fu(27\))50 1227 y(where)973 1324 y Fr(L)1035 1338 y Fp(\017)1070 1324 y Fr(h)p Fu(\()p Fr(x)1209 1338 y Fv(1)1250 1324 y Fr(;)15 b(:)g(:)g(:)i(;)e(c)1491 1338 y Fp(l)1519 1324 y Fu(\)\()p Fr(y)s Fu(\))26 b(=)f(0)p Fr(;)107 b(y)28 b Fs(2)d Fr(D)s Fs(n)c([)2335 1287 y Fp(l)2335 1347 y(i)p Fv(=1)2477 1324 y Fr(@)5 b(B)2599 1338 y Fp(\017)2634 1324 y Fu(\()p Fr(x)2721 1338 y Fp(i)2751 1324 y Fu(\))1082 1497 y Fr(h)p Fu(\()p Fr(c)1208 1511 y Fv(1)1250 1497 y Fr(;)15 b(:)g(:)g(:)i(;)e(c)1491 1511 y Fp(l)1519 1497 y Fu(\)\()p Fr(y)s Fu(\))26 b(=)f Fr(c)1833 1511 y Fp(i)1864 1497 y Fr(;)106 b(y)28 b Fs(2)d Fr(@)5 b(B)2276 1511 y Fp(\017)2311 1497 y Fu(\()p Fr(x)2398 1511 y Fp(i)2429 1497 y Fu(\))3477 1408 y(\(3)p Fr(:)p Fu(28\))50 1671 y(It)21 b(is)f(easy)h(to)h(see)g(that)f(the)g(result)f(di\013ers)g (only)g(b)m(y)g(a)i(constan)m(t)g(factor)g(from)e(that)i(in)d(the)j (non-degenerate)50 1816 y(case)31 b(stated)g(in)f(the)g(prop)s (osition.)50 2034 y Ft(Uniform)51 b(estimates)g(on)h(principal)g (eigenfunctions.)85 b Fu(The)45 b(pro)s(of)f(of)h(Prop)s(osition)e(3.2) j(has)50 2180 y(already)37 b(pro)m(vided)f(us)g(with)g(an)h(appro)m (ximation)g(for)g(the)g(principal)d(eigenfunction,)k(namely)f Fr(h)3505 2194 y Fp(B)s(;D)3646 2175 y Fk(c)3685 2180 y Fu(.)50 2325 y(W)-8 b(e)37 b(ha)m(v)m(e)g(seen)f(that)g(in)e Fr(L)1005 2292 y Fv(2)1082 2325 y Fu(this)g(appro)m(ximation)h(is)g(go) s(o)s(d)g(on)h(the)g(order)f Fr(\017)2774 2292 y Fp(\013=)p Fv(2)2899 2325 y Fu(.)57 b(W)-8 b(e)37 b(will)c(no)m(w)j(sho)m(w)50 2471 y(that)31 b(this)e(appro)m(ximation)g(is)h(also)g(uniformly)e(go)s (o)s(d.)50 2689 y Fh(Prop)s(osition)50 b(3.3:)59 b Fo(Under)42 b(the)g(hyp)-5 b(othesis)44 b(of)e(Pr)-5 b(op)g(osition)44 b(3.2,)g(the)e(princip)-5 b(al)43 b(eigenfunction,)50 2834 y Fr(\036)104 2848 y Fp(D)170 2834 y Fo(,)32 b(of)h Fr(L)399 2801 y Fp(D)460 2774 y Fk(c)399 2857 y Fp(\017)499 2834 y Fo(,)f(normalize)-5 b(d)36 b(such)c(that)i Fu(inf)1524 2848 y Fp(y)r Fj(2)p Fp(@)t(B)1732 2834 y Fr(\036)1786 2848 y Fp(D)1878 2834 y Fu(=)25 b(1)p Fo(,)33 b(satis\014es)841 3095 y Fr(h)893 3109 y Fp(B)s(;D)1034 3090 y Fk(c)1073 3095 y Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fr(\036)1367 3109 y Fp(D)1433 3095 y Fu(\()p Fr(y)s Fu(\))h Fs(\024)f Fr(h)1725 3109 y Fp(B)s(;D)1866 3090 y Fk(c)1905 3095 y Fu(\()p Fr(y)s Fu(\)\(1)c(+)f Fr(C)7 b(\017)2324 3057 y Fp(\013=)p Fv(2)2449 3095 y Fu(\)\(1)21 b(+)f Fr(e)2718 3057 y Fj(\000)p Fp(\016)r(=\017)2883 3095 y Fu(\))559 b(\(3)p Fr(:)p Fu(29\))108 3355 y Fe(Pro)s(of:)46 b Fu(Let)30 b(us)e(\014rst)g(assume)h(that)g Fr(D)j Fu(is)c(b)s(ounded.)38 b(Observ)m(e)29 b(that)g(set)h Fr(\016)s(f)2792 3322 y Fp(\025)2865 3355 y Fu(=)25 b Fr(f)3016 3322 y Fp(\025)3080 3355 y Fs(\000)17 b Fr(f)3223 3322 y Fv(0)3263 3355 y Fu(.)40 b(Then)28 b Fr(\016)s(f)3662 3322 y Fp(\025)50 3501 y Fu(satis\014es)i(the)g(Diric)m(hlet)f(problem)1228 3724 y Fr(L)1290 3738 y Fp(\017)1324 3724 y Fr(\016)s(f)1422 3687 y Fp(\025)1470 3724 y Fu(\()p Fr(y)s Fu(\))d(=)f Fr(\025 )1825 3687 y Fp(\025)1873 3724 y Fu(\()p Fr(y)s Fu(\))p Fr(;)107 b(y)28 b Fs(2)d Fr(D)s Fs(n)p Fr(@)5 b(B)1324 3897 y(\016)s(f)1422 3859 y Fp(\025)1470 3897 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(0)p Fr(;)107 b(y)28 b Fs(2)d Fr(@)5 b(B)1324 4070 y(\016)s(f)1422 4032 y Fp(\025)1470 4070 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(0)p Fr(;)107 b(y)28 b Fs(2)d Fr(D)2124 4032 y Fp(c)3477 3895 y Fu(\(3)p Fr(:)p Fu(30\))50 4289 y(Th)m(us)k(w)m(e)i(can)g(write)809 4490 y Fr(\016)s(f)907 4457 y Fp(\025)954 4490 y Fu(\()p Fr(y)s Fu(\))p 765 4531 351 4 v 765 4614 a Fr(h)817 4628 y Fp(B)s(;D)958 4609 y Fk(c)997 4614 y Fu(\()p Fr(y)s Fu(\))1152 4552 y(=)1248 4428 y Fi(Z)1299 4634 y Fp(D)r Fj(n)p Fp(B)1638 4490 y Fu(1)p 1486 4531 V 1486 4614 a Fr(h)1538 4628 y Fp(B)s(;D)1679 4609 y Fk(c)1718 4614 y Fu(\()p Fr(y)s Fu(\))1847 4552 y Fr(G)1919 4568 y Fp(D)r Fj(n)p Fp(B)2080 4552 y Fu(\()p Fr(y)s(;)15 b(z)t Fu(\))p Fr(h)2336 4566 y Fp(B)s(;D)2477 4547 y Fk(c)2517 4552 y Fu(\()p Fr(z)t Fu(\))2688 4490 y Fr(\016)s(f)2786 4457 y Fp(\025)2834 4490 y Fu(\()p Fr(z)t Fu(\))p 2644 4531 349 4 v 2644 4614 a Fr(h)2696 4628 y Fp(B)s(;D)2837 4609 y Fk(c)2877 4614 y Fu(\()p Fr(z)t Fu(\))3477 4552 y(\(3)p Fr(:)p Fu(31\))50 4866 y(Assume)30 b(that)h Fr(M)k Fs(\021)25 b Fu(sup)940 4888 y Fp(y)r Fj(2)p Fp(D)r Fj(n)p Fp(B)1267 4822 y(\036)1311 4831 y Fk(D)1367 4822 y Fv(\()p Fp(y)r Fv(\))p 1215 4845 300 4 v 1215 4898 a Fp(h)1258 4909 y Fk(B)r(;D)1380 4896 y(c)1419 4898 y Fv(\()p Fp(y)r Fv(\))1551 4866 y Fr(<)g Fs(1)p Fu(.)40 b(Then)30 b(\(3.31\))i (together)g(with)d(\(3.24\))k(implies)27 b(that)613 5165 y Fr(M)36 b Fs(\024)25 b Fu(1)20 b(+)g Fr(C)7 b(\017)1098 5128 y Fp(\013=)p Fv(2)1243 5165 y Fu(+)20 b Fr(\025M)79 b Fu(sup)1500 5246 y Fp(y)r Fj(2)p Fp(D)r Fj(n)p Fp(B)1760 5042 y Fi(Z)1811 5248 y Fp(D)r Fj(n)p Fp(B)2151 5104 y Fu(1)p 1998 5144 351 4 v 1998 5228 a Fr(h)2050 5242 y Fp(B)s(;D)2191 5223 y Fk(c)2230 5228 y Fu(\()p Fr(y)s Fu(\))2359 5165 y Fr(G)2431 5182 y Fp(D)r Fj(n)p Fp(B)2592 5165 y Fu(\()p Fr(y)s(;)15 b(z)t Fu(\))p Fr(h)2848 5179 y Fp(B)s(;D)2989 5161 y Fk(c)3029 5165 y Fu(\()p Fr(z)t Fu(\))737 5393 y(=)25 b(1)20 b(+)g Fr(C)7 b(\017)1098 5355 y Fp(\013=)p Fv(2)1243 5393 y Fu(+)20 b Fr(\025M)79 b Fu(sup)1500 5473 y Fp(y)r Fj(2)p Fp(D)r Fj(n)p Fp(B)1760 5393 y Fq(E)1815 5407 y Fp(y)1880 5393 y Fu([)p Fr(\034)1945 5407 y Fp(B)2007 5393 y Fs(j)p Fr(\034)2072 5407 y Fp(B)2160 5393 y Fs(\024)25 b Fr(\034)2296 5407 y Fp(D)2357 5388 y Fk(c)2396 5393 y Fu(])3477 5293 y(\(3)p Fr(:)p Fu(32\))p eop %%Page: 17 17 17 16 bop 3633 295 a Fm(17)50 513 y Fu(Using)26 b(the)h(represen)m (tation)g(of)g(the)h(conditional)d(mean)i(time)g(from)f(Prop)s(osition) f(6.1)j(of)34 b([BEGK3],)29 b(one)50 658 y(sho)m(ws)h(that)1207 813 y(sup)1153 893 y Fp(y)r Fj(2)p Fp(D)r Fj(n)p Fp(B)1413 813 y Fq(E)1468 827 y Fp(y)1533 813 y Fu([)p Fr(\034)1598 827 y Fp(B)1660 813 y Fs(j)p Fr(\034)1725 827 y Fp(B)1813 813 y Fs(\024)25 b Fr(\034)1949 827 y Fp(D)2010 808 y Fk(c)2049 813 y Fu(])g(=)g(1)p Fr(=)2289 789 y Fu(\026)2285 813 y Fr(\025)q Fu(\()p Fr(D)s Fs(n)p Fr(B)5 b Fu(\))871 b(\(3)p Fr(:)p Fu(33\))50 1063 y(so)30 b(that)889 1223 y Fr(M)36 b Fs(\024)1286 1162 y Fu(1)21 b(+)f Fr(C)7 b(\017)1552 1129 y Fp(\013=)p Fv(2)p 1119 1203 725 4 v 1119 1289 a Fu(1)21 b Fs(\000)1280 1265 y Fu(\026)1276 1289 y Fr(\025)p Fu(\()p Fr(D)s Fu(\))p Fr(=)1526 1265 y Fu(\026)1522 1289 y Fr(\025)q Fu(\()p Fr(D)s Fs(n)p Fr(B)5 b Fu(\))1880 1223 y Fs(\024)25 b Fu(\(1)c(+)f Fr(C)7 b(\017)2277 1186 y Fp(\013=)p Fv(2)2401 1223 y Fu(\)\(1)22 b(+)d Fr(e)2670 1186 y Fj(\000)p Fp(\016)r(=\017)2835 1223 y Fu(\))607 b(\(3)p Fr(:)p Fu(34\))50 1462 y(Since)29 b(b)m(y)h(construction)g Fr(h)986 1476 y Fp(B)s(;D)1127 1457 y Fk(c)1166 1462 y Fu(\()p Fr(y)s Fu(\))c Fs(\024)f Fr(\036)1460 1476 y Fp(D)1526 1462 y Fu(\()p Fr(y)s Fu(\),)31 b(the)g(assertion)f(of)h(the)f(prop)s(osition)e(follo)m(ws.)159 1680 y(It)i(remains)e(to)i(justify)e(the)i(assumption)e Fr(M)35 b(<)25 b Fs(1)p Fu(.)41 b(Ho)m(w)m(ev)m(er,)32 b(this)c(is)h(easy)-8 b(.)41 b(First,)30 b Fr(\036)3183 1694 y Fp(D)3278 1680 y Fu(is)f(b)s(ounded)50 1825 y(and)39 b Fr(C)308 1792 y Fv(2)348 1825 y Fu(\()p Fr(D)s Fu(\).)69 b(Th)m(us,)919 1781 y Fp(\036)963 1790 y Fk(D)1020 1781 y Fv(\()p Fp(y)r Fv(\))p 868 1805 300 4 v 868 1857 a Fp(h)911 1868 y Fk(B)r(;D)1033 1855 y(c)1071 1857 y Fv(\()p Fp(y)r Fv(\))1217 1825 y Fu(ma)m(y)41 b(only)d(div)m(erge)i(when)f Fr(h)2254 1839 y Fp(B)s(;D)2395 1821 y Fk(c)2434 1825 y Fu(\()p Fr(y)s Fu(\))i Fs(#)g Fu(0.)68 b(Ho)m(w)m(ev)m(er,)45 b(since)39 b Fr(h)3505 1839 y Fp(B)s(;D)3646 1821 y Fk(c)3685 1825 y Fu(,)50 1971 y(is)g(harmonic)g(and)h(non-negativ)m(e)h(on)g(the) f(b)s(oundary)-8 b(,)42 b(it)e(is)f(strictly)g(p)s(ositiv)m(e)h(on)g Fr(D)j Fu(b)m(y)d(the)h(strong)50 2116 y(maxim)m(um)26 b(principle.)36 b(Th)m(us)27 b(its)f(explosion)g(can)i(o)s(ccur)f(only) f(at)i(the)g(b)s(oundary)d(of)i Fr(D)j Fu(where)d Fr(h)3411 2130 y Fp(B)s(;D)3552 2112 y Fk(c)3591 2116 y Fu(\()p Fr(y)s Fu(\))50 2262 y(tends)37 b(to)h(zero.)62 b(Moreo)m(v)m(er,)41 b(its)c(normal)f(deriv)-5 b(ativ)m(e)37 b(on)g Fr(@)5 b(D)40 b Fu(is)c(strictly)g(\(and)h(since)3159 2239 y(\026)3138 2262 y Fr(D)j Fu(is)c(compact,)50 2407 y(uniformly\))25 b(p)s(ositiv)m(e)i(\(see)i(e.g.)41 b(Section)28 b(5,)i(Prop)s(osition)c (2.2)j(of)35 b([T)-8 b(a)m(y]\).)42 b(Therefore)3114 2363 y Fp(\036)3158 2372 y Fk(D)3215 2363 y Fv(\()p Fp(y)r Fv(\))p 3063 2386 V 3063 2439 a Fp(h)3106 2450 y Fk(B)r(;D)3228 2437 y(c)3266 2439 y Fv(\()p Fp(y)r Fv(\))3401 2407 y Fu(remains)50 2553 y(b)s(ounded)28 b(also)i(when)g Fr(y)e Fs(!)d Fr(@)5 b(D)s Fu(.)159 2771 y(Therefore)30 b(the)h(prop)s (osition)d(is)h(pro)m(v)m(en)i(if)1697 2748 y(\026)1676 2771 y Fr(D)i Fu(is)d(compact.)159 2989 y(In)j(the)g(non-compact)i (case,)h(w)m(e)e(can)f(obtain)g(a)h(similar)d(result)i(for)g(the)h (suprem)m(um)e(o)m(v)m(er)i(compact)50 3135 y(subsets)23 b(\000)i Fs(\032)g Fq(R)596 3102 y Fp(d)644 3135 y Fu(,)h(using)c(the)j (rapid)d(deca)m(y)j(of)f(the)g(Green's)h(function)e(in)f(regions)i (where)f Fr(F)13 b Fu(\()p Fr(y)s Fu(\))25 b(is)e(getting)50 3280 y(v)m(ery)31 b(large.)40 b Fs(})50 3498 y Ft(Eigenfunction)33 b(and)f(their)g(zeros.)40 b Fu(W)-8 b(e)30 b(are)e(no)m(w)g(ready)g(to) h(deriv)m(e)f(the)g(crucial)f(a)h(priori)e(estimates)50 3644 y(on)c(eigenfunctions)e(of)i Fr(L)910 3658 y Fp(\017)966 3644 y Fu(\(p)s(ossibly)e(on)h(some)i(domain)d(\012\).)38 b(Assume)21 b(that)i Fr(\025)f Fu(is)e(an)i(exp)s(onen)m(tially)e (small)50 3789 y(eigen)m(v)-5 b(alue)33 b(of)g Fr(L)655 3803 y Fp(\017)723 3789 y Fu(and)g(let)g Fr(\036)1091 3756 y Fp(\025)1172 3789 y Fu(denote)h(a)g(corresp)s(onding)d (eigenfunction.)48 b(Then)32 b(w)m(e)i(can)g(decomp)s(ose)50 3935 y(\012)27 b(in)m(to)h(op)s(en)g(subsets)f Fr(D)930 3949 y Fp(i)989 3935 y Fu(suc)m(h)g(that)i(either)e Fr(\036)1695 3902 y Fp(\025)1743 3935 y Fu(\()p Fr(y)s Fu(\))f Fr(>)f Fu(0)j(for)g(all)f Fr(y)h Fs(2)d Fr(D)2551 3949 y Fp(i)2582 3935 y Fu(,)j(or)g Fr(\036)2798 3902 y Fp(\025)2846 3935 y Fu(\()p Fr(y)s Fu(\))e Fr(<)f Fu(0)j(for)g(all)f Fr(y)h Fs(2)d Fr(D)3654 3949 y Fp(i)3685 3935 y Fu(,)50 4080 y(while)j Fr(\036)341 4047 y Fp(\025)389 4080 y Fu(\()p Fr(y)s Fu(\))e(=)f(0,)31 b(if)e Fr(y)f Fs(2)d Fr(@)5 b(D)1100 4094 y Fp(i)1131 4080 y Fu(.)159 4298 y(Ob)m(viously)-8 b(,)35 b(the)h(restriction)e(of)i Fr(\036)1375 4265 y Fp(\025)1458 4298 y Fu(to)g Fr(D)1649 4312 y Fp(i)1715 4298 y Fu(is)e(the)i(principal)c(eigenfunction)i Fr(\036)2974 4312 y Fp(D)3033 4321 y Fk(i)3103 4298 y Fu(of)h Fr(D)3286 4312 y Fp(i)3317 4298 y Fu(,)i(and)d Fr(\025)i Fu(is)50 4444 y(the)30 b(principal)d(eigen)m(v)-5 b(alue)31 b(of)f(an)m(y)h(of)g (the)f(sets)h Fr(D)1808 4458 y Fp(i)1838 4444 y Fu(.)41 b(This)29 b(en)m(tails)h(that)50 4662 y Fh(Lemma)39 b(3.4:)54 b Fo(Assume)34 b(that)h Fr(\025)27 b Fs(\024)g Fr(e)1437 4629 y Fj(\000)p Fp(a=\017)1605 4662 y Fo(,)34 b(for)h(some)f Fr(a)27 b Fs(\025)g Fr(a)2268 4676 y Fv(0)2309 4662 y Fo(,)34 b(with)g Fr(a)2617 4676 y Fv(0)2685 4662 y Fr(>)27 b Fu(0)34 b Fo(indep)-5 b(endent)36 b(of)e Fr(\017)p Fo(.)44 b(L)-5 b(et)50 4807 y Fr(D)125 4821 y Fp(i)184 4807 y Fo(b)g(e)28 b(the)h(c)-5 b(orr)g(esp)g(onding)31 b(sets)e(de\014ne)-5 b(d)30 b(ab)-5 b(ove.)41 b(Then,)29 b(e)-5 b(ach)29 b(set)g Fr(D)2441 4821 y Fp(i)2500 4807 y Fo(c)-5 b(ontains)30 b(at)f(le)-5 b(ast)29 b(one)g(minimum)50 4953 y Fr(x)102 4967 y Fp(i)172 4953 y Fo(of)41 b Fr(F)13 b Fu(\()p Fr(x)p Fu(\))40 b Fo(for)h(which)g Fu(cap)1088 4969 y Fp(@)t(B)1186 4978 y Fk(\017)1218 4969 y Fv(\()p Fp(x)1287 4978 y Fk(i)1316 4969 y Fv(\))1349 4953 y Fu(\()p Fr(D)1462 4920 y Fp(c)1459 4976 y(i)1499 4953 y Fu(\))p Fr(=)p Fs(k)p Fr(h)1676 4969 y Fp(B)1731 4978 y Fk(\017)1765 4969 y Fv(\()p Fp(x)1834 4978 y Fk(i)1864 4969 y Fv(\))p Fp(;D)1975 4947 y Fk(c)1973 4992 y(i)2014 4953 y Fs(k)2059 4920 y Fv(2)2059 4975 y(2)2139 4953 y Fs(\031)e Fr(\025)p Fo(;)k(for)e(al)5 b(l)41 b(other)g(minima)g Fr(x)3301 4967 y Fp(ij)3405 4953 y Fo(of)g Fr(F)53 b Fo(in)50 5098 y Fr(D)125 5112 y Fp(i)155 5098 y Fo(,)46 b Fu(cap)380 5115 y Fp(B)435 5124 y Fk(\017)468 5115 y Fv(\()p Fp(x)537 5124 y Fk(ij)596 5115 y Fv(\))629 5098 y Fu(\()p Fr(D)742 5065 y Fp(c)739 5122 y(i)779 5098 y Fu(\))p Fr(=)p Fs(k)p Fr(h)956 5115 y Fp(B)1011 5124 y Fk(\017)1045 5115 y Fv(\()p Fp(x)1114 5124 y Fk(ij)1173 5115 y Fv(\))p Fp(;D)1284 5093 y Fk(c)1282 5137 y(i)1323 5098 y Fs(k)1368 5065 y Fv(2)1368 5121 y(2)1453 5098 y Fs(\025)e Fr(\025)p Fo(.)74 b(In)43 b(p)-5 b(articular,)63 b Fu(dist)o(\()p Fr(x)2558 5112 y Fp(i)2589 5098 y Fr(;)15 b(D)2707 5065 y Fp(c)2704 5122 y(i)2744 5098 y Fu(\))45 b Fs(\025)f Fr(\032)g(>)g Fu(0)p Fo(,)i(for)e(some)g Fr(\032)50 5244 y Fo(indep)-5 b(endent)34 b(of)f Fr(\017)p Fo(.)159 5462 y(Mor)-5 b(e)g(over,)30 b(if)e Fr(x)719 5476 y Fp(ij)808 5462 y Fs(2)d Fr(D)969 5476 y Fp(i)1028 5462 y Fo(is)j(a)g(minimum)h (of)g Fr(F)41 b Fo(such)28 b(that)45 b Fu(dist)o(\()p Fr(x)2437 5476 y Fp(ij)2501 5462 y Fr(;)15 b(D)2619 5429 y Fp(c)2616 5485 y(i)2657 5462 y Fu(\))25 b Fs(\024)g Fu(10)2903 5396 y Fs(p)p 2980 5396 37 4 v 2980 5462 a Fr(\017)p Fo(,)k(then)f(ther)-5 b(e)29 b(exists)p eop %%Page: 18 18 18 17 bop 50 295 a Fm(18)50 513 y Fo(a)33 b(c)-5 b(onstant)34 b Fr(C)39 b Fo(such)33 b(that)h(for)f(al)5 b(l)33 b Fr(y)28 b Fs(2)d Fr(B)1491 534 y Fv(5)1527 488 y Fj(p)p 1587 488 31 4 v 46 x Fp(\017)1622 513 y Fu(\()p Fr(x)1709 527 y Fp(ij)1773 513 y Fu(\))c Fs(\\)f Fr(D)1985 527 y Fp(i)2015 513 y Fo(,)1242 771 y Fr(\036)1296 734 y Fp(\025)1343 771 y Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fr(C)7 b(\017)1692 734 y Fj(\000)p Fp(d)1791 771 y Fu(cap)1942 788 y Fp(B)1997 797 y Fk(\017)2030 788 y Fv(\()p Fp(x)2099 797 y Fk(ij)2158 788 y Fv(\))2191 771 y Fu(\()p Fr(B)2295 785 y Fp(\017)2330 771 y Fu(\()p Fr(x)2417 785 y Fp(i)2447 771 y Fu(\)\))960 b(\(3)p Fr(:)p Fu(35\))50 1248 y Fe(Pro)s(of:)45 b Fu(The)25 b(\014rst)h(t)m(w)m(o)h(assertions)f(follo)m(w)f(from)h (Lemma)g(2.5)h(and)f(Prop)s(osition)e(3.2)j(\(plus)e(the)h(remark)50 1394 y(follo)m(wing)k(its)h(pro)s(of)7 b(\).)43 b(The)31 b(last)h(assertion)f(follo)m(ws)g(since)f(b)m(y)i(Prop)s(osition)d(3.3) k(and)e(the)h(estimate)g(on)50 1539 y(the)e(equilibrium)c(p)s(oten)m (tial)k(from)g(Prop)s(osition)e(4.3)k(of)37 b([BEGK3],)1329 1821 y Fr(\036)1383 1783 y Fp(\025)1431 1821 y Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fr(C)1753 1753 y Fu(cap)1904 1769 y Fp(B)1959 1778 y Fk(\017)1992 1769 y Fv(\()p Fp(y)r Fv(\))2092 1753 y Fu(\()p Fr(B)2196 1767 y Fp(\017)2231 1753 y Fu(\()p Fr(x)2318 1767 y Fp(i)2349 1753 y Fu(\)\))p 1753 1800 667 4 v 1824 1883 a(cap)1975 1899 y Fp(B)2030 1908 y Fk(\017)2063 1899 y Fv(\()p Fp(y)r Fv(\))2163 1883 y Fu(\()p Fr(D)2276 1852 y Fp(c)2273 1908 y(i)2313 1883 y Fu(\))3477 1821 y(\(3)p Fr(:)p Fu(36\))50 2110 y(But)30 b(since)45 b(dist)o(\()p Fr(y)s(;)15 b(D)816 2077 y Fp(c)813 2134 y(i)854 2110 y Fu(\))25 b Fs(\024)g Fu(15)1100 2045 y Fs(p)p 1177 2045 37 4 v 1177 2110 a Fr(\017)p Fu(,)30 b(and)g Fr(y)j Fu(closer)e(than)f(5)2035 2045 y Fs(p)p 2111 2045 V 65 x Fr(\017)g Fu(from)g(a)h(lo)s(cal)e (minim)m(um,)f Fr(F)13 b Fu(\()p Fr(z)3267 2077 y Fj(\003)3310 2110 y Fu(\()p Fr(y)s(;)i(D)3511 2077 y Fp(c)3508 2134 y(i)3548 2110 y Fu(\)\))21 b Fs(\000)50 2256 y Fr(F)13 b Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fr(C)7 b(\017)p Fu(.)40 b(Th)m(us)29 b(Prop)s(osition)f(4.7)k(of)37 b([BEGK3])32 b(yields)1398 2515 y(cap)1550 2531 y Fp(B)1605 2540 y Fk(\017)1637 2531 y Fv(\()p Fp(y)r Fv(\))1737 2515 y Fu(\()p Fr(D)1850 2477 y Fp(c)1847 2537 y(i)1888 2515 y Fu(\))25 b Fs(\025)g Fr(C)7 b(\017)2153 2477 y Fp(d)p Fj(\000)p Fv(1)p Fp(=)p Fv(2)3477 2515 y Fu(\(3)p Fr(:)p Fu(37\))50 2773 y(Finally)-8 b(,)40 b(cap)539 2790 y Fp(B)594 2799 y Fk(\017)627 2790 y Fv(\()p Fp(y)r Fv(\))727 2773 y Fu(\()p Fr(B)831 2787 y Fp(\017)866 2773 y Fu(\()p Fr(x)953 2787 y Fp(i)984 2773 y Fu(\)\))h Fs(\031)e Fu(cap)1357 2790 y Fp(B)1412 2799 y Fk(\017)1444 2790 y Fv(\()p Fp(x)1513 2799 y Fk(ij)1572 2790 y Fv(\))1605 2773 y Fu(\()p Fr(B)1709 2787 y Fp(\017)1744 2773 y Fu(\()p Fr(x)1831 2787 y Fp(i)1862 2773 y Fu(\)\))h(follo)m(ws)e(e.g.)68 b(from)39 b(the)g(explicit)f (form)m(ulae)h(ob-)50 2919 y(tained)30 b(in)f([BEGK3].)42 b(This)28 b(pro)m(v)m(es)j(the)g(lemma.)p Fs(})50 3137 y Ft(Exp)s(onen)m(tially)i(small)g(eigen)m(v)-6 b(alues)34 b(and)g(their)f(eigenfunctions.)41 b Fu(Let)30 b(us)e(no)m(w)h(order)g (all)f(min-)50 3283 y(ima)i Fr(x)279 3297 y Fp(i)339 3283 y Fu(of)h Fr(F)43 b Fu(in)29 b(suc)m(h)h(a)h(w)m(a)m(y)g(that)783 3541 y Fr(F)13 b Fu(\()p Fr(z)935 3504 y Fj(\003)977 3541 y Fu(\()p Fr(x)1064 3555 y Fp(i)p Fv(+1)1187 3541 y Fr(;)i Fs(M)1336 3555 y Fp(i)1367 3541 y Fu(\)\))21 b Fs(\000)f Fr(F)13 b Fu(\()p Fr(x)1707 3555 y Fp(i)p Fv(+1)1829 3541 y Fu(\))26 b Fs(\024)f Fr(F)13 b Fu(\()p Fr(z)2138 3504 y Fj(\003)2180 3541 y Fu(\()p Fr(x)2267 3555 y Fp(i)2298 3541 y Fr(;)i Fs(M)2447 3555 y Fp(i)p Fj(\000)p Fv(1)2571 3541 y Fu(\)\))21 b Fs(\000)f Fr(F)13 b Fu(\()p Fr(x)2911 3555 y Fp(i)2941 3541 y Fu(\))501 b(\(3)p Fr(:)p Fu(38\))50 3800 y(for)44 b Fr(i)k Fu(=)f(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(n)30 b Fs(\000)f Fu(1,)48 b(where)43 b Fs(M)1335 3814 y Fp(i)1413 3800 y Fu(=)48 b Fs(f)p Fr(x)1629 3814 y Fv(1)1670 3800 y Fr(;)15 b(:)g(:)g(:)i(;)e(x)1924 3814 y Fp(i)1954 3800 y Fs(g)p Fu(.)82 b(W)-8 b(e)45 b(put)f(moreo)m(v)m(er)h Fs(M)2971 3814 y Fv(0)3060 3800 y Fs(\021)j Fu(\012)3245 3767 y Fp(c)3281 3800 y Fu(.)82 b(W)-8 b(e)45 b(also)50 3946 y(set)39 b Fr(B)269 3960 y Fp(i)340 3946 y Fs(\021)g Fr(B)519 3960 y Fp(\017)554 3946 y Fu(\()p Fr(x)641 3960 y Fp(i)671 3946 y Fu(\))h(and)e Fs(S)986 3960 y Fp(i)1057 3946 y Fu(=)h Fs([)1228 3913 y Fp(i)1228 3969 y(j)t Fv(=1)1358 3946 y Fr(B)1427 3960 y Fp(i)1457 3946 y Fu(.)67 b(Note)41 b(that)e(considerable)f (simpli\014cations)e(o)s(ccur)j(when)f(all)50 4091 y(inequalities)28 b(in)h(\(3.38\))j(are)f(strict,)f(and)g(w)m(e)h(will)d(only)h(consider) h(this)f(case)i(here.)159 4309 y(Supp)s(ose)g(that)i(w)m(e)g(w)m(an)m (t)g(to)g(compute)g(eigen)m(v)-5 b(alues)32 b(b)s(elo)m(w)2297 4285 y(\026)2293 4309 y Fr(\025)p Fu(\(\012)p Fs(nS)2547 4323 y Fp(k)2592 4309 y Fu(\))d Fs(\021)2759 4285 y Fu(\026)2756 4309 y Fr(\025)2809 4323 y Fp(k)2853 4309 y Fu(.)47 b(W)-8 b(e)34 b(kno)m(w)e(that)i(if)d Fr(\036)3662 4276 y Fp(\025)50 4455 y Fu(is)h(an)i(eigenfunction)e(with)h Fr(\025)d(<)1235 4431 y Fu(\026)1231 4455 y Fr(\025)1284 4469 y Fp(k)1329 4455 y Fu(,)k(then)g(it)f(can)h(b)s(e)f(represen)m(ted)g(as)h(the)g (solution)e(of)i(the)g(Diric)m(hlet)50 4600 y(problem)1211 4713 y(\()p Fr(L)1308 4727 y Fp(\017)1364 4713 y Fs(\000)19 b Fr(\025)p Fu(\))p Fr(f)1597 4675 y Fp(\025)1645 4713 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(0)p Fr(;)106 b(y)28 b Fs(2)d Fu(\012)p Fs(n)p Fr(@)5 b Fs(S)2439 4727 y Fp(k)1543 4886 y Fr(f)1598 4848 y Fp(\025)1645 4886 y Fu(\()p Fr(y)s Fu(\))26 b(=)f Fr(\036)1939 4848 y Fp(\025)1986 4886 y Fu(\()p Fr(y)s Fu(\))p Fr(;)107 b(y)28 b Fs(2)d Fr(@)5 b Fs(S)2503 4900 y Fp(k)3477 4797 y Fu(\(3)p Fr(:)p Fu(39\))50 5058 y(Th)m(us,)23 b(as)h(in)d(the)i(analysis)f(of)h(principle)c(eigen) m(v)-5 b(alues)23 b(ab)s(o)m(v)m(e,)j(the)d(condition)f(on)h Fr(\025)g Fu(will)d(b)s(e)i(the)h(existence)50 5203 y(of)30 b(a)h(non-trivial)d Fr(\036)730 5170 y Fp(\025)808 5203 y Fu(on)j Fr(@)5 b Fs(S)1043 5217 y Fp(k)1118 5203 y Fu(suc)m(h)30 b(that)h(the)g(surface)f(measure)369 5462 y Fr(dy)s(e)506 5424 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)788 5462 y Fu(\()p Fr(L)885 5476 y Fp(\017)940 5462 y Fs(\000)20 b Fr(\025)p Fu(\))p Fr(f)1174 5424 y Fp(\025)1221 5462 y Fu(\()p Fr(y)s Fu(\))26 b(=)f Fr(e)1503 5424 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1784 5462 y Fu(\()p Fr(@)1867 5478 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2013 5462 y Fr(f)2068 5424 y Fp(\025)2115 5462 y Fu(\()p Fr(y)s Fu(\))21 b(+)f Fr(@)2393 5478 y Fj(\000)p Fp(n)p Fv(\()p Fp(y)r Fv(\))2595 5462 y Fr(f)2650 5424 y Fp(\025)2697 5462 y Fu(\()p Fr(y)s Fu(\)\))p Fr(d\033)2949 5476 y Fj(S)2994 5486 y Fk(k)3039 5462 y Fu(\()p Fr(y)s Fu(\))320 b(\(3)p Fr(:)p Fu(40\))p eop %%Page: 19 19 19 18 bop 3633 295 a Fm(19)50 513 y Fu(v)-5 b(anishes.)37 b(A)25 b(necessary)h(condition)d(for)i(this)f(to)i(happ)s(en)d(is)h(of) h(course)g(the)g(v)-5 b(anishing)23 b(of)i(the)g(total)h(mass)50 658 y(on)k(eac)m(h)i(of)e(the)h(surfaces)f Fr(@)5 b(B)1105 672 y Fp(i)1135 658 y Fu(,)31 b Fr(i)26 b Fs(\024)f Fr(k)s Fu(,)30 b(i.e.)850 896 y Fi(Z)900 1102 y Fp(@)t(B)998 1111 y Fk(i)1046 1020 y Fr(e)1088 982 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1369 1020 y Fu(\()p Fr(@)1452 1036 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1599 1020 y Fr(f)1654 982 y Fp(\025)1700 1020 y Fu(\()p Fr(y)s Fu(\))21 b(+)f Fr(@)1978 1036 y Fj(\000)p Fp(n)p Fv(\()p Fp(y)r Fv(\))2181 1020 y Fr(f)2236 982 y Fp(\025)2283 1020 y Fu(\()p Fr(y)s Fu(\)\))p Fr(d\033)2535 1034 y Fj(S)2580 1044 y Fk(k)2625 1020 y Fu(\()p Fr(y)s Fu(\))26 b(=)e(0)568 b(\(3)p Fr(:)p Fu(41\))50 1395 y(No)m(w)33 b(if)e(w)m(e)i(knew)f(a)h(priori)d(that)j(all)e(minima)f Fr(x)1754 1409 y Fp(i)1784 1395 y Fu(,)j Fr(i)c Fs(\024)g Fr(k)35 b Fu(lie)c(w)m(ell)h(within)e(the)i(in)m(terior)f(of)i(the)f (sets)h Fr(D)3679 1409 y Fp(i)50 1540 y Fu(on)28 b(whic)m(h)e Fr(\036)485 1507 y Fp(\025)561 1540 y Fu(has)i(constan)m(t)h(sign,)f(w) m(e)g(could)f(use)h(Lemma)g(3.1)h(as)f(b)s(efore)g(in)e(the)i(analysis) f(of)h(principle)50 1686 y(eigen)m(v)-5 b(alues)26 b(to)g(sho)m(w)g (that)g Fr(f)1090 1653 y Fp(\025)1163 1686 y Fu(is)f(close)h(to)h(the)f (solution)f(of)h(the)g(problem)e(\(3.40\))k(where)e(the)g(b)s(oundary) 50 1831 y(conditions)g(are)i(replaced)f(b)m(y)h(constan)m(t)h(v)-5 b(alues)27 b Fr(c)1782 1845 y Fp(i)1813 1831 y Fu(.)39 b(Unfortunately)27 b(w)m(e)i(do)e(not)h(kno)m(w)g(this.)39 b(W)-8 b(e)28 b(kno)m(w,)50 1977 y(ho)m(w)m(ev)m(er,)37 b(that)e(eac)m(h)h(connected)g(comp)s(onen)m(t)f Fr(D)1812 1991 y Fp(i)1877 1977 y Fu(con)m(tains)g(at)g(least)g(one)g(suc)m(h)f (minim)m(um,)f(while)g(at)50 2122 y(those)e(minima)d(that)j(lie)e (close)i(to)g(the)g(b)s(oundary)d(of)i Fr(D)2007 2136 y Fp(i)2038 2122 y Fu(,)g Fr(\036)2147 2089 y Fp(\025)2226 2122 y Fu(is)f(v)m(ery)i(small)e(\(b)m(y)h(Lemma)h(3.4\).)159 2340 y(In)f(fact)h(w)m(e)g(ha)m(v)m(e)g(the)g(follo)m(wing)e(dic)m (hotom)m(y:)41 b(Let)31 b Fr(c)2015 2354 y Fp(i)2071 2340 y Fu(=)25 b(inf)2277 2354 y Fp(y)r Fj(2)p Fp(B)2420 2363 y Fk(i)2469 2340 y Fr(\036)2523 2307 y Fp(\025)2571 2340 y Fu(\()p Fr(y)s Fu(\).)41 b(Then)29 b Fo(either)17 2558 y Fu(\(i\))46 b Fs(j)15 b Fu(sup)336 2581 y Fp(y)r Fj(2)p Fp(B)479 2590 y Fk(i)528 2558 y Fr(\036)582 2525 y Fp(\025)630 2558 y Fu(\()p Fr(y)s Fu(\))p Fr(=c)832 2572 y Fp(i)883 2558 y Fs(\000)20 b Fu(1)p Fs(j)26 b(\024)f Fr(C)7 b(\017)1275 2525 y Fp(\013=)p Fv(2)1400 2558 y Fu(,)30 b Fo(or)-8 2777 y Fu(\(ii\))45 b(there)30 b(exists)g(1)c Fs(\024)f Fr(j)31 b Fs(\024)25 b Fr(k)34 b Fu(suc)m(h)c(that)h(sup)1592 2799 y Fp(y)r Fj(2)p Fp(B)1735 2808 y Fk(i)1784 2777 y Fs(j)p Fr(\036)1863 2744 y Fp(\025)1911 2777 y Fu(\()p Fr(y)s Fu(\))p Fs(j)p Fr(=)p Fs(j)p Fr(c)2163 2791 y Fp(j)2203 2777 y Fs(j)26 b(\024)f Fr(C)7 b(\017)2459 2744 y Fj(\000)p Fp(d)2557 2777 y Fu(cap)2709 2791 y Fp(B)2764 2800 y Fk(i)2797 2777 y Fu(\()p Fr(B)2901 2791 y Fp(j)2940 2777 y Fu(\).)159 2995 y(W)-8 b(e)25 b(no)m(w)g(consider)e (all)g(p)s(ossible)e(cases:)39 b(Let)25 b Fr(J)34 b Fs(\032)25 b(f)p Fu(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(k)s Fs(g)25 b Fu(b)s(e)e(the)i(set)g(of)f(indices)e(where)i(\(i\))g(holds,)50 3140 y(and)30 b(let)g Fr(J)408 3154 y Fp(j)477 3140 y Fu(b)s(e)g(the)g(subset)g(of)h(indices)d Fr(j)36 b Fu(where)30 b(\(ii\))f(holds)g(with)g Fr(j)5 b Fu(.)42 b(Giv)m(en)30 b(suc)m(h)g(a)h(partition,)e(w)m(e)i(set)1295 3497 y Fr(f)1350 3459 y Fp(\025)1422 3497 y Fu(=)1518 3410 y Fi(X)1519 3605 y Fp(j)t Fj(2)p Fp(J)1664 3497 y Fr(c)1703 3511 y Fp(j)1757 3396 y Fi(\020)1811 3497 y Fr(h)1863 3459 y Fp(\025)1863 3521 y(B)1918 3530 y Fk(j)1953 3521 y Fp(;)p Fj(S)2020 3531 y Fk(k)2059 3521 y Fj(n)p Fp(B)2151 3530 y Fk(j)2210 3497 y Fu(+)19 b Fr( )2362 3459 y Fp(\025)2359 3519 y(j)2411 3396 y Fi(\021)3477 3497 y Fu(\(3)p Fr(:)p Fu(42\))50 3924 y(where)39 b(the)i Fr(h)541 3891 y Fp(\025)541 3948 y(j)630 3924 y Fs(\021)g Fr(h)794 3891 y Fp(\025)794 3953 y(B)849 3962 y Fk(j)884 3953 y Fp(;)p Fj(S)951 3963 y Fk(k)990 3953 y Fj(n)p Fp(B)1082 3962 y Fk(j)1160 3924 y Fu(are)g(the)f Fr(\025)p Fu(-equilibrium)c(p)s(oten)m(tials)j(\(see)i ([BEGK3],)j(Section)c(2\),)k(i.e.)50 4069 y(solutions)29 b(of)h(\()p Fr(L)630 4083 y Fp(\017)685 4069 y Fs(\000)20 b Fr(\025)p Fu(\))p Fr(h)916 4083 y Fp(j)981 4069 y Fu(=)25 b Fr(O)33 b Fu(with)c(b)s(oundary)f(conditions)h(1)i(on)f Fr(@)5 b(B)2554 4083 y Fp(j)2623 4069 y Fu(and)30 b(0)g(on)h Fr(@)5 b Fu(\()p Fs(S)3145 4083 y Fp(k)3190 4069 y Fs(n)p Fr(B)3304 4083 y Fp(j)3343 4069 y Fu(\).)159 4288 y(Then)29 b Fr( )458 4255 y Fp(\025)455 4311 y(j)537 4288 y Fu(satis\014es,)h (for)g Fr(j)h Fs(2)25 b Fr(J)1013 4625 y Fu(\()p Fr(L)1110 4639 y Fp(\017)1165 4625 y Fs(\000)20 b Fr(\025)p Fu(\))p Fr( )1406 4588 y Fp(\025)1403 4648 y(j)1454 4625 y Fu(\()p Fr(y)s Fu(\))26 b(=)f(0)p Fr(;)107 b(y)28 b Fs(2)d Fu(\012)p Fs(n)p Fr(@)5 b Fs(S)2249 4639 y Fp(k)1344 4798 y Fr( )1406 4761 y Fp(\025)1403 4821 y(j)1454 4798 y Fu(\()p Fr(y)s Fu(\))26 b(=)f Fr(\036)1748 4761 y Fp(\025)1796 4798 y Fu(\()p Fr(y)s Fu(\))p Fr(=c)1998 4812 y Fp(j)2058 4798 y Fs(\000)20 b Fu(1)p Fr(;)106 b(y)28 b Fs(2)d Fr(@)5 b(B)2606 4812 y Fp(j)1344 4971 y Fr( )1406 4933 y Fp(\025)1403 4993 y(j)1454 4971 y Fu(\()p Fr(y)s Fu(\))26 b(=)f Fr(\036)1748 4933 y Fp(\025)1796 4971 y Fu(\()p Fr(y)s Fu(\))p Fr(=c)1998 4985 y Fp(j)2037 4971 y Fr(;)107 b(y)28 b Fs(2)d Fr(@)5 b(B)2450 4985 y Fp(l)2478 4971 y Fr(;)15 b(l)27 b Fs(2)e Fr(J)2708 4985 y Fp(j)1344 5144 y Fr( )1406 5106 y Fp(\025)1403 5166 y(j)1454 5144 y Fu(\()p Fr(y)s Fu(\))h(=)f(0)p Fr(;)107 b(y)28 b Fs(2)d Fr(@)5 b(B)2152 5158 y Fp(i)2182 5144 y Fr(;)15 b(i)26 b Fs(62)f Fr(J)2415 5158 y Fp(j)3477 4884 y Fu(\(3)p Fr(:)p Fu(43\))50 5462 y(W)-8 b(e)23 b(no)m(w)f(pro)s(ceed)g(as)g(in)f(the)h(analysis)f(of)h(principle)d (eigen)m(v)-5 b(alues,)24 b(i.e.)38 b(w)m(e)22 b(write)f(as)i (necessary)f(condition)p eop %%Page: 20 20 20 19 bop 50 295 a Fm(20)50 513 y Fu(for)30 b Fr(\025)g Fu(to)h(b)s(e)f(an)g(eigen)m(v)-5 b(alue)31 b(that)g(for)f(all)f Fr(i)d Fu(=)f(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(k)s Fu(,)301 695 y(0)25 b(=)467 571 y Fi(Z)518 778 y Fp(@)t(B)616 787 y Fk(i)664 695 y Fr(e)706 657 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)987 695 y Fr(h)1039 709 y Fp(i)1070 695 y Fu(\()p Fr(y)s Fu(\))1203 621 y Fi(\000)1245 695 y Fr(@)1293 711 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1439 695 y Fr(f)1494 657 y Fp(\025)1541 695 y Fu(\()p Fr(y)s Fu(\))21 b(+)e Fr(@)1818 711 y Fj(\000)p Fp(n)p Fv(\()p Fp(y)r Fv(\))2021 695 y Fr(f)2076 657 y Fp(\025)2123 695 y Fu(\()p Fr(y)s Fu(\))2241 621 y Fi(\001)2298 695 y Fr(d\033)2397 709 y Fp(@)t Fj(S)2485 719 y Fk(k)2529 695 y Fu(\()p Fr(y)s Fu(\))371 947 y(=)467 823 y Fi(Z)518 1030 y Fp(@)t Fj(S)606 1040 y Fk(k)665 947 y Fr(e)707 909 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)988 947 y Fr(@)1036 963 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1182 947 y Fr(h)1234 961 y Fp(i)1264 947 y Fu(\()p Fr(y)s Fu(\))p Fr(f)1437 909 y Fp(\025)1485 947 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)1702 961 y Fp(@)t Fj(S)1790 971 y Fk(k)1834 947 y Fu(\()p Fr(y)s Fu(\))366 1200 y Fs(\000)468 1139 y Fr(\025)p 468 1180 54 4 v 476 1263 a(\017)547 1077 y Fi(Z)653 1200 y Fr(dy)s(e)790 1163 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1072 1200 y Fr(h)1124 1214 y Fp(i)1154 1200 y Fu(\()p Fr(y)s Fu(\))p Fr(f)1327 1163 y Fp(\025)1375 1200 y Fu(\()p Fr(y)s Fu(\))371 1474 y(=)467 1388 y Fi(X)468 1583 y Fp(j)t Fj(2)p Fp(J)614 1474 y Fr(c)653 1488 y Fp(j)691 1319 y Fi(")744 1350 y(Z)795 1557 y Fp(@)t(B)893 1566 y Fk(j)946 1474 y Fr(e)988 1437 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1269 1474 y Fr(@)1317 1491 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1463 1474 y Fr(h)1515 1488 y Fp(i)1546 1474 y Fu(\()p Fr(y)s Fu(\)\(1)21 b(+)f Fr( )1918 1437 y Fp(\025)1915 1497 y(j)1966 1474 y Fu(\()p Fr(y)s Fu(\)\))p Fr(d\033)2218 1488 y Fp(@)t Fj(S)2306 1498 y Fk(k)2351 1474 y Fu(\()p Fr(y)s Fu(\))639 1764 y(+)738 1678 y Fi(X)730 1874 y Fp(l)p Fj(2)p Fp(J)842 1883 y Fk(j)892 1641 y Fi(Z)942 1847 y Fp(@)t(B)1040 1857 y Fk(l)1086 1764 y Fr(e)1128 1727 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1409 1764 y Fr(@)1457 1781 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1603 1764 y Fr(h)1655 1778 y Fp(i)1685 1764 y Fu(\()p Fr(y)s Fu(\))p Fr(\036)1857 1727 y Fp(\025)1906 1764 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)2123 1778 y Fp(@)t Fj(S)2211 1788 y Fk(k)2255 1764 y Fu(\()p Fr(y)s Fu(\))366 2124 y Fs(\000)468 2062 y Fr(\025)p 468 2103 V 476 2186 a(\017)547 1941 y Fi(0)547 2105 y(@)627 2000 y(Z)733 2124 y Fr(dy)s(e)870 2086 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1151 2124 y Fr(h)1203 2138 y Fp(i)1234 2124 y Fu(\()p Fr(y)s Fu(\)\()p Fr(h)1439 2086 y Fp(\025)1439 2146 y(j)1488 2124 y Fu(\()p Fr(y)s Fu(\))20 b(+)g Fr( )1779 2086 y Fp(\025)1776 2146 y(j)1827 2124 y Fu(\()p Fr(y)s Fu(\)\))i(+)2100 2037 y Fi(X)2092 2233 y Fp(l)p Fj(2)p Fp(J)2204 2242 y Fk(j)2254 2124 y Fr(dy)s(e)2391 2086 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2672 2124 y Fr(h)2724 2138 y Fp(i)2755 2124 y Fu(\()p Fr(y)s Fu(\))p Fr(f)2928 2086 y Fp(\025)2976 2124 y Fu(\()p Fr(y)s Fu(\))3094 1941 y Fi(1)3094 2105 y(A)3174 1968 y(#)3477 1441 y Fu(\(3)p Fr(:)p Fu(44\))50 2373 y(Note)32 b(that)e(b)m(y)h(the)f(b)s(ounds)e (\(i\))j(and)f(\(ii\),)231 2437 y Fi(\014)231 2492 y(\014)231 2546 y(\014)231 2601 y(\014)231 2655 y(\014)231 2710 y(\014)261 2500 y(Z)311 2706 y Fp(@)t(B)409 2715 y Fk(j)463 2623 y Fr(e)505 2586 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)786 2623 y Fr(@)834 2640 y Fp(n)p Fv(\()p Fp(y)r Fv(\))980 2623 y Fr(h)1032 2637 y Fp(i)1062 2623 y Fu(\()p Fr(y)s Fu(\))p Fr( )1242 2586 y Fp(\025)1239 2646 y(j)1291 2623 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)1508 2637 y Fp(@)t Fj(S)1596 2647 y Fk(k)1640 2623 y Fu(\()p Fr(y)s Fu(\))21 b(+)1878 2537 y Fi(X)1870 2733 y Fp(l)p Fj(2)p Fp(J)1982 2742 y Fk(j)2032 2500 y Fi(Z)2082 2706 y Fp(@)t(B)2180 2716 y Fk(l)2226 2623 y Fr(e)2268 2586 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2549 2623 y Fr(@)2597 2640 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2743 2623 y Fr(h)2795 2637 y Fp(i)2825 2623 y Fu(\()p Fr(y)s Fu(\))p Fr(\036)2997 2586 y Fp(\025)3046 2623 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)3263 2637 y Fp(@)t Fj(S)3351 2647 y Fk(k)3395 2623 y Fu(\()p Fr(y)s Fu(\))3513 2437 y Fi(\014)3513 2492 y(\014)3513 2546 y(\014)3513 2601 y(\014)3513 2655 y(\014)3513 2710 y(\014)241 2960 y Fs(\024)337 2801 y Fi(\014)337 2855 y(\014)337 2910 y(\014)337 2964 y(\014)337 3019 y(\014)367 2836 y(Z)417 3043 y Fp(@)t(B)515 3052 y Fk(j)569 2960 y Fr(e)611 2922 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)892 2960 y Fr(@)940 2976 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1086 2960 y Fr(h)1138 2974 y Fp(i)1168 2960 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)1385 2974 y Fp(@)t Fj(S)1473 2984 y Fk(k)1518 2960 y Fu(\()p Fr(y)s Fu(\))1636 2801 y Fi(\014)1636 2855 y(\014)1636 2910 y(\014)1636 2964 y(\014)1636 3019 y(\014)236 3296 y Fs(\002)327 3113 y Fi(0)327 3277 y(@)406 3296 y Fr(C)d(\017)515 3259 y Fp(\013=)p Fv(2)660 3296 y Fu(+)758 3210 y Fi(X)751 3405 y Fp(l)p Fj(2)p Fp(J)863 3414 y Fk(j)913 3296 y Fr(C)g(\017)1022 3259 y Fj(\000)p Fp(d)1120 3296 y Fu(cap)1272 3310 y Fp(B)1327 3320 y Fk(l)1358 3296 y Fu(\()p Fr(B)1462 3310 y Fp(j)1500 3296 y Fu(\))1550 3137 y Fi(\014)1550 3192 y(\014)1550 3246 y(\014)1550 3301 y(\014)1550 3355 y(\014)1596 3145 y(R)1639 3251 y Fp(@)t(B)1737 3261 y Fk(l)1782 3219 y Fr(e)1824 3186 y Fj(\000)p Fp(F)j Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2105 3219 y Fr(@)2153 3235 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2299 3219 y Fr(h)2351 3233 y Fp(i)2382 3219 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)2599 3233 y Fp(@)t Fj(S)2687 3243 y Fk(k)2731 3219 y Fu(\()p Fr(y)s Fu(\))p 1592 3275 1262 4 v 1592 3287 a Fi(R)1635 3392 y Fp(@)t(B)1733 3401 y Fk(j)1786 3360 y Fr(e)1828 3334 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2109 3360 y Fr(@)2157 3377 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2303 3360 y Fr(h)2355 3374 y Fp(i)2386 3360 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)2603 3374 y Fp(@)t Fj(S)2691 3384 y Fk(k)2735 3360 y Fu(\()p Fr(y)s Fu(\))2865 3137 y Fi(\014)2865 3192 y(\014)2865 3246 y(\014)2865 3301 y(\014)2865 3355 y(\014)2895 3113 y(1)2895 3277 y(A)3477 3514 y Fu(\(3)p Fr(:)p Fu(45\))50 3660 y(A)m(t)28 b(this)d(p)s(oin)m(t)h(it)h(is)f(con) m(v)m(enien)m(t)i(to)f(realize)g(that)g(Green's)h(\014rst)e(iden)m(tit) m(y)g(and)g(the)h(fact)h(that)g(the)f Fr(h)3531 3674 y Fp(i)3588 3660 y Fu(are)50 3805 y(harmonic,)i(implies)f(that)155 3864 y Fi(\014)155 3918 y(\014)155 3973 y(\014)155 4027 y(\014)155 4082 y(\014)186 3899 y(Z)236 4106 y Fp(@)t(B)334 4115 y Fk(j)388 4023 y Fr(e)430 3985 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)711 4023 y Fr(h)763 4037 y Fp(j)801 4023 y Fu(\()p Fr(y)s Fu(\))p Fr(@)967 4039 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1114 4023 y Fr(h)1166 4037 y Fp(i)1196 4023 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)1413 4037 y Fp(B)1468 4046 y Fk(j)1507 4023 y Fu(\()p Fr(y)s Fu(\))1625 3864 y Fi(\014)1625 3918 y(\014)1625 3973 y(\014)1625 4027 y(\014)1625 4082 y(\014)1682 4023 y Fu(=)25 b Fr(e)1820 3985 y Fj(\000)p Fv(1)1933 3891 y Fi(\014)1933 3945 y(\014)1933 4000 y(\014)1933 4055 y(\014)1963 3899 y(Z)2013 4106 y Fm(ext)13 b Fj(S)2176 4116 y Fk(k)2235 4023 y Fr(dy)s(e)2372 3985 y Fj(\000)p Fp(F)d Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2669 4023 y Fu(\()p Fs(r)p Fr(h)2832 4037 y Fp(j)2870 4023 y Fu(\()p Fr(y)s Fu(\))p Fr(;)15 b Fs(r)p Fr(h)3156 4037 y Fp(i)3188 4023 y Fu(\()p Fr(y)s Fu(\)\))3342 3891 y Fi(\014)3342 3945 y(\014)3342 4000 y(\014)3342 4055 y(\014)1682 4285 y Fs(\024)25 b Fr(\017)1815 4248 y Fj(\000)p Fv(1)1912 4180 y Fi(q)p 2003 4180 1124 4 v 105 x Fu(cap)2154 4299 y Fp(B)2209 4308 y Fk(i)2243 4285 y Fu(\()p Fs(S)2333 4299 y Fp(k)2378 4285 y Fs(n)p Fr(B)2492 4299 y Fp(i)2523 4285 y Fu(\)cap)2709 4299 y Fp(B)2764 4308 y Fk(j)2803 4285 y Fu(\()p Fs(S)2893 4299 y Fp(k)2938 4285 y Fs(n)p Fr(B)3052 4299 y Fp(j)3091 4285 y Fu(\))3477 4125 y(\(3)p Fr(:)p Fu(46\))50 4444 y(where)30 b(the)g(last)g(inequalit)m(y)f(uses)h(the)h(Cauc)m(h)m(y-Sc) m(h)m(w)m(artz)h(inequalit)m(y)-8 b(.)40 b(Noting)30 b(further)f(that,)i(since)1061 4621 y(cap)1212 4635 y Fp(B)1267 4645 y Fk(l)1298 4621 y Fu(\()p Fr(B)1402 4635 y Fp(j)1441 4621 y Fu(\))25 b(=)g(cap)1749 4635 y Fp(B)1804 4644 y Fk(j)1842 4621 y Fu(\()p Fr(B)1946 4635 y Fp(l)1974 4621 y Fu(\))h Fs(\024)f Fu(cap)2282 4635 y Fp(B)2337 4644 y Fk(j)2376 4621 y Fu(\()p Fs(S)2466 4635 y Fp(k)2511 4621 y Fs(n)p Fr(B)2625 4635 y Fp(j)2664 4621 y Fu(\))778 b(\(3)p Fr(:)p Fu(47\))50 4798 y(w)m(e)31 b(can)f(b)s(ound)f(\(3.45\))j (from)e(ab)s(o)m(v)m(e)i(b)m(y)323 4862 y Fi(\014)323 4916 y(\014)323 4971 y(\014)323 5025 y(\014)323 5080 y(\014)353 4897 y(Z)404 5104 y Fp(@)t(B)502 5113 y Fk(j)555 5021 y Fr(e)597 4983 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)878 5021 y Fr(@)926 5037 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1072 5021 y Fr(h)1124 5035 y Fp(i)1155 5021 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)1372 5035 y Fp(@)t Fj(S)1460 5045 y Fk(k)1504 5021 y Fu(\()p Fr(y)s Fu(\))1622 4862 y Fi(\014)1622 4916 y(\014)1622 4971 y(\014)1622 5025 y(\014)1622 5080 y(\014)328 5357 y Fs(\002)419 5175 y Fi(0)419 5338 y(@)498 5357 y Fr(C)d(\017)607 5320 y Fp(\013=)p Fv(2)752 5357 y Fu(+)851 5271 y Fi(X)843 5467 y Fp(l)p Fj(2)p Fp(J)955 5476 y Fk(j)1005 5357 y Fr(C)g(\017)1114 5320 y Fj(\000)p Fp(d)1212 5357 y Fu(cap)1364 5371 y Fp(B)1419 5380 y Fk(j)1457 5357 y Fu(\()p Fs(S)1547 5371 y Fp(k)1593 5357 y Fs(n)p Fr(B)1707 5371 y Fp(j)1745 5357 y Fu(\))1795 5198 y Fi(\014)1795 5253 y(\014)1795 5307 y(\014)1795 5362 y(\014)1795 5416 y(\014)1870 5216 y(p)p 1961 5216 1106 4 v 78 x Fu(cap)2112 5308 y Fp(B)2167 5317 y Fk(i)2201 5294 y Fu(\()p Fs(S)2291 5308 y Fp(k)2336 5294 y Fs(n)p Fr(B)2450 5308 y Fp(i)2480 5294 y Fu(\)cap)2667 5308 y Fp(B)2722 5318 y Fk(l)2753 5294 y Fu(\()p Fs(S)2843 5308 y Fp(k)2888 5294 y Fs(n)p Fr(B)3002 5308 y Fp(l)3031 5294 y Fu(\))p 1837 5336 1262 4 v 1837 5348 a Fi(R)1880 5454 y Fp(@)t(B)1978 5463 y Fk(j)2031 5421 y Fr(e)2073 5395 y Fj(\000)p Fp(F)j Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2354 5421 y Fr(@)2402 5438 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2548 5421 y Fr(h)2600 5435 y Fp(i)2631 5421 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)2848 5435 y Fp(@)t Fj(S)2936 5445 y Fk(k)2980 5421 y Fu(\()p Fr(y)s Fu(\))3110 5198 y Fi(\014)3110 5253 y(\014)3110 5307 y(\014)3110 5362 y(\014)3110 5416 y(\014)3140 5175 y(1)3140 5338 y(A)3477 5203 y Fu(\(3)p Fr(:)p Fu(48\))p eop %%Page: 21 21 21 20 bop 3633 295 a Fm(21)50 513 y Fu(In)29 b(particular,)g(in)g(the)i (case)g(when)f Fr(i)25 b Fu(=)g Fr(j)5 b Fu(,)32 b(this)d(simpli\014es) e(to)554 826 y(cap)705 840 y Fp(B)760 849 y Fk(j)799 826 y Fu(\()p Fs(S)889 840 y Fp(k)934 826 y Fs(n)p Fr(B)1048 840 y Fp(j)1086 826 y Fu(\))1136 643 y Fi(0)1136 807 y(@)1216 826 y Fr(C)7 b(\017)1325 788 y Fp(\013=)p Fv(2)1470 826 y Fu(+)1569 739 y Fi(X)1561 935 y Fp(l)p Fj(2)p Fp(J)1673 944 y Fk(j)1723 826 y Fr(C)g(\017)1832 788 y Fj(\000)p Fp(d)1930 744 y Fi(p)p 2021 744 1106 4 v 82 x Fu(cap)2173 840 y Fp(B)2228 849 y Fk(i)2261 826 y Fu(\()p Fs(S)2351 840 y Fp(k)2396 826 y Fs(n)p Fr(B)2510 840 y Fp(i)2541 826 y Fu(\)cap)2728 840 y Fp(B)2783 850 y Fk(l)2814 826 y Fu(\()p Fs(S)2904 840 y Fp(k)2949 826 y Fs(n)p Fr(B)3063 840 y Fp(l)3091 826 y Fu(\))3127 643 y Fi(1)3127 807 y(A)3477 826 y Fu(\(3)p Fr(:)p Fu(49\))159 1217 y(F)-8 b(or)31 b(the)f(terms)g(in)f(the)h(last)g(line)e(of)j(\(3.44\))h(w)m(e) e(obtain)g(in)f(complete)h(analogy)h(to)g(the)f(deriv)-5 b(ation)29 b(of)50 1362 y(the)h(b)s(ounds)f(\(3.20\))j(and)e(\(3.21\))j (that)384 1494 y Fi(Z)490 1618 y Fr(dy)s(e)627 1580 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)908 1618 y Fr(h)960 1632 y Fp(i)991 1618 y Fu(\()p Fr(y)s Fu(\)\()p Fr(h)1196 1580 y Fp(\025)1196 1640 y(j)1245 1618 y Fu(\()p Fr(y)s Fu(\))20 b Fs(\000)g Fr(h)1526 1632 y Fp(j)1565 1618 y Fu(\()p Fr(y)s Fu(\))h(+)f Fr( )1857 1580 y Fp(\025)1854 1640 y(j)1905 1618 y Fu(\))g(+)2059 1531 y Fi(X)2051 1727 y Fp(l)p Fj(2)p Fp(J)2163 1736 y Fk(j)2213 1494 y Fi(Z)2319 1618 y Fr(dy)s(e)2456 1580 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2737 1618 y Fr(h)2789 1632 y Fp(i)2820 1618 y Fu(\()p Fr(y)s Fu(\))p Fr(f)2993 1580 y Fp(\025)3040 1618 y Fu(\()p Fr(y)s Fu(\))394 1914 y(=)25 b Fr(O)s Fu(\()p Fr(\017)634 1877 y Fp(\013=)p Fv(2)759 1914 y Fu(\)\(1)c(+)f Fr(O)s Fu(\()p Fr(e)1135 1877 y Fj(\000)p Fp(\016)r(=\017)1300 1914 y Fu(\)\))1385 1791 y Fi(Z)1492 1914 y Fr(dy)s(e)1629 1877 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1910 1914 y Fr(h)1962 1928 y Fp(i)1993 1914 y Fu(\()p Fr(y)s Fu(\))p Fr(h)2163 1928 y Fp(j)2202 1914 y Fu(\()p Fr(y)s Fu(\))389 2215 y(+)20 b Fr(O)567 2033 y Fi(0)567 2196 y(@)646 2215 y Fr(C)7 b(\017)755 2178 y Fj(\000)p Fp(d)877 2129 y Fi(X)869 2325 y Fp(l)p Fj(2)p Fp(J)981 2334 y Fk(j)1031 2215 y Fu(cap)1182 2229 y Fp(B)1237 2239 y Fk(l)1268 2215 y Fu(\()p Fr(B)1372 2229 y Fp(j)1411 2215 y Fu(\))1461 2092 y Fi(Z)1567 2215 y Fr(dy)s(e)1704 2178 y Fj(\000)p Fp(F)j Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1985 2215 y Fr(h)2037 2229 y Fp(i)2068 2215 y Fu(\()p Fr(y)s Fu(\))p Fr(h)2238 2229 y Fp(j)2277 2215 y Fu(\()p Fr(y)s Fu(\)\(1)22 b(+)e Fr(e)2630 2178 y Fj(\000)p Fp(\016)r(=\017)2794 2215 y Fu(\))2829 2033 y Fi(1)2829 2196 y(A)3477 1948 y Fu(\(3)p Fr(:)p Fu(50\))50 2529 y(But)30 b(note)h(that)588 2761 y(cap)740 2775 y Fp(B)795 2785 y Fk(l)826 2761 y Fu(\()p Fr(B)930 2775 y Fp(j)968 2761 y Fu(\))1018 2638 y Fi(Z)1125 2761 y Fr(dy)s(e)1262 2724 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1543 2761 y Fr(h)1595 2775 y Fp(i)1626 2761 y Fu(\()p Fr(y)s Fu(\))p Fr(h)1796 2775 y Fp(j)1835 2761 y Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fu(cap)2226 2775 y Fp(B)2281 2785 y Fk(l)2312 2761 y Fu(\()p Fr(B)2416 2775 y Fp(j)2455 2761 y Fu(\))p Fs(k)p Fr(h)2587 2775 y Fp(i)2618 2761 y Fs(k)2663 2775 y Fv(2)2705 2761 y Fs(k)p Fr(h)2802 2775 y Fp(l)2830 2761 y Fs(k)2875 2775 y Fv(2)1979 3006 y Fs(\024)2091 2944 y(k)p Fr(h)2188 2958 y Fp(l)2217 2944 y Fs(k)2262 2958 y Fv(2)p 2086 2985 223 4 v 2086 3068 a Fs(k)p Fr(h)2183 3082 y Fp(j)2222 3068 y Fs(k)2267 3082 y Fv(2)2319 3006 y Fu(cap)2471 3020 y Fp(B)2526 3030 y Fk(l)2556 3006 y Fu(\()p Fr(B)2660 3020 y Fp(j)2699 3006 y Fu(\))p Fs(k)p Fr(h)2831 3020 y Fp(i)2863 3006 y Fs(k)2908 3020 y Fv(2)2949 3006 y Fs(k)p Fr(h)3046 3020 y Fp(j)3085 3006 y Fs(k)3130 3020 y Fv(2)1979 3216 y Fs(\024)g Fr(e)2117 3178 y Fj(\000)p Fp(\016)r(=\017)2281 3216 y Fs(k)p Fr(h)2378 3230 y Fp(i)2410 3216 y Fs(k)2455 3230 y Fv(2)2496 3216 y Fs(k)p Fr(h)2593 3230 y Fp(j)2632 3216 y Fs(k)2677 3230 y Fv(2)3477 2965 y Fu(\(3)p Fr(:)p Fu(51\))159 3503 y(Let)31 b(us)e(de\014ne)h(the)h (classical)e(capacit)m(y)j(matrix)1832 3470 y Fv(4)1902 3503 y Fs(C)k Fu(with)29 b(elemen)m(ts)879 3754 y Fs(C)927 3768 y Fp(ij)1016 3754 y Fs(\021)c(C)1165 3707 y Fv(\()p Fp(k)r Fv(\))1160 3779 y Fp(ij)1292 3754 y Fs(\021)g Fr(\017)1440 3630 y Fi(Z)1491 3837 y Fp(@)t(B)1589 3846 y Fk(j)1642 3754 y Fr(e)1684 3716 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1965 3754 y Fr(h)2017 3768 y Fp(j)2056 3754 y Fu(\()p Fr(y)s Fu(\))p Fr(@)2222 3770 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2368 3754 y Fr(h)2420 3768 y Fp(i)2451 3754 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)2668 3768 y Fp(B)2723 3777 y Fk(j)2762 3754 y Fu(\()p Fr(y)s Fu(\))50 4027 y(and)30 b(its)f(normalized)g(v)m(ersion)1377 4204 y Fs(K)1446 4218 y Fp(ij)1535 4204 y Fs(\021)c(K)1701 4157 y Fv(\()p Fp(k)r Fv(\))1700 4230 y Fp(ij)1829 4204 y Fs(\021)2076 4131 y(C)2129 4084 y Fv(\()p Fp(k)r Fv(\))2124 4156 y Fp(ij)p 1935 4183 437 4 v 1935 4267 a Fs(k)p Fr(h)2032 4281 y Fp(i)2063 4267 y Fs(k)2108 4281 y Fv(2)2150 4267 y Fs(k)p Fr(h)2247 4281 y Fp(j)2286 4267 y Fs(k)2331 4281 y Fv(2)3477 4204 y Fu(\(3)p Fr(:)p Fu(52\))50 4435 y(Note)32 b(that)e(this)g(matrix)f(is)h(symmetric)1444 4402 y Fv(5)1484 4435 y Fu(.)41 b(If)30 b(w)m(e)g(in)m(tro)s(duce)g (the)g(matrices)50 4729 y Fr(A)118 4743 y Fp(ij)207 4729 y Fs(\021)314 4643 y Fr(\017)366 4570 y Fi(R)409 4675 y Fp(@)t(B)507 4684 y Fk(j)560 4643 y Fr(e)602 4610 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)884 4643 y Fr(@)932 4659 y Fp(n)p Fv(\()p Fp(y)r Fv(\))1077 4643 y Fr(h)1129 4657 y Fp(i)1160 4643 y Fu(\()p Fr(y)s Fu(\))p Fr( )1340 4610 y Fp(\025)1337 4666 y(j)1389 4643 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)1606 4657 y Fp(@)t Fj(S)1694 4667 y Fk(k)1738 4643 y Fu(\()p Fr(y)s Fu(\))21 b(+)1968 4575 y Fi(P)2064 4670 y Fp(l)p Fj(2)p Fp(J)2176 4679 y Fk(j)2230 4643 y Fr(\017)2282 4570 y Fi(R)2325 4675 y Fp(@)t(B)2423 4685 y Fk(l)2469 4643 y Fr(e)2511 4610 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2792 4643 y Fr(@)2840 4659 y Fp(n)p Fv(\()p Fp(y)r Fv(\))2985 4643 y Fr(h)3037 4657 y Fp(i)3068 4643 y Fu(\()p Fr(y)s Fu(\))p Fr(\036)3240 4610 y Fp(\025)3289 4643 y Fu(\()p Fr(y)s Fu(\))p Fr(d\033)3506 4657 y Fp(@)t Fj(S)3594 4667 y Fk(k)3638 4643 y Fu(\()p Fr(y)s Fu(\))p 314 4708 3443 4 v 1817 4791 a Fs(k)p Fr(h)1914 4805 y Fp(i)1945 4791 y Fs(k)1990 4805 y Fv(2)2031 4791 y Fs(k)p Fr(h)2128 4805 y Fp(j)2167 4791 y Fs(k)2212 4805 y Fv(2)3477 4894 y Fu(\(3)p Fr(:)p Fu(53\))p 50 4998 546 4 v 159 5060 a Fn(4)197 5086 y Fm(The)23 b(matrix)h Fd(C)j Fm(is)d(a)g(classical)i (ob)t(ject)d(in)h(electrostatics,)k(the)c(diagonal)f(elemen)n(ts)h(b)r (eing)g(called)h(capacities,)h(and)d(the)50 5176 y(o\013-diagonal)f (ones)h(co)r(e\016cien)n(ts)j(of)c(induction)i([Jac)n(k].)34 b(The)23 b(o\013-diagonal)f(co)r(e\016cien)n(ts)j(represen)n(t)g(the)e (c)n(harge)g(induced)h(in)50 5267 y(the)i Fc(i)p Fm(-th)g(ball)g(when)g (the)g Fc(j)t Fm(-th)g(has)g(p)r(oten)n(tial)g(one)f(and)h(all)g (others)g(are)g(at)g(p)r(oten)n(tial)g(zero.)159 5345 y Fn(5)197 5371 y Fm(One)35 b(could)h(also)f(in)n(tro)r(duce)h(a)f (matrix)1446 5360 y Fi(b)1437 5371 y Fd(K)1496 5382 y Fb(ij)1593 5371 y Fm(=)i Fd(C)1731 5382 y Fb(ij)1791 5371 y Fc(=)p Fd(k)p Fc(h)1914 5382 y Fb(i)1942 5371 y Fd(k)1981 5345 y Fn(2)1981 5395 y(2)2054 5371 y Fm(whic)n(h)e(then)h (w)n(ould)f(b)r(e)g(a)g(sto)r(c)n(hastic)i(matrix)d(\(resp.)50 5462 y(sub-sto)r(c)n(hastic,)28 b(if)d(Diric)n(hlet)j(b)r(oundary)d (conditions)i(are)e(imp)r(osed)i(on)e(\012)2273 5436 y Fb(c)2308 5462 y Fm(\).)p eop %%Page: 22 22 22 21 bop 50 295 a Fm(22)324 583 y Fr(B)393 597 y Fp(ij)483 583 y Fs(\021)25 b Fu(1)-23 b(I)634 597 y Fp(i)p Fj(6)p Fv(=)p Fp(j)765 429 y Fi(R)841 503 y Fr(dy)s(e)978 470 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1259 503 y Fr(h)1311 517 y Fp(i)1342 503 y Fu(\()p Fr(y)s Fu(\)\()p Fr(h)1547 470 y Fp(\025)1547 526 y(j)1595 503 y Fu(\()p Fr(y)s Fu(\))21 b(+)f Fr( )1887 470 y Fp(\025)1884 526 y(j)1935 503 y Fu(\))h(+)2082 434 y Fi(P)2178 530 y Fp(l)p Fj(2)p Fp(J)2290 539 y Fk(j)2344 503 y Fr(dy)s(e)2481 470 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2762 503 y Fr(h)2814 517 y Fp(i)2845 503 y Fu(\()p Fr(y)s Fu(\))p Fr( )3025 470 y Fp(\025)3022 526 y(j)3073 503 y Fu(\()p Fr(y)s Fu(\))p 765 562 2428 4 v 1760 646 a Fs(k)p Fr(h)1857 660 y Fp(i)1888 646 y Fs(k)1933 660 y Fv(2)1974 646 y Fs(k)p Fr(h)2071 660 y Fp(j)2111 646 y Fs(k)2156 660 y Fv(2)3477 583 y Fu(\(3)p Fr(:)p Fu(54\))50 849 y(and)241 1170 y Fr(D)316 1184 y Fp(j)t(j)413 1170 y Fs(\021)520 1016 y Fi(R)596 1089 y Fr(dy)s(e)733 1056 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1014 1089 y Fr(h)1066 1103 y Fp(j)1105 1089 y Fu(\()p Fr(y)s Fu(\)\()p Fr(h)1310 1056 y Fp(\025)1310 1113 y(j)1359 1089 y Fu(\()p Fr(y)s Fu(\))21 b Fs(\000)f Fr(h)1641 1103 y Fp(j)1679 1089 y Fu(\()p Fr(y)s Fu(\))h(+)f Fr( )1971 1056 y Fp(\025)1968 1113 y(j)2019 1089 y Fu(\))h(+)2166 1021 y Fi(P)2261 1117 y Fp(l)p Fj(2)p Fp(J)2373 1126 y Fk(j)2428 1089 y Fr(dy)s(e)2565 1056 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2846 1089 y Fr(h)2898 1103 y Fp(j)2937 1089 y Fu(\()p Fr(y)s Fu(\))p Fr(\036)3109 1056 y Fp(\025)3109 1113 y(j)3157 1089 y Fu(\()p Fr(y)s Fu(\))p 520 1149 2756 4 v 1787 1232 a Fs(k)p Fr(h)1884 1246 y Fp(j)1923 1232 y Fs(k)1968 1201 y Fv(2)1968 1257 y(2)3477 1170 y Fu(\(3)p Fr(:)p Fu(55\))50 1486 y(Then)29 b(the)i(conditions)e(\(3.44\))j(for)e Fr(\025)h Fu(can)f(b)s(e)g (written)g(as)1006 1782 y(0)25 b(=)1172 1696 y Fi(X)1174 1891 y Fp(j)t Fj(2)p Fp(J)1321 1782 y Fu(^)-47 b Fr(c)1358 1796 y Fp(j)1412 1782 y Fu(\()p Fs(K)1516 1796 y Fp(ij)1601 1782 y Fs(\000)19 b Fr(\025\016)1784 1796 y Fp(ij)1869 1782 y Fu(+)h Fr(A)2028 1796 y Fp(ij)2112 1782 y Fs(\000)g Fr(\025)p Fu(\()p Fr(D)2366 1796 y Fp(j)t(j)2459 1782 y Fu(+)g Fr(B)2619 1796 y Fp(ij)2683 1782 y Fu(\)\))724 b(\(3)p Fr(:)p Fu(56\))50 2140 y(where)35 b(^)-47 b Fr(c)355 2154 y Fp(j)425 2140 y Fu(=)30 b Fs(k)p Fr(h)623 2154 y Fp(j)663 2140 y Fs(k)708 2154 y Fv(2)749 2140 y Fr(c)788 2154 y Fp(j)826 2140 y Fu(.)51 b(T)-8 b(o)35 b(sho)m(w)e(that)i(all)e (the)h(o\013-diagonal)f(terms)h(in)f Fr(B)2690 2154 y Fp(ij)2787 2140 y Fu(are)i(small,)e(w)m(e)h(still)e(need)50 2286 y(to)f(sho)m(w)f(that)h(the)g(normalized)e(functions)g Fr(h)1649 2300 y Fp(i)1709 2286 y Fu(and)h Fr(h)1938 2300 y Fp(j)2007 2286 y Fu(are)h(almost)f(orthogonal.)50 2504 y Fh(Lemma)38 b(3.5:)51 b Fo(Ther)-5 b(e)33 b(is)g(a)g(c)-5 b(onstant)34 b Fr(C)e(<)25 b Fs(1)32 b Fo(such)h(that)336 2811 y Fu(max)363 2870 y Fp(i)p Fj(6)p Fv(=)p Fp(j)531 2675 y Fi(R)607 2748 y Fr(dy)s(e)744 2715 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1025 2748 y Fr(h)1077 2762 y Fp(j)1116 2748 y Fu(\()p Fr(y)s Fu(\))p Fr(h)1286 2762 y Fp(i)1317 2748 y Fu(\()p Fr(y)s Fu(\))p 531 2790 905 4 v 765 2873 a Fs(k)p Fr(h)862 2887 y Fp(i)893 2873 y Fs(k)938 2887 y Fv(2)980 2873 y Fs(k)p Fr(h)1077 2887 y Fp(j)1116 2873 y Fs(k)1161 2887 y Fv(2)1472 2811 y Fs(\024)25 b Fr(C)7 b(\017)1677 2774 y Fj(\000)p Fv(\()p Fp(d)p Fv(+1\))p Fp(=)p Fv(2)2013 2811 y Fu(max)2085 2868 y Fp(i)2197 2811 y Fr(e)2239 2774 y Fj(\000)p Fv([)p Fp(F)j Fv(\()p Fp(z)2439 2746 y Fg(\003)2477 2774 y Fv(\()p Fp(x)2546 2783 y Fk(i)2575 2774 y Fp(;)p Fj(S)2642 2784 y Fk(k)2681 2774 y Fj(n)p Fp(B)2773 2783 y Fk(i)2802 2774 y Fv(\)\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)3040 2783 y Fk(i)3070 2774 y Fv(\)])p Fp(=\017)1472 3021 y Fs(\024)25 b Fr(e)1610 2984 y Fj(\000)p Fp(\022)r(=\017)3477 2883 y Fu(\(3)p Fr(:)p Fu(57\))50 3276 y Fo(for)33 b(some)g Fr(\022)28 b(>)d Fu(0)p Fo(.)50 3494 y Fe(Pro)s(of:)47 b Fu(Note)32 b(\014rst)e(that)h(the)f(terms)g(in)f(the)i(denominator)f (in)f(\(3.57\))j(are)f(b)s(ounded)d(via)274 3676 y Fi(Z)380 3800 y Fr(dy)s(e)517 3763 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)798 3800 y Fr(h)850 3763 y Fv(2)850 3823 y Fp(j)891 3800 y Fu(\()p Fr(y)s Fu(\))26 b Fs(\025)1131 3676 y Fi(Z)1181 3883 y Fp(B)1236 3866 y Fg(p)p 1286 3866 28 3 v 34 x Fk(\017)1319 3883 y Fv(\()p Fp(x)1388 3892 y Fk(j)1422 3883 y Fv(\))1470 3800 y Fr(dy)s(e)1607 3763 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1904 3699 y Fi(\020)1958 3800 y Fu(1)21 b Fs(\000)f Fr(C)7 b(\017)2224 3763 y Fj(\000)p Fv(1)p Fp(=)p Fv(2)2394 3800 y Fr(e)2436 3763 y Fj(\000)p Fv([)p Fp(F)j Fv(\()p Fp(z)2636 3735 y Fg(\003)2673 3763 y Fv(\()p Fp(x)2742 3772 y Fk(j)2777 3763 y Fp(;)p Fj(S)2844 3773 y Fk(k)2883 3763 y Fj(n)p Fp(B)2975 3772 y Fk(j)3009 3763 y Fv(\)\)])p Fp(=\017)3159 3699 y Fi(\021)3213 3718 y Fv(2)1035 4031 y Fu(=)25 b Fr(C)7 b(\017)1240 3993 y Fp(d=)p Fv(2)1355 4031 y Fr(e)1397 3993 y Fj(\000)p Fp(F)j Fv(\()p Fp(x)1579 4002 y Fk(j)1614 3993 y Fv(\))p Fp(=\017)3477 3892 y Fu(\(3)p Fr(:)p Fu(58\))50 4285 y(On)29 b(the)i(other)g(hand,)e(for)h Fr(i)c Fs(6)p Fu(=)f Fr(j)5 b Fu(,)122 4463 y Fi(Z)229 4586 y Fr(dy)s(e)366 4549 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)647 4586 y Fr(h)699 4600 y Fp(j)737 4586 y Fu(\()p Fr(y)s Fu(\))p Fr(h)907 4600 y Fp(i)939 4586 y Fu(\()p Fr(y)s Fu(\))25 b(=)1178 4463 y Fi(Z)1229 4669 y Fp(y)r Fv(:)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fj(\024)p Fv(max\()p Fp(F)g Fv(\()p Fp(z)1781 4651 y Fg(\003)1820 4669 y Fv(\()p Fp(x)1889 4678 y Fk(i)1918 4669 y Fp(;)p Fj(S)1985 4679 y Fk(k)2024 4669 y Fj(n)p Fp(B)2116 4678 y Fk(i)2145 4669 y Fv(\)\))p Fp(;F)g Fv(\()p Fp(z)2345 4651 y Fg(\003)2383 4669 y Fv(\()p Fp(x)2452 4678 y Fk(j)2487 4669 y Fp(;)p Fj(S)2554 4679 y Fk(k)2593 4669 y Fj(n)p Fp(B)2685 4678 y Fk(j)2719 4669 y Fv(\)\)\))2824 4586 y Fr(dy)s(e)2961 4549 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)3242 4586 y Fr(h)3294 4600 y Fp(j)3333 4586 y Fu(\()p Fr(y)s Fu(\))p Fr(h)3503 4600 y Fp(i)3534 4586 y Fu(\()p Fr(y)s Fu(\))127 4847 y(+)218 4724 y Fi(Z)269 4930 y Fp(y)r Fv(:)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(>)p Fv(max\()p Fp(F)g Fv(\()p Fp(z)821 4912 y Fg(\003)860 4930 y Fv(\()p Fp(x)929 4939 y Fk(i)958 4930 y Fp(;)p Fj(S)1025 4940 y Fk(k)1064 4930 y Fj(n)p Fp(B)1156 4939 y Fk(i)1185 4930 y Fv(\)\))p Fp(;F)g Fv(\()p Fp(z)1385 4912 y Fg(\003)1423 4930 y Fv(\()p Fp(x)1492 4939 y Fk(j)1527 4930 y Fp(;)p Fj(S)1594 4940 y Fk(k)1633 4930 y Fj(n)p Fp(B)1725 4939 y Fk(j)1759 4930 y Fv(\)\)\))1864 4847 y Fr(dy)s(e)2001 4810 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2282 4847 y Fr(h)2334 4861 y Fp(j)2373 4847 y Fu(\()p Fr(y)s Fu(\))p Fr(h)2543 4861 y Fp(i)2574 4847 y Fu(\()p Fr(y)s Fu(\))3477 5025 y(\(3)p Fr(:)p Fu(59\))50 5171 y(In)25 b(the)h(second)h(in)m(tegral)e(w)m(e)i (just)e(use)h(that)h Fr(h)1630 5185 y Fp(i)1660 5171 y Fu(\()p Fr(y)s Fu(\))f Fs(\024)f Fu(1;)j(b)m(y)e(our)g(general)g (assumptions)e(on)i Fr(F)13 b Fu(,)27 b(this)e(giv)m(es)50 5316 y(a)33 b(b)s(ound)d Fr(C)7 b(e)524 5283 y Fj(\000)12 b Fv(max\()p Fp(F)e Fv(\()p Fp(z)877 5256 y Fg(\003)915 5283 y Fv(\()p Fp(x)984 5292 y Fk(i)1014 5283 y Fp(;)p Fj(S)1081 5293 y Fk(k)1120 5283 y Fj(n)p Fp(B)1212 5292 y Fk(i)1241 5283 y Fv(\)\))p Fp(;F)g Fv(\()p Fp(z)1441 5256 y Fg(\003)1479 5283 y Fv(\()p Fp(x)1548 5292 y Fk(j)1583 5283 y Fp(;)p Fj(S)1650 5293 y Fk(k)1689 5283 y Fj(n)p Fp(B)1781 5292 y Fk(j)1815 5283 y Fv(\)\)\))p Fp(=\017)1972 5316 y Fu(.)47 b(In)32 b(the)h(\014rst)e(in)m(tegral)i(w)m(e)g(use)f (the)h(b)s(ounds)d(on)50 5462 y(the)c(equilibrium)20 b(p)s(oten)m(tial)26 b(from)f(Corollary)f(4.8)j(of)33 b([BEGK3].)40 b(Note)27 b(that)f(for)g(an)m(y)g Fr(y)s Fu(,)g(at)h(most)f(one)g(of)p eop %%Page: 23 23 23 22 bop 3633 295 a Fm(23)50 513 y Fu(the)30 b(factors)i Fr(h)555 527 y Fp(i)585 513 y Fu(\()p Fr(y)s Fu(\))f(or)f Fr(h)897 527 y Fp(j)936 513 y Fu(\()p Fr(y)s Fu(\))h(can)g(b)s(e)e (close)i(to)g(one.)41 b(Th)m(us)29 b(ev)m(en)j(the)e(roughest)h (estimate)g(yields)d(that)3655 480 y Fv(6)591 626 y Fi(Z)641 833 y Fp(y)r Fv(:)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fj(\024)p Fv(max)q(\()p Fp(F)g Fv(\()p Fp(z)1194 815 y Fg(\003)1232 833 y Fv(\()p Fp(x)1301 842 y Fk(i)1330 833 y Fp(;)p Fj(S)1397 843 y Fk(k)1436 833 y Fj(n)p Fp(B)1528 842 y Fk(i)1557 833 y Fv(\)\))p Fp(;F)g Fv(\()p Fp(z)1757 815 y Fg(\003)1795 833 y Fv(\()p Fp(x)1864 842 y Fk(j)1899 833 y Fp(;)p Fj(S)1966 843 y Fk(k)2005 833 y Fj(n)p Fp(B)2097 842 y Fk(j)2131 833 y Fv(\)\)\))2236 750 y Fr(dy)s(e)2373 712 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2654 750 y Fr(h)2706 764 y Fp(j)2745 750 y Fu(\()p Fr(y)s Fu(\))p Fr(h)2915 764 y Fp(i)2946 750 y Fu(\()p Fr(y)s Fu(\))601 1011 y Fs(\024)697 887 y Fi(Z)747 1094 y Fp(y)r Fv(:)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fj(\024)p Fv(max)q(\()p Fp(F)g Fv(\()p Fp(z)1300 1075 y Fg(\003)1338 1094 y Fv(\()p Fp(x)1407 1103 y Fk(i)1436 1094 y Fp(;)p Fj(S)1503 1104 y Fk(k)1543 1094 y Fj(n)p Fp(B)1635 1103 y Fk(i)1663 1094 y Fv(\)\))p Fp(;F)g Fv(\()p Fp(z)1863 1075 y Fg(\003)1902 1094 y Fv(\()p Fp(x)1971 1103 y Fk(j)2005 1094 y Fp(;)p Fj(S)2072 1104 y Fk(k)2111 1094 y Fj(n)p Fp(B)2203 1103 y Fk(j)2237 1094 y Fv(\)\)\))2342 1011 y Fr(dy)s(e)2479 973 y Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\))p Fp(=\017)869 1234 y Fs(\002)20 b Fr(C)7 b(\017)1069 1197 y Fj(\000)p Fv(1)p Fp(=)p Fv(2)1239 1234 y Fr(e)1281 1197 y Fj(\000)p Fv([max\()p Fp(F)j Fv(\()p Fp(z)1643 1169 y Fg(\003)1681 1197 y Fv(\()p Fp(x)1750 1206 y Fk(i)1779 1197 y Fp(;)p Fj(S)1846 1207 y Fk(k)1885 1197 y Fj(n)p Fp(B)1977 1206 y Fk(i)2006 1197 y Fv(\)\))p Fp(;F)g Fv(\()p Fp(z)2206 1169 y Fg(\003)2244 1197 y Fv(\()p Fp(x)2313 1206 y Fk(j)2348 1197 y Fp(;)p Fj(S)2415 1207 y Fk(k)2454 1197 y Fj(n)p Fp(B)2546 1206 y Fk(j)2580 1197 y Fv(\)\)\))p Fj(\000)p Fp(F)g Fv(\()p Fp(y)r Fv(\)])p Fp(=\017)601 1407 y Fs(\024)25 b Fr(C)7 b(\017)806 1369 y Fj(\000)p Fv(1)p Fp(=)p Fv(2)976 1407 y Fs(jf)p Fr(y)29 b Fu(:)d Fr(F)13 b Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fu(max\()p Fr(F)13 b Fu(\()p Fr(z)1838 1369 y Fj(\003)1881 1407 y Fu(\()p Fr(x)1968 1421 y Fp(i)1998 1407 y Fr(;)i Fs(S)2093 1421 y Fp(k)2138 1407 y Fs(n)p Fr(B)2252 1421 y Fp(i)2283 1407 y Fu(\)\))p Fr(;)g(F)e Fu(\()p Fr(z)2545 1369 y Fj(\003)2589 1407 y Fu(\()p Fr(x)2676 1421 y Fp(j)2714 1407 y Fr(;)i Fs(S)2809 1421 y Fp(k)2854 1407 y Fs(n)p Fr(B)2968 1421 y Fp(j)3007 1407 y Fu(\)\)\))p Fs(gj)869 1580 y(\002)20 b Fr(e)1002 1542 y Fj(\000)12 b Fv(max\()p Fp(F)e Fv(\()p Fp(z)1355 1515 y Fg(\003)1393 1542 y Fv(\()p Fp(x)1462 1551 y Fk(i)1492 1542 y Fp(;)p Fj(S)1559 1552 y Fk(k)1598 1542 y Fj(n)p Fp(B)1690 1551 y Fk(i)1719 1542 y Fv(\)\))p Fp(;F)g Fv(\()p Fp(z)1919 1515 y Fg(\003)1957 1542 y Fv(\()p Fp(x)2026 1551 y Fk(j)2060 1542 y Fp(;)p Fj(S)2127 1552 y Fk(k)2166 1542 y Fj(n)p Fp(B)2258 1551 y Fk(j)2293 1542 y Fv(\)\)\))p Fp(=\017)3477 1142 y Fu(\(3)p Fr(:)p Fu(60\))50 1762 y(Com)m(bining)24 b(this)h(upp)s(er)g(b)s(ound)f(with)h(the)i(lo)m(w)m (er)f(b)s(ound)e(w)m(e)j(arriv)m(e)f(at)h(the)g(assertion)f(of)h(the)f (lemma.)p Fs(})159 1980 y Fu(W)-8 b(e)31 b(can)g(no)m(w)g(collect)f (the)h(estimates)g(on)f(these)h(matrix)f(elemen)m(ts:)1624 2204 y Fs(j)p Fr(B)1718 2218 y Fp(ij)1782 2204 y Fs(j)25 b(\024)g Fr(e)1970 2166 y Fj(\000)p Fp(\022)r(=\017)3477 2204 y Fu(\(3)p Fr(:)p Fu(61\))1603 2428 y Fs(j)p Fr(D)1703 2442 y Fp(j)t(j)1776 2428 y Fs(j)h(\024)e Fr(C)7 b(\017)2031 2390 y Fp(\013=)p Fv(2)3477 2428 y Fu(\(3)p Fr(:)p Fu(62\))50 2623 y(and)30 b(for)g(all)f Fr(i;)15 b(j)5 b Fu(,)1182 2769 y Fs(j)p Fr(A)1275 2783 y Fp(ij)1339 2769 y Fs(j)25 b(\024)g(jK)1579 2783 y Fp(ij)1644 2769 y Fs(j)p Fr(C)7 b(\017)1778 2731 y Fp(\013=)p Fv(2)1923 2769 y Fu(+)20 b Fr(e)2056 2731 y Fj(\000)p Fp(\016)r(=\017)2221 2691 y Fi(p)p 2312 2691 267 4 v 78 x Fs(K)2381 2783 y Fp(ii)2437 2769 y Fs(K)2506 2783 y Fp(j)t(j)3477 2769 y Fu(\(3)p Fr(:)p Fu(63\))50 2964 y(where)30 b(the)g(last)g(b)s(ound)f(uses)h (\(3.45\))i(together)g(with)d(\(3.46\))k(and)d(\(3.47\))i(and)e(the)h (fact)g(that)50 3036 y Fi(p)p 141 3036 1119 4 v 74 x Fu(cap)292 3124 y Fp(B)347 3133 y Fk(j)386 3110 y Fu(\()p Fs(S)476 3124 y Fp(k)521 3110 y Fs(n)p Fr(B)635 3124 y Fp(j)674 3110 y Fu(\)cap)860 3124 y Fp(B)915 3134 y Fk(l)946 3110 y Fu(\()p Fs(S)1036 3124 y Fp(k)1081 3110 y Fs(n)p Fr(B)1195 3124 y Fp(l)1224 3110 y Fu(\))25 b Fs(\024)g Fr(e)1422 3077 y Fj(\000)p Fp(\016)r(=\017)1617 3110 y Fu(for)30 b(some)h Fr(\016)e(>)c Fu(0.)159 3328 y(W)-8 b(e)31 b(collect)g(the)g(results)e(obtained)h(so)g(far)h(as)50 3546 y Fh(Theorem)51 b(3.6:)78 b Fo(L)-5 b(et)43 b Fs(S)1028 3560 y Fp(k)1116 3546 y Fs(\021)g([)1291 3513 y Fp(k)1291 3570 y(i)p Fv(=1)1413 3546 y Fr(B)1482 3560 y Fp(\017)1517 3546 y Fu(\()p Fr(x)1604 3560 y Fp(i)1634 3546 y Fu(\))g Fo(and)h(let)2040 3522 y Fu(\026)2036 3546 y Fr(\025)2089 3560 y Fp(k)2177 3546 y Fo(denote)f(the)g(princip)-5 b(al)44 b(eigenvalue)e(of)h(the)50 3691 y(op)-5 b(er)g(ator)48 b Fr(L)482 3705 y Fp(\017)562 3691 y Fo(with)e(Dirichlet)f(c)-5 b(onditions)47 b(on)f Fs(S)1808 3705 y Fp(k)1898 3691 y Fo(\(and)g(p)-5 b(ossibly)47 b(an)f(additional)h(set)f Fu(\012)p Fo(\).)79 b(Then)45 b(a)50 3837 y(numb)-5 b(er)44 b Fr(\025)i(<)603 3813 y Fu(\026)600 3837 y Fr(\025)653 3851 y Fp(k)741 3837 y Fo(may)f(b)-5 b(e)43 b(an)i(eigenvalue)e(of)h (the)g(op)-5 b(er)g(ator)47 b Fr(L)2375 3851 y Fp(\017)2410 3837 y Fo(,)f(if)d(ther)-5 b(e)45 b(exists)f(a)g(nonempty)i(set)50 3982 y Fr(J)34 b Fs(\032)25 b(f)p Fu(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(k)s Fs(g)33 b Fo(such)g(that,)h(if)e Fs(G)5 b Fu(\()p Fr(\025)p Fu(\))34 b Fo(denotes)f(the)g Fs(j)p Fr(J)9 b Fs(j)21 b(\002)f(j)p Fr(J)9 b Fs(j)33 b Fo(matrix)h(with)g(elements)1097 4206 y Fs(G)1151 4220 y Fp(ij)1215 4206 y Fu(\()p Fr(\025)p Fu(\))26 b Fs(\021)f(K)1529 4220 y Fp(ij)1614 4206 y Fu(+)20 b Fr(A)1773 4220 y Fp(ij)1857 4206 y Fs(\000)g Fr(\025)p Fu(\()p Fr(\016)2076 4220 y Fp(ij)2161 4206 y Fu(+)g Fr(D)2327 4220 y Fp(ii)2403 4206 y Fu(+)g Fr(B)2563 4220 y Fp(ij)2627 4206 y Fu(\))815 b(\(3)p Fr(:)p Fu(64\))50 4430 y Fr(i;)15 b(j)31 b Fs(2)25 b Fr(J)9 b Fo(,)1599 4575 y Fu(det)15 b(\()q Fs(G)5 b Fu(\()p Fr(\025)p Fu(\)\))26 b(=)f(0)1317 b(\(3)p Fr(:)p Fu(65\))50 4989 y Ft(Remark:)44 b Fu(Note)29 b(that)e(the)h(matrix)f Fs(G)32 b Fu(dep)s(ends)26 b(on)h(the)g(constan)m(ts)i Fr(c)2493 5003 y Fp(i)2523 4989 y Fu(;)g(ho)m(w)m(ev)m(er,)g(this)d(will)f(not)i(b)s(other)50 5135 y(us:)37 b(in)25 b(fact,)i(w)m(e)g(will)22 b(only)j(use)g(Theorem) h(3.6)g(to)h(deriv)m(e)e(conditions)f(on)h Fr(\025)h Fu(uniformly)d(in)h(all)g(symmetric)50 5280 y(matrices)30 b Fr(A;)15 b(B)5 b(;)15 b(D)34 b Fu(satisfying)29 b(the)h(b)s (ounds\(3.61\),)h(\(3.62\))i(and)d(\(3.63\).)p 50 5375 546 4 v 159 5436 a Fn(6)197 5462 y Fm(See)c(the)g(pro)r(of)f(of)g (\(3.79\))g(for)g(more)h(details.)p eop %%Page: 24 24 24 23 bop 50 295 a Fm(24)159 513 y Fu(The)29 b(usefulness)f(of)j(this)e (theorem)h(arises)g(from)f(the)i(fact)g(that)f(w)m(e)h(can)f(con)m (trol)h(the)f(eigen)m(v)-5 b(alues)30 b(to)50 658 y(a)h(v)m(ery)f(go)s (o)s(d)h(precision)d(in)h(terms)h(of)h(the)g Fo(c)-5 b(ap)g(acity)34 b(matrix)p Fu(.)50 876 y Fh(Theorem)45 b(3.7:)68 b Fo(Under)39 b(the)g(same)h(hyp)-5 b(othesis)41 b(as)e(in)g(the)g(pr)-5 b(e)g(c)g(e)g(ding)41 b(the)-5 b(or)g(em,)42 b(if)d Fr(\025)d(<)3376 852 y Fu(\026)3372 876 y Fr(\025)3425 890 y Fp(k)3509 876 y Fo(is)i(an)50 1022 y(eigenvalue)28 b(of)i Fr(L)651 1036 y Fp(\017)685 1022 y Fo(,)g(then)g(for)f(some)h(nonempty)h(set)e Fr(J)34 b Fs(\032)25 b(f)p Fu(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(k)s Fs(g)30 b Fo(ther)-5 b(e)30 b(exists)f(exists)h(an)f(eigenvalue)50 1167 y Fr(\026)f Fo(of)g(the)h Fs(j)p Fr(J)9 b Fs(j)i(\002)g(j)p Fr(J)e Fs(j)p Fo(-matrix)29 b Fs(K)1083 1134 y Fp(J)1145 1167 y Fu(+)11 b Fr(A)1295 1134 y Fp(J)1373 1167 y Fo(\(wher)-5 b(e)30 b(by)e Fr(A)1845 1134 y Fp(J)1924 1167 y Fo(we)g(understand)i (the)f(matrix)g(made)g(of)g(the)f(indic)-5 b(es)50 1313 y Fr(A)118 1327 y Fp(ij)182 1313 y Fo(,)32 b Fr(i;)15 b(j)32 b Fs(2)25 b Fr(J)9 b Fo(,)32 b(etc.\))42 b(such)33 b(that)h Fr(\025)25 b Fu(=)g Fr(\026)1445 1239 y Fi(\000)1487 1313 y Fu(1)20 b(+)g Fr(O)1730 1239 y Fi(\000)1772 1313 y Fr(e)1814 1280 y Fj(\000)p Fp(D)r(=\017)2005 1313 y Fr(;)15 b(\025=)p Fu(\()2182 1289 y(\026)2178 1313 y Fr(\025)2231 1327 y Fp(k)2297 1313 y Fs(\000)20 b Fr(\025)p Fu(\))2476 1239 y Fi(\001\001)2559 1313 y Fo(.)50 1531 y Fe(Pro)s(of:)47 b Fu(The)30 b(pro)s(of)g(will)d(rely)j(on)g(Theorem)g (7.1)i(that)f(w)m(e)g(pro)m(v)m(e)g(in)e(the)h(app)s(endix.)39 b(Since)1134 1774 y Fs(G)5 b Fu(\()p Fr(\025)p Fu(\))26 b(=)f Fs(K)1508 1736 y Fp(J)1580 1774 y Fu(+)20 b Fr(A)1739 1736 y Fp(J)1810 1774 y Fs(\000)g Fr(\025)p Fu(\(1)-23 b(I)21 b(+)f(\()p Fr(B)2265 1736 y Fp(J)2336 1774 y Fu(+)g Fr(D)2505 1736 y Fp(J)2555 1774 y Fu(\)\))817 b(\(3)p Fr(:)p Fu(66\))r(\))50 2017 y(to)42 b(apply)d(Theorem)i(7.1)h(requires) d(us)i(to)g(b)s(ound)e(the)j(norm)e(of)h Fr(B)2480 1984 y Fp(J)2557 2017 y Fu(+)27 b Fr(D)2733 1984 y Fp(J)2784 2017 y Fu(.)73 b(As)40 b(a)i(consequence)g(of)50 2162 y(estimates)f(\(3.54\))i(and)d(\(3.66\),)45 b(the)c(preceding)f (Lemmata,)k(w)m(e)d(see)g(that)g(the)g(matrix)f Fr(B)3305 2129 y Fp(J)3382 2162 y Fu(+)27 b Fr(D)3558 2129 y Fp(J)3649 2162 y Fu(is)50 2307 y(indeed)i(b)s(ounded)f(in)h(norm)h(b)m(y)1316 2550 y Fs(k)p Fr(B)1435 2513 y Fp(J)1506 2550 y Fu(+)20 b Fr(D)1675 2513 y Fp(J)1725 2550 y Fs(k)26 b(\024)f Fr(C)7 b(\017)2001 2513 y Fp(\013=)p Fv(2)2146 2550 y Fu(+)20 b Fr(e)2279 2513 y Fj(\000)p Fp(\016)r(=\017)3477 2550 y Fu(\(3)p Fr(:)p Fu(67\))50 2793 y(for)30 b(some)h Fr(\016)e(>)c Fu(0.)41 b(The)30 b(theorem)h(follo)m(ws)e(no)m(w)h(from) g(Theorem)g(7.1)i(of)e(the)h(app)s(endix.)p Fs(})159 3011 y Fu(It)e(remains)e(to)j(estimate)g(the)f(eigen)m(v)-5 b(alues)28 b(of)h(the)g(matrix)g Fs(K)2311 2978 y Fp(J)2379 3011 y Fu(+)17 b Fs(A)2540 2978 y Fp(J)2590 3011 y Fu(.)40 b(W)-8 b(e)30 b(will)d(do)h(this)g(only)g(in)g(the)50 3157 y(non-degenerate)38 b(situation)d(when)h(all)f(\\depths")h(of)h (the)g(v)-5 b(alleys)36 b Fr(x)2464 3171 y Fp(i)2530 3157 y Fu(are)h(distinct,)g(i.e.)59 b(when)36 b(for)g(all)50 3302 y Fr(i)25 b(<)g(k)34 b Fu(the)c(inequalities)e(\(3.38\))33 b(are)e(strict.)159 3520 y(Let)g(us)e(\014rst)h(consider)f(the)i(case)g Fr(J)k Fu(=)25 b Fs(f)p Fu(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(n)p Fs(g)p Fu(.)50 3739 y Fh(Lemma)38 b(3.8:)51 b Fo(L)-5 b(et)33 b Fs(K)911 3753 y Fp(ij)1007 3739 y Fo(b)-5 b(e)33 b(the)g(normalize)-5 b(d)35 b(c)-5 b(ap)g(acity)35 b(matrix)f(and)f (assume)g(that)1484 3981 y Fu(max)1507 4041 y Fp(i)c Fu(0)32 b Fo(\(uniformly)g(on)g (c)-5 b(omp)g(act)33 b(subsets)f(if)f Fu(\012)f Fo(is)i(unb)-5 b(ounde)g(d\))33 b(.)50 4825 y Fe(Pro)s(of:)51 b Fu(It)31 b(is)g(easy)i(to)f(see)h(that)f(if)f Fr(k)g Fs(2)c Fr(J)9 b Fu(,)33 b(then)e(Lemma)h(3.8)h(together)g(with)e(the)h(b)s(ounds)d (on)j Fr(B)k Fu(and)50 4970 y Fr(D)h Fu(implies)32 b(that)k(a)f (solution)e(of)i(\(3.56\))i(with)f(^)-47 b Fr(c)1731 4984 y Fp(k)1809 4970 y Fu(=)32 b(1)j(m)m(ust)g(satisfy)f Fs(j)r Fu(^)-47 b Fr(c)2574 4984 y Fp(j)2613 4970 y Fs(j)33 b(\024)f Fr(e)2816 4937 y Fj(\000)p Fp(\016)r(=\017)3015 4970 y Fu(for)j(all)f Fr(j)k Fs(6)p Fu(=)32 b Fr(k)s Fu(.)54 b(By)50 5115 y(\(3.42\),)33 b(this)c(implies)f(that)932 5360 y Fr(\036)986 5323 y Fp(\025)986 5383 y(k)1034 5360 y Fu(\()p Fr(y)s Fu(\))e(=)1284 5299 y Fr(h)1336 5266 y Fp(\025)1336 5325 y(k)1385 5299 y Fu(\()p Fr(y)s Fu(\))20 b(+)g Fr(\036)1668 5266 y Fp(\025)1668 5325 y(k)1716 5299 y Fu(\()p Fr(y)s Fu(\))p 1284 5339 551 4 v 1445 5423 a Fs(k)p Fr(h)1542 5437 y Fp(k)1588 5423 y Fs(k)1633 5437 y Fv(2)1866 5360 y Fu(+)1994 5274 y Fi(X)1957 5473 y Fp(j)t Fj(2)p Fp(J)6 b Fj(n)p Fp(k)2180 5360 y Fu(^)-47 b Fr(c)2217 5374 y Fp(j)2267 5289 y Fr(h)2319 5256 y Fp(\025)2319 5312 y(j)2367 5289 y Fu(\()p Fr(y)s Fu(\))20 b(+)g Fr(\036)2650 5256 y Fp(\025)2650 5315 y(k)2698 5289 y Fu(\()p Fr(y)s Fu(\))p 2267 5339 V 2431 5423 a Fs(k)p Fr(h)2528 5437 y Fp(j)2567 5423 y Fs(k)2612 5437 y Fv(2)3477 5360 y Fu(\(3)p Fr(:)p Fu(76\))p eop %%Page: 26 26 26 25 bop 50 295 a Fm(26)50 513 y Fu(Using)29 b(the)i(same)f(argumen)m (ts)h(as)f(in)f(the)h(pro)s(of)g(of)g(Prop)s(osition)e(3.3,)k(and)d (the)i(b)s(ounds)c(on)k Fr(\036)3336 480 y Fp(\025)3403 513 y Fs(\000)20 b Fr(c)3533 527 y Fp(J)3614 513 y Fu(on)50 658 y(the)30 b(b)s(oundaries)e Fr(@)5 b(B)791 672 y Fp(j)830 658 y Fu(,)30 b(w)m(e)h(get)h(that)f(for)f Fr(j)h Fs(2)25 b Fr(J)61 857 y Fs(j)p Fr(\036)140 824 y Fp(\025)140 881 y(j)188 857 y Fu(\()p Fr(y)s Fu(\))c Fs(\000)e Fr(h)469 871 y Fp(j)508 857 y Fu(\()p Fr(y)s Fu(\))p Fs(j)p 61 908 592 4 v 245 991 a(k)p Fr(h)342 1005 y Fp(j)381 991 y Fs(k)426 1005 y Fv(2)688 929 y Fs(\024)25 b Fr(C)7 b(\017)893 891 y Fp(\013=)p Fv(2)1035 867 y Fr(h)1087 881 y Fp(j)1126 867 y Fu(\()p Fr(y)s Fu(\))p 1029 908 223 4 v 1029 991 a Fs(k)p Fr(h)1126 1005 y Fp(j)1165 991 y Fs(k)1210 1005 y Fv(2)1277 929 y Fu(+)1371 842 y Fi(X)1363 1038 y Fp(l)p Fj(2)p Fp(J)1475 1047 y Fk(j)1536 867 y Fu(cap)1687 881 y Fp(B)1742 891 y Fk(l)1773 867 y Fu(\()p Fr(B)1877 881 y Fp(j)1916 867 y Fu(\))p Fs(k)p Fr(h)2048 881 y Fp(l)2077 867 y Fs(k)2122 881 y Fv(2)p 1536 908 628 4 v 1738 991 a Fs(k)p Fr(h)1835 1005 y Fp(j)1874 991 y Fs(k)1919 1005 y Fv(2)2191 867 y Fr(h)2243 881 y Fp(l)2272 867 y Fu(\()p Fr(y)s Fu(\))p 2185 908 212 4 v 2185 991 a Fs(k)p Fr(h)2282 1005 y Fp(l)2310 991 y Fs(k)2355 1005 y Fv(2)2433 929 y Fs(\024)25 b Fr(C)7 b(\017)2638 891 y Fp(\013=)p Fv(2)2780 867 y Fr(h)2832 881 y Fp(j)2871 867 y Fu(\()p Fr(y)s Fu(\))p 2773 908 223 4 v 2773 991 a Fs(k)p Fr(h)2870 1005 y Fp(j)2910 991 y Fs(k)2955 1005 y Fv(2)3022 929 y Fu(+)3115 842 y Fi(X)3108 1038 y Fp(l)p Fj(2)p Fp(J)3220 1047 y Fk(j)3269 929 y Fr(e)3311 891 y Fj(\000)p Fp(\016)r(=\017)3494 867 y Fr(h)3546 881 y Fp(l)3574 867 y Fu(\()p Fr(y)s Fu(\))p 3487 908 212 4 v 3487 991 a Fs(k)p Fr(h)3584 1005 y Fp(l)3613 991 y Fs(k)3658 1005 y Fv(2)3477 1142 y Fu(\(3)p Fr(:)p Fu(77\))50 1288 y(Com)m(bining)22 b(these)j (estimates)g(w)m(e)g(arriv)m(e)f(at)i(\(3.75\).)41 b(Note)25 b(that)g(this)f(\014nal)f(estimate)i(do)s(es)f(not)h(actually)50 1433 y(dep)s(end)37 b(on)i(the)h(c)m(hoice)g(of)f Fr(J)9 b Fu(.)68 b(Since)38 b(it)h(is)f(imp)s(ossible)e(that)k(t)m(w)m(o)g (functions)e(satisfying)g(\(3.75\))j(are)50 1579 y(orthogonal,)31 b(it)f(follo)m(ws)f(that)i Fr(\025)1169 1593 y Fp(k)1244 1579 y Fu(is)f(a)g(simple)f(eigen)m(v)-5 b(alue.)40 b Fs(})159 1797 y Fu(No)m(w)29 b(w)m(e)g(can)g(further)f(explore)g(the)h (eigen)m(v)-5 b(alues)28 b(b)s(elo)m(w)2163 1773 y(\026)2159 1797 y Fr(\025)2212 1811 y Fp(k)r Fj(\000)p Fv(1)2350 1797 y Fu(,)h(etc.,)i(with)c(the)i(same)g(results.)39 b(Th)m(us)50 1942 y(at)26 b(the)g(end)f(of)h(the)g(pro)s(cedure)e(w)m (e)i(arriv)m(e)g(at)g(the)g(conclusion)e(that)i Fr(L)2473 1956 y Fp(\017)2534 1942 y Fu(can)g(ha)m(v)m(e)g(at)h(most)f(the)g Fr(n)f Fu(simple)50 2088 y(eigen)m(v)-5 b(alues)29 b(giv)m(en)g(b)m(y)g (the)h(v)-5 b(alues)28 b(of)i(the)f(preceding)g(corollary)f(b)s(elo)m (w)h(the)g(v)-5 b(alues)29 b Fr(C)7 b(\017)3138 2055 y Fp(d)p Fj(\000)p Fv(1)3273 2088 y Fu(.)40 b(But)29 b(since)50 2233 y(w)m(e)h(kno)m(w)f(that)h(there)f(m)m(ust)h(b)s(e)e Fr(n)h Fu(suc)m(h)g(eigen)m(v)-5 b(alues,)30 b(w)m(e)g(conclude)e(that) i(all)f(these)g(candidates)g(are)h(in)50 2379 y(fact)h(eigen)m(v)-5 b(alues.)41 b(This)28 b(yields)h(the)h(follo)m(wing)f(prop)s(osition:) 50 2597 y Fh(Prop)s(osition)34 b(3.10:)39 b Fo(Assume)29 b(that)i(al)5 b(l)30 b(ine)-5 b(qualities)29 b(\(3.38\))j(ar)-5 b(e)30 b(strict)g(for)g(al)5 b(l)30 b Fr(i)c Fu(=)f(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(n)p Fo(.)40 b(Then)50 2742 y(the)33 b(sp)-5 b(e)g(ctrum)34 b(of)f Fr(L)752 2756 y Fp(\017)819 2742 y Fo(b)-5 b(elow)34 b Fr(\017)1098 2709 y Fp(d)p Fj(\000)p Fv(1)1266 2742 y Fo(c)-5 b(onsists)33 b(of)g Fr(n)f Fo(simple)i(eigenvalues)e(that)i(satisfy:)276 2986 y Fr(\025)329 3000 y Fp(k)399 2986 y Fu(=)505 2924 y(cap)657 2938 y Fp(B)712 2948 y Fk(k)756 2924 y Fu(\()p Fs(S)846 2938 y Fp(k)r Fj(\000)p Fv(1)984 2924 y Fu(\))p 505 2965 515 4 v 648 3048 a Fs(k)p Fr(h)745 3062 y Fp(k)791 3048 y Fs(k)836 3017 y Fv(2)836 3072 y(2)1031 2986 y Fu(\(1)21 b(+)f Fr(O)s Fu(\()p Fr(\017)1367 2948 y Fp(\013=)p Fv(2)1512 2986 y Fu(+)g Fr(e)1645 2948 y Fj(\000)p Fp(\016)r(=\017)1810 2986 y Fu(\)\))93 b Fr(;)15 b(k)29 b Fu(=)c(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(n)399 3257 y Fu(=)25 b(cap)646 3271 y Fp(B)701 3281 y Fk(k)745 3257 y Fu(\()p Fs(S)835 3271 y Fp(k)r Fj(\000)p Fv(1)973 3257 y Fu(\))1020 3116 y Fi(p)p 1111 3116 552 4 v 78 x Fu(det\()p Fs(r)1348 3167 y Fv(2)1389 3194 y Fr(F)13 b Fu(\()p Fr(x)1547 3208 y Fp(k)1592 3194 y Fu(\)\))p 1020 3236 643 4 v 1213 3276 a Fs(p)p 1289 3276 138 4 v 75 x Fu(2)p Fr(\031)s(\017)1427 3291 y Fp(d)1674 3257 y Fr(e)1716 3219 y Fp(F)d Fv(\()p Fp(x)1841 3229 y Fk(k)1881 3219 y Fv(\))p Fp(=\017)1997 3156 y Fi(\020)2051 3257 y Fu(1)21 b(+)f Fr(O)2295 3156 y Fi(\020)2349 3257 y Fr(\017)2386 3219 y Fv(1)p Fp(=)p Fv(2)2500 3257 y Fs(j)15 b Fu(ln)f Fr(\017)p Fs(j)p Fr(;)h(\017)2770 3219 y Fp(\013=)p Fv(2)2896 3257 y Fr(;)g(e)2978 3219 y Fj(\000)p Fp(\016)r(=\017)3143 3156 y Fi(\021\021)399 3519 y Fu(=)660 3458 y(1)p 505 3499 356 4 v 505 3582 a Fq(E)560 3596 y Fp(x)601 3606 y Fk(k)652 3582 y Fr(\034)692 3596 y Fj(S)737 3606 y Fk(k)q Fg(\000)p Fl(1)871 3519 y Fu(\(1)21 b(+)f Fr(O)s Fu(\()p Fr(\017)1207 3482 y Fp(\013=)p Fv(2)1353 3519 y Fu(+)g Fr(e)1486 3482 y Fj(\000)p Fp(\016)r(=\017)1650 3519 y Fu(\)\))3477 3257 y(\(3)p Fr(:)p Fu(78\))50 3777 y Fo(The)33 b(c)-5 b(orr)g(esp)g(onding)36 b(eigenfunctions)c(satisfy)h(\(3.75\).)50 3995 y Fe(Pro)s(of:)61 b Fu(W)-8 b(e)38 b(ha)m(v)m(e)g(seen)e(in)f(fact)j(that)f Fr(\025)1529 4009 y Fp(k)1609 3995 y Fu(=)e Fs(K)1785 3948 y Fv(\()p Fp(k)r Fv(\))1784 4022 y Fp(k)r(k)1902 3921 y Fi(\000)1944 3995 y Fu(1)21 b(+)e Fr(O)s Fu(\()p Fr(e)2249 3962 y Fj(\000)p Fp(\022)r(=\017)2415 3995 y Fr(;)c(\017)2492 3962 y Fp(\013=)p Fv(2)2618 3995 y Fu(\))2653 3921 y Fi(\001)2695 3995 y Fu(,)38 b(whic)m(h)d(pro)m(vides) h(the)g(\014rst)50 4140 y(assertion)f(of)h(Prop)s(osition)d(3.10.)58 b(It)35 b(remains)f(to)j(iden)m(tify)d(the)h(eigen)m(v)-5 b(alues)36 b(with)e(the)h(in)m(v)m(erse)h(mean)50 4286 y(times.)k(This)29 b(follo)m(ws)g(from)h(Prop)s(osition)e(6.1)k(in)d ([BEGK3],)j(pro)m(vided)d(w)m(e)i(can)f(sho)m(w)h(that)1079 4404 y Fi(Z)1185 4528 y Fr(dy)s(e)1322 4491 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1604 4528 y Fr(h)1656 4491 y Fv(2)1656 4551 y Fp(k)1701 4528 y Fu(\()p Fr(y)s Fu(\))25 b Fs(\030)1940 4404 y Fi(Z)2046 4528 y Fr(dy)s(e)2183 4491 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2465 4528 y Fr(h)2517 4542 y Fp(k)2562 4528 y Fu(\()p Fr(y)s Fu(\))797 b(\(3)p Fr(:)p Fu(79\))50 4770 y(In)29 b(fact,)j(w)m(e)f(will)d(sho)m(w)i(more,)h(namely)e(that)i (b)s(oth)f(sides)f(of)i(\(3.79\))h(are)f(asymptotically)f(equal)f(to) 1365 5055 y Fr(e)1407 5017 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(x)1589 5027 y Fk(k)1629 5017 y Fv(\))p Fp(=\017)1934 4918 y Fs(p)p 2010 4918 138 4 v 75 x Fu(2)p Fr(\031)s(\017)2148 4933 y Fp(d)p 1741 5034 643 4 v 1741 5052 a Fi(p)p 1832 5052 552 4 v 78 x Fu(det\()p Fs(r)2069 5103 y Fv(2)2110 5130 y Fr(F)j Fu(\()p Fr(x)2268 5144 y Fp(k)2313 5130 y Fu(\)\))3477 5055 y(\(3)p Fr(:)p Fu(80\))50 5316 y(W)-8 b(e)31 b(m)m(ust)e(sho)m(w)h(that)g(the)g(main)e(con)m(tribution)h(of)g (the)h(in)m(tegrals)f(comes)i(from)e(a)h(small)e(neigh)m(b)s(orho)s(o)s (d)50 5462 y(of)39 b Fr(x)214 5476 y Fp(k)259 5462 y Fu(,)i(whic)m(h)d(yields)f(the)i(con)m(tribution)f(\(3.80\).)68 b(It)40 b(is)e(clear)h(that)g(all)f(con)m(tributions)f(from)i(the)g (set)p eop %%Page: 27 27 27 26 bop 3633 295 a Fm(27)50 513 y Fr(y)42 b Fu(:)d Fr(F)13 b Fu(\()p Fr(y)s Fu(\))40 b Fr(>)f(F)13 b Fu(\()p Fr(x)698 527 y Fp(k)743 513 y Fu(\))26 b(+)g Fr(\017)p Fs(j)15 b Fu(ln)f Fr(\017)p Fs(j)39 b Fu(giv)m(e)g(only)f(sub-leading)f (corrections.)66 b(T)-8 b(o)39 b(treat)h(the)f(complemen)m(t)g(of)50 658 y(this)34 b(set,)j(w)m(e)e(use)g(the)g(b)s(ounds)e(on)i(the)h (equilibrium)30 b(p)s(oten)m(tial)k(of)i(Eq.)54 b(\(4.27\))38 b(in)33 b([BEGK3].)56 b(Up)35 b(to)50 804 y(p)s(olynomial)d(factors)37 b(in)d Fr(\017)p Fu(,)i(it)f(implies)e(that)j(the)f(in)m(tegrand)g(on)g (the)h(righ)m(t-hand)e(side)g(of)h(\(3.79\))j(\(and)50 949 y(a)g(fortiori)f(on)g(the)h(left-hand)f(side\))g(in)g(the)h (connected)h(comp)s(onen)m(ts)f(of)g(this)e(lev)m(el)i(set)g(that)h(do) e(not)50 1095 y(con)m(tain)31 b Fr(x)423 1109 y Fp(k)498 1095 y Fu(is)e(smaller)g(than)1206 1334 y Fr(e)1248 1296 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\)+)p Fp(F)g Fv(\()p Fp(z)1655 1269 y Fg(\003)1693 1296 y Fv(\()p Fp(y)r(;B)1837 1306 y Fk(k)1877 1296 y Fv(\)\))p Fj(\000)p Fp(F)g Fv(\()p Fp(z)2112 1269 y Fg(\003)2150 1296 y Fv(\()p Fp(y)r(;)p Fj(S)2284 1306 y Fk(k)q Fg(\000)p Fl(1)2404 1296 y Fv(\)\)])p Fp(=\017)3477 1334 y Fu(\(3)p Fr(:)p Fu(81\))50 1573 y(If)34 b Fr(y)k Fu(is)d(in)e(the)j(comp)s(onen)m(t)f (of)g(the)h(lev)m(el)f(set)g(that)h(con)m(tains)f(the)h(minim)m(um)c Fr(x)2882 1587 y Fp(j)2920 1573 y Fu(,)37 b(and)d Fr(j)39 b(<)33 b(k)s Fu(,)k(w)m(e)e(see)50 1718 y(that)c(this)e(is)g(equal)h (to)1576 1864 y Fr(e)1618 1826 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(z)1797 1799 y Fg(\003)1835 1826 y Fv(\()p Fp(x)1904 1835 y Fk(j)1938 1826 y Fp(;B)2015 1836 y Fk(k)2055 1826 y Fv(\)\))p Fp(=\017)3477 1864 y Fu(\(3)p Fr(:)p Fu(82\))50 2066 y(whic)m(h)31 b(is)h(exp)s(onen)m(tially)e(smaller)h (than)i(exp)o(\()p Fs(\000)p Fr(F)13 b Fu(\()p Fr(x)1901 2080 y Fp(k)1947 2066 y Fu(\))p Fr(=\017)p Fu(\),)34 b(indep)s(enden)m(t)c(of)j Fr(y)s Fu(.)47 b(If)32 b Fr(j)i(>)29 b(k)s Fu(,)k(w)m(e)g(still)e(get)50 2211 y(the)f(same)h(result)e(if)h Fr(F)13 b Fu(\()p Fr(y)s Fu(\))26 b Fs(\025)f Fr(F)13 b Fu(\()p Fr(z)1234 2178 y Fj(\003)1276 2211 y Fu(\()p Fr(x)1363 2225 y Fp(j)1402 2211 y Fr(;)i Fs(S)1497 2225 y Fp(k)r Fj(\000)p Fv(1)1635 2211 y Fu(\)\).)41 b(Otherwise,)30 b(w)m(e)g(can)h(write)f(\(3.81\))i(as)540 2451 y Fr(e)582 2413 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)993 2422 y Fk(j)1028 2413 y Fv(\)])p Fp(=\017)1149 2451 y Fr(e)1191 2413 y Fj(\000)p Fv([)p Fp(F)g Fv(\()p Fp(x)1394 2423 y Fk(k)1434 2413 y Fv(\)+\()p Fp(F)g Fv(\()p Fp(z)1668 2386 y Fg(\003)1707 2413 y Fv(\()p Fp(x)1776 2422 y Fk(j)1810 2413 y Fp(;B)1887 2423 y Fk(k)1926 2413 y Fv(\)\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)2164 2423 y Fk(k)2205 2413 y Fv(\)\))p Fj(\000)p Fv(\()p Fp(F)g Fv(\()p Fp(z)2468 2386 y Fg(\003)2507 2413 y Fv(\()p Fp(x)2576 2422 y Fk(j)2611 2413 y Fp(;)p Fj(S)2678 2423 y Fk(k)q Fg(\000)p Fl(1)2797 2413 y Fv(\)\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)3035 2422 y Fk(j)3070 2413 y Fv(\)\)])p Fp(=\017)3477 2451 y Fu(\(3)p Fr(:)p Fu(83\))50 2690 y(W)-8 b(e)31 b(will)d(argue)j(that)888 2929 y Fr(F)13 b Fu(\()p Fr(z)1040 2891 y Fj(\003)1083 2929 y Fu(\()p Fr(x)1170 2943 y Fp(j)1208 2929 y Fr(;)i(B)1317 2943 y Fp(k)1363 2929 y Fu(\)\))20 b Fs(\000)g Fr(F)13 b Fu(\()p Fr(x)1702 2943 y Fp(k)1747 2929 y Fu(\))26 b Fr(>)f(F)13 b Fu(\()p Fr(z)2056 2891 y Fj(\003)2099 2929 y Fu(\()p Fr(x)2186 2943 y Fp(j)2224 2929 y Fr(;)i Fs(S)2319 2943 y Fp(k)r Fj(\000)p Fv(1)2457 2929 y Fu(\)\))21 b Fs(\000)f Fr(F)13 b Fu(\()p Fr(x)2797 2943 y Fp(j)2836 2929 y Fu(\))606 b(\(3)p Fr(:)p Fu(84\))50 3168 y(Assume)30 b(the)g(con)m(trary)-8 b(.)42 b(Note)32 b(that)f(trivially)1198 3407 y Fr(F)13 b Fu(\()p Fr(z)1350 3370 y Fj(\003)1393 3407 y Fu(\()p Fr(x)1480 3421 y Fp(j)1518 3407 y Fr(;)i Fs(S)1613 3421 y Fp(k)r Fj(\000)p Fv(1)1752 3407 y Fu(\)\))26 b Fs(\025)f Fr(F)13 b Fu(\()p Fr(z)2096 3370 y Fj(\003)2138 3407 y Fu(\()p Fr(x)2225 3421 y Fp(j)2264 3407 y Fr(;)i Fs(S)2359 3421 y Fp(j)t Fj(\000)p Fv(1)2491 3407 y Fu(\)\))916 b(\(3)p Fr(:)p Fu(85\))50 3646 y(while)869 3792 y Fr(F)13 b Fu(\()p Fr(z)1021 3754 y Fj(\003)1063 3792 y Fu(\()p Fr(x)1150 3806 y Fp(j)1189 3792 y Fr(;)i(B)1298 3806 y Fp(k)1343 3792 y Fu(\)\))26 b(=)f Fr(F)13 b Fu(\()p Fr(z)1687 3754 y Fj(\003)1729 3792 y Fu(\()p Fr(x)1816 3806 y Fp(k)1861 3792 y Fr(;)i(B)1970 3806 y Fp(j)2009 3792 y Fu(\)\))26 b Fs(\024)f Fr(F)13 b Fu(\()p Fr(z)2353 3754 y Fj(\003)2395 3792 y Fu(\()p Fr(x)2482 3806 y Fp(k)2527 3792 y Fr(;)i Fs(S)2622 3806 y Fp(j)2661 3792 y Fs(n)p Fr(B)2775 3806 y Fp(k)2820 3792 y Fu(\)\))587 b(\(3)p Fr(:)p Fu(86\))50 3994 y(Therefore,)30 b(our)g(assumption)f(implies)e (that)819 4233 y Fr(F)13 b Fu(\()p Fr(z)971 4196 y Fj(\003)1013 4233 y Fu(\()p Fr(x)1100 4247 y Fp(j)1139 4233 y Fr(;)i Fs(S)1234 4247 y Fp(j)t Fj(\000)p Fv(1)1366 4233 y Fu(\)\))21 b Fs(\000)f Fr(F)13 b Fu(\()p Fr(x)1706 4247 y Fp(j)1744 4233 y Fu(\))26 b Fs(\024)f Fr(F)13 b Fu(\()p Fr(z)2053 4196 y Fj(\003)2095 4233 y Fu(\()p Fr(x)2182 4247 y Fp(k)2227 4233 y Fr(;)i Fs(S)2322 4247 y Fp(j)2361 4233 y Fs(n)p Fr(B)2475 4247 y Fp(k)2520 4233 y Fu(\)\))21 b Fs(\000)f Fr(F)13 b Fu(\()p Fr(x)2860 4247 y Fp(k)2905 4233 y Fu(\))537 b(\(3)p Fr(:)p Fu(87\))50 4472 y(whic)m(h)30 b(a)i(momen)m(ts)f (re\015ection)g(sho)m(ws)g(to)i(b)s(e)d(in)g(con)m(tradiction)h(with)f (the)i(conditions)e(\(3.38\))j(at)f(stage)50 4618 y Fr(j)5 b Fu(.)41 b(In)30 b(other)h(w)m(ords,)f(if)g(our)g(assumption)e(w)m(as) j(true,)g(then)f(the)h(set)g Fr(B)2490 4632 y Fp(k)2565 4618 y Fu(w)m(ould)e(ha)m(v)m(e)j(had)e(to)h(yields)e(the)50 4763 y(largest)d(eigen)m(v)-5 b(alue)26 b(at)h(stage)g Fr(j)5 b Fu(,)28 b(i.e.)39 b(it)26 b(w)m(ould)e(ha)m(v)m(e)j(had)f(to)h (b)s(e)e(lab)s(elled)e Fr(B)2705 4777 y Fp(j)2744 4763 y Fu(.)39 b(Th)m(us)25 b(\(3.84\))j(m)m(ust)e(hold.)159 4981 y(Since)36 b(b)m(y)g(assumption)f(the)i(inequalities)d(are)j (strict)g(\(whic)m(h)f(is)f(more)i(than)g(w)m(e)g(need\),)h(it)f(follo) m(ws)50 5127 y(that)31 b(indeed)524 5272 y Fi(Z)630 5395 y Fr(dy)s(e)767 5358 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1048 5395 y Fr(h)1100 5409 y Fp(k)1146 5395 y Fu(\()p Fr(y)s Fu(\))25 b(=)g Fr(e)1427 5358 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(x)1609 5368 y Fk(k)1650 5358 y Fv(\))p Fp(=\017)1955 5259 y Fs(p)p 2031 5259 138 4 v 75 x Fu(2)p Fr(\031)s(\017)2168 5274 y Fp(d)p 1761 5375 643 4 v 1761 5393 a Fi(p)p 1852 5393 552 4 v 77 x Fu(det\()p Fs(r)2089 5444 y Fv(2)2130 5470 y Fr(F)j Fu(\()p Fr(x)2288 5484 y Fp(k)2333 5470 y Fu(\)\))2430 5295 y Fi(\020)2484 5395 y Fu(1)21 b(+)f Fr(O)2728 5295 y Fi(\020)2782 5395 y Fr(\017)2819 5358 y Fv(1)p Fp(=)p Fv(2)2933 5395 y Fs(j)15 b Fu(ln)g Fr(\017)p Fs(j)3127 5295 y Fi(\021\021)3477 5395 y Fu(\(3)p Fr(:)p Fu(88\))p eop %%Page: 28 28 28 27 bop 50 295 a Fm(28)50 513 y Fu(and)30 b(of)h(course)g(the)g(same) h(b)s(ound)c(holds)i(when)g Fr(h)1805 527 y Fp(k)1881 513 y Fu(is)g(replaced)g(b)m(y)h Fr(h)2511 480 y Fv(2)2511 538 y Fp(k)2556 513 y Fu(.)42 b(This)29 b(concludes)h(the)h(pro)s(of)g (of)50 658 y(the)f(theorem.)p Fs(})50 876 y Ft(Impro)m(v)m(ed)j(error)h (estimates.)k Fu(T)-8 b(o)30 b(conclude)e(the)i(pro)s(ofs)e(of)h (Theorems)g(1.1)h(and)f(1.2)h(w)m(e)g(only)e(need)50 1022 y(to)35 b(impro)m(v)m(e)g(the)g(error)f(estimates.)54 b(In)34 b(the)h(pro)s(ofs)f(of)h(this)f(section)g(w)m(e)i(ha)m(v)m(e)f (pro)s(duced)f(error)g(terms)50 1167 y(from)42 b(t)m(w)m(o)j(sources:) 66 b(the)43 b(exp)s(onen)m(tially)f(small)f(errors)i(resulting)e(from)h (the)i(p)s(erturbation)d(around)50 1313 y Fr(\025)j Fu(=)g(0)e(and)f (the)h(not)f(p)s(erfect)h(orthogonalit)m(y)g(of)g(the)g(functions)e Fr(h)2502 1327 y Fp(i)2533 1313 y Fu(,)k(and)d(the)h(m)m(uc)m(h)g (larger)f(errors)50 1458 y(of)35 b(order)f Fr(\017)437 1425 y Fp(\013=)p Fv(2)596 1458 y Fu(that)h(resulted)f(from)g(the)h(a)g (priori)d(con)m(trol)j(on)f(the)h(regularit)m(y)f(of)g(the)h (eigenfunctions)50 1604 y(obtained)d(from)g(the)g(H\177)-45 b(older)32 b(estimate)h(of)g(Lemma)g(3.1.)48 b(In)31 b(the)i(ligh)m(t)f(of)g(the)h(estimates)g(obtained)f(on)50 1749 y(the)i(eigenfunctions)f(these)i(can)g(no)m(w)f(b)s(e)g(impro)m(v) m(ed)f(successiv)m(ely)h(\(as)h(in)e(the)i(pro)s(of)e(of)i(Theorem)f (3.1)50 1895 y(of)k([BEGK3]\).)46 b(Notice)32 b(\014rst)f(that)h(the)f (eigenfunction)f(corresp)s(onding)g(to)i(the)f(minim)m(um)e Fr(x)3334 1909 y Fp(k)3410 1895 y Fu(is)h(small)50 2040 y(enough)c(at)h(all)e(the)i(minima)d Fr(x)1117 2054 y Fp(l)1145 2040 y Fu(,)j Fr(l)g(<)e(k)30 b Fu(that)d(w)m(e)g(can)f (actually)g(tak)m(e)i Fr(J)34 b Fu(=)25 b Fs(f)p Fr(k)s Fs(g)j Fu(and)d Fr(J)3014 2054 y Fp(k)3085 2040 y Fu(=)g Fs(f)p Fu(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(k)g Fs(\000)d Fu(1)p Fs(g)50 2185 y Fu(in)29 b(\(3.43\),)k(\(3.44\).)43 b(Then)29 b(w)m(e)i(kno)m(w)f(from)g(Corollary)f(3.9)j(that)1013 2418 y(osc)1134 2434 y Fp(y)r Fj(2)p Fp(B)1277 2451 y Fl(4)1308 2417 y Fg(p)p 1358 2417 28 3 v 34 x Fk(\017)1391 2434 y Fv(\()p Fp(x)1460 2444 y Fk(k)1500 2434 y Fv(\))1533 2418 y Fr(\036)1587 2432 y Fp(k)1632 2418 y Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fr(C)7 b(\017)1981 2380 y Fp(\013=)p Fv(2)2249 2418 y Fu(sup)2120 2498 y Fp(y)r Fj(2)p Fp(B)2263 2515 y Fl(4)2294 2481 y Fg(p)p 2344 2481 V 34 x Fk(\017)2377 2498 y Fv(\()p Fp(x)2446 2508 y Fk(k)2486 2498 y Fv(\))2530 2418 y Fr(\036)2584 2432 y Fp(k)2629 2418 y Fu(\()p Fr(y)s Fu(\))730 b(\(3)p Fr(:)p Fu(89\))50 2698 y(whic)m(h)23 b(impro)m(v)m(es)i(the)g(a)g(priori)d(estimate)k(\(3.5\).)41 b(Then)24 b(the)h(H\177)-45 b(older)24 b(estimate)h(Lemma)g(4.1)h(in)e ([BEGK3])50 2843 y(giv)m(es)31 b(the)f(impro)m(v)m(emen)m(t)157 3075 y(osc)279 3092 y Fp(y)r Fj(2)p Fp(B)422 3101 y Fk(\017)455 3092 y Fv(\()p Fp(x)524 3102 y Fk(k)564 3092 y Fv(\))597 3075 y Fr(\036)651 3089 y Fp(k)696 3075 y Fu(\()p Fr(y)s Fu(\))c Fs(\024)f Fr(C)7 b(\017)1045 3038 y Fp(\013=)p Fv(2)1184 2975 y Fi(\020)1239 3075 y Fr(C)g(\017)1348 3038 y Fp(\013=)p Fv(2)1493 3075 y Fu(+)19 b Fr(\025)1636 3089 y Fp(k)1681 3075 y Fr(\017)1718 3038 y Fv(\()p Fp(d)p Fv(+1\))p Fp(=)p Fv(2)1983 2975 y Fi(\021)2181 3075 y Fu(sup)2052 3156 y Fp(y)r Fj(2)p Fp(B)2195 3173 y Fl(4)2226 3139 y Fg(p)p 2276 3139 V 34 x Fk(\017)2309 3156 y Fv(\()p Fp(x)2378 3166 y Fk(k)2418 3156 y Fv(\))2462 3075 y Fr(\036)2516 3089 y Fp(k)2560 3075 y Fu(\()p Fr(y)s Fu(\))26 b Fs(\024)f Fr(C)7 b(\017)2909 3038 y Fp(\013)3104 3075 y Fu(sup)2976 3156 y Fp(y)r Fj(2)p Fp(B)3119 3173 y Fl(4)3150 3139 y Fg(p)p 3199 3139 V 3199 3173 a Fk(\017)3232 3156 y Fv(\()p Fp(x)3301 3166 y Fk(k)3341 3156 y Fv(\))3385 3075 y Fr(\036)3439 3089 y Fp(k)3484 3075 y Fu(\()p Fr(y)s Fu(\))3477 3268 y(\(3)p Fr(:)p Fu(90\))50 3414 y(o)m(v)m(er)33 b(the)g(estimate)g(\(3.3\).)48 b(This)30 b(allo)m(ws)i(to)h(replace)f (all)f(errors)h(of)g(order)g Fr(\017)2705 3381 y Fp(\013=)p Fv(2)2862 3414 y Fu(b)m(y)g(errors)g(of)g(order)g Fr(\017)3633 3381 y Fp(\013)3685 3414 y Fu(.)50 3559 y(This)d(pro)s(cedure)h(can)h (b)s(e)f(iterated)i Fr(m)e Fu(times)h(to)g(get)i(errors)d(of)h(order)f Fr(\017)2568 3526 y Fp(m\013=)p Fv(2)2789 3559 y Fu(un)m(til)f(these)j (are)f(as)g(small)50 3705 y(as)f(the)h(exp)s(onen)m(tially)e(small)f (errors.)159 3923 y(Finally)e(w)m(e)i(w)m(ould)f(lik)m(e)g(to)i(impro)m (v)m(e)f(the)g(precision)e(with)h(whic)m(h)g(w)m(e)h(relate)h(the)f (eigen)m(v)-5 b(alues)28 b(to)h(the)50 4068 y(in)m(v)m(erse)h(mean)h (exit)f(times.)40 b(This)29 b(precision)f(is)i(so)g(far)g(limited)e(b)m (y)j(the)f(precision)f(with)g(whic)m(h)1374 4312 y Fq(E)1428 4326 y Fp(x)1469 4336 y Fk(k)1520 4312 y Fr(\034)1560 4326 y Fj(S)1605 4336 y Fk(k)q Fg(\000)p Fl(1)1754 4312 y Fs(\031)1861 4250 y Fu(cap)2012 4264 y Fp(B)2067 4274 y Fk(k)2112 4250 y Fu(\()p Fs(S)2202 4264 y Fp(k)r Fj(\000)p Fv(1)2340 4250 y Fu(\))p 1861 4291 515 4 v 2004 4374 a Fs(k)p Fr(h)2101 4388 y Fp(k)2146 4374 y Fs(k)2191 4388 y Fv(2)3477 4312 y Fu(\(3)p Fr(:)p Fu(91\))50 4561 y(holds.)44 b(F)-8 b(rom)33 b(Prop)s(osition)d(6.1)j(of)40 b([BEGK3])33 b(w)m(e)g(kno)m(w)f(that)h(this)e(precision)f(is)h (limited)f(only)h(b)m(y)i(the)50 4706 y(v)-5 b(ariation)29 b(of)i Fq(E)590 4720 y Fp(x)641 4706 y Fr(\034)681 4720 y Fj(S)726 4730 y Fk(k)q Fg(\000)p Fl(1)880 4706 y Fu(on)g Fr(B)1076 4720 y Fp(k)1120 4706 y Fu(.)41 b(T)-8 b(o)31 b(impro)m(v)m(e)f(this,)g(w)m(e)h(need)f(to)h(con)m(trol)g(the)1254 4908 y(cap)1405 4922 y Fp(B)1460 4932 y Fk(k)1504 4908 y Fu(\()p Fs(S)1594 4922 y Fp(k)r Fj(\000)p Fv(1)1733 4908 y Fu(\))p 1254 4948 V 1396 5032 a Fs(k)p Fr(h)1493 5046 y Fp(k)1539 5032 y Fs(k)1584 5046 y Fv(2)1799 4969 y Fs(\000)1901 4902 y Fu(cap)2052 4918 y Fp(B)2107 4927 y Fk(\017)2140 4918 y Fv(\()p Fp(x)p Fv(\))2242 4902 y Fu(\()p Fs(S)2332 4916 y Fp(k)r Fj(\000)p Fv(1)2471 4902 y Fu(\))p 1901 4948 606 4 v 1923 5032 a Fs(k)p Fr(h)2020 5048 y Fp(B)2075 5057 y Fk(\017)2108 5048 y Fv(\()p Fp(x)p Fv(\))p Fp(;)p Fj(S)2272 5058 y Fk(k)q Fg(\000)p Fl(1)2397 5032 y Fs(k)2442 5046 y Fv(2)3477 4969 y Fu(\(3)p Fr(:)p Fu(92\))50 5229 y(No)m(w)g(it)f(is)f(v)m(ery)i(simple)d(so)j(see)g (that)g(if)e Fr(x)c Fs(2)g Fr(B)1693 5204 y Fj(p)p 1753 5204 31 4 v 47 x Fp(\017)1788 5229 y Fu(\()p Fr(x)1875 5243 y Fp(k)1920 5229 y Fu(\),)31 b(then)1146 5462 y Fs(j)p Fr(h)1223 5478 y Fp(B)1278 5487 y Fk(\017)1311 5478 y Fv(\()p Fp(x)p Fv(\))p Fp(;)p Fj(S)1475 5488 y Fk(k)q Fg(\000)p Fl(1)1600 5462 y Fu(\()p Fr(y)s Fu(\))20 b Fs(\000)g Fr(h)1881 5476 y Fp(k)1926 5462 y Fu(\()p Fr(y)s Fu(\))p Fs(j)27 b(\024)d Fr(e)2233 5424 y Fj(\000)p Fp(\016)r(=\017)2398 5462 y Fr(h)2450 5476 y Fp(k)2495 5462 y Fu(\()p Fr(y)s Fu(\))864 b(\(3)p Fr(:)p Fu(93\))p eop %%Page: 29 29 29 28 bop 3633 295 a Fm(29)50 513 y Fu(Namely)-8 b(,)147 710 y Fs(j)p Fr(h)224 727 y Fp(B)279 736 y Fk(\017)312 727 y Fv(\()p Fp(x)p Fv(\))p Fp(;)p Fj(S)476 737 y Fk(k)q Fg(\000)p Fl(1)601 710 y Fu(\()p Fr(y)s Fu(\))20 b Fs(\000)g Fr(h)882 724 y Fp(k)927 710 y Fu(\()p Fr(y)s Fu(\))p Fs(j)172 883 y(\024)25 b Fq(P)323 897 y Fp(y)383 809 y Fi(\002)420 883 y Fs(f)p Fr(\034)505 897 y Fp(B)560 907 y Fk(k)630 883 y Fr(<)g(\034)766 897 y Fj(S)811 907 y Fk(k)q Fg(\000)p Fl(1)935 883 y Fs(g)c(\\)e(f)p Fr(\034)1166 897 y Fj(S)1211 907 y Fk(k)q Fg(\000)p Fl(1)1361 883 y Fr(<)25 b(\034)1497 899 y Fp(B)1552 908 y Fk(\017)1584 899 y Fv(\()p Fp(x)p Fv(\))1687 883 y Fs(g)1732 809 y Fi(\003)1790 883 y Fu(+)20 b Fq(P)1936 897 y Fp(y)1996 809 y Fi(\002)2033 883 y Fs(f)p Fr(\034)2118 899 y Fp(B)2173 908 y Fk(\017)2206 899 y Fv(\()p Fp(x)p Fv(\))2334 883 y Fr(<)25 b(\034)2470 897 y Fj(S)2515 907 y Fk(k)q Fg(\000)p Fl(1)2639 883 y Fs(g)20 b(\\)g(f)p Fr(\034)2870 897 y Fj(S)2915 907 y Fk(k)q Fg(\000)p Fl(1)3065 883 y Fr(<)25 b(\034)3201 897 y Fp(B)3256 907 y Fk(k)3300 883 y Fr(\034)3340 899 y Fp(B)3395 908 y Fk(\017)3427 899 y Fv(\()p Fp(x)p Fv(\))3529 883 y Fs(g)3574 809 y Fi(\003)3477 993 y Fu(\(3)p Fr(:)p Fu(94\))50 1138 y(But)30 b(b)m(y)h(the)f(Mark)m(o)m(v)i(prop)s (ert)m(y)537 1336 y Fq(P)592 1350 y Fp(y)652 1262 y Fi(\002)690 1336 y Fs(f)p Fr(\034)775 1350 y Fp(B)830 1360 y Fk(k)899 1336 y Fr(<)25 b(\034)1035 1350 y Fj(S)1080 1360 y Fk(k)q Fg(\000)p Fl(1)1204 1336 y Fs(g)c(\\)f(f)p Fr(\034)1436 1350 y Fj(S)1481 1360 y Fk(k)q Fg(\000)p Fl(1)1630 1336 y Fr(<)25 b(\034)1766 1352 y Fp(B)1821 1361 y Fk(\017)1853 1352 y Fv(\()p Fp(x)p Fv(\))1956 1336 y Fs(g)2001 1262 y Fi(\003)563 1508 y Fs(\024)g Fq(P)714 1522 y Fp(y)773 1435 y Fi(\002)811 1508 y Fr(\034)851 1522 y Fp(B)906 1532 y Fk(k)975 1508 y Fr(<)g(\034)1111 1522 y Fj(S)1156 1532 y Fk(k)q Fg(\000)p Fl(1)1280 1435 y Fi(\003)1339 1508 y Fu(max)1333 1567 y Fp(z)s Fj(2)p Fp(B)1475 1577 y Fk(k)1529 1508 y Fq(P)1599 1435 y Fi(\002)1638 1508 y Fr(\034)1678 1522 y Fj(S)1723 1532 y Fk(k)q Fg(\000)p Fl(1)1872 1508 y Fr(<)g(\034)2008 1525 y Fp(B)2063 1534 y Fk(\017)2095 1525 y Fv(\()p Fp(x)p Fv(\))2197 1435 y Fi(\003)2261 1508 y Fs(\024)g Fr(e)2399 1471 y Fj(\000)p Fp(\016)r(=\017)2563 1508 y Fq(P)2618 1522 y Fp(y)2678 1435 y Fi(\002)2715 1508 y Fr(\034)2755 1522 y Fp(B)2810 1532 y Fk(k)2880 1508 y Fr(<)g(\034)3016 1522 y Fj(S)3061 1532 y Fk(k)q Fg(\000)p Fl(1)3184 1435 y Fi(\003)3477 1445 y Fu(\(3)p Fr(:)p Fu(95\))50 1757 y(The)30 b(second)g(summand)f (in)g(\(3.94\))j(is)e(b)s(ounded)e(in)h(the)i(same)f(w)m(a)m(y)-8 b(.)159 1975 y(This)28 b(implies)g(of)j(course)f(that)1151 2214 y Fs(k)p Fr(h)1248 2230 y Fp(B)1303 2239 y Fk(\017)1337 2230 y Fv(\()p Fp(x)p Fv(\))p Fp(;)p Fj(S)1501 2240 y Fk(k)q Fg(\000)p Fl(1)1625 2214 y Fs(k)1670 2228 y Fv(2)1732 2214 y Fs(\000)20 b(k)p Fr(h)1920 2228 y Fp(k)1965 2214 y Fs(k)2010 2228 y Fv(2)2077 2214 y Fs(\024)25 b Fr(e)2215 2176 y Fj(\000)p Fp(\016)r(=\017)2379 2214 y Fs(k)p Fr(h)2476 2228 y Fp(k)2522 2214 y Fs(k)2567 2228 y Fv(2)3477 2214 y Fu(\(3)p Fr(:)p Fu(96\))50 2453 y(W)-8 b(e)31 b(only)e(need)h(a)g (similar)e(estimate)i(for)g(capacities.)41 b(While)29 b(this)g(ma)m(y)h(app)s(ear)g(more)g(di\016cult)e(at)i(\014rst)50 2598 y(sigh)m(t,)38 b(w)m(e)f(can)f(tak)m(e)i(adv)-5 b(an)m(tage)38 b(of)f(the)f(fact)i(that)f(as)f(long)g(as)2356 2574 y(\026)2352 2598 y Fr(\025)p Fu(\(\()p Fr(B)2544 2612 y Fp(\017)2579 2598 y Fu(\()p Fr(x)p Fu(\))25 b Fs([)f(S)2866 2612 y Fp(k)r Fj(\000)p Fv(1)3004 2598 y Fu(\))3039 2565 y Fp(c)3076 2598 y Fu(\))36 b Fs(\035)f Fr(\025)3326 2612 y Fp(k)3371 2598 y Fu(,)j(w)m(e)e(can)50 2744 y(replace)30 b Fr(B)427 2758 y Fp(k)502 2744 y Fu(in)f(the)i(pro)s (of)e(of)i(Prop)s(osition)d(3.10)k(without)e(further)f(c)m(hanges)i(b)m (y)f Fr(B)2963 2758 y Fp(\017)2998 2744 y Fu(\()p Fr(x)p Fu(\).)41 b(Th)m(us)383 3005 y Fr(\025)436 3019 y Fp(k)506 3005 y Fu(=)613 2937 y(cap)764 2954 y Fp(B)819 2963 y Fk(\017)851 2954 y Fv(\()p Fp(x)p Fv(\))954 2937 y Fu(\()p Fs(S)1044 2951 y Fp(k)r Fj(\000)p Fv(1)1182 2937 y Fu(\))p 613 2984 606 4 v 635 3067 a Fs(k)p Fr(h)732 3084 y Fp(B)787 3093 y Fk(\017)820 3084 y Fv(\()p Fp(x)p Fv(\))p Fp(;)p Fj(S)984 3094 y Fk(k)q Fg(\000)p Fl(1)1109 3067 y Fs(k)1154 3036 y Fv(2)1154 3092 y(2)1244 2904 y Fi(\020)1298 3005 y Fu(1)21 b(+)e Fr(O)s Fu(\()p Fr(e)1603 2968 y Fj(\000)p Fp(\016)r(=\017)1768 3005 y Fu(\))1803 2904 y Fi(\021)1883 3005 y Fu(=)1990 2944 y(cap)2142 2958 y Fp(B)2197 2968 y Fk(k)2241 2944 y Fu(\()p Fs(S)2331 2958 y Fp(k)r Fj(\000)p Fv(1)2469 2944 y Fu(\))p 1990 2984 515 4 v 2133 3067 a Fs(k)p Fr(h)2230 3081 y Fp(k)2275 3067 y Fs(k)2320 3036 y Fv(2)2320 3092 y(2)2530 2904 y Fi(\020)2585 3005 y Fu(1)h(+)g Fr(O)s Fu(\()p Fr(e)2890 2968 y Fj(\000)p Fp(\016)r(=\017)3055 3005 y Fu(\))3090 2904 y Fi(\021)3477 3005 y Fu(\(3)p Fr(:)p Fu(97\))50 3272 y(whic)m(h)29 b(implies)f(together)j(with)e(\(3.96\))k(that)818 3511 y Fs(j)p Fu(cap)995 3527 y Fp(B)1050 3536 y Fk(\017)1082 3527 y Fv(\()p Fp(x)p Fv(\))1185 3511 y Fu(\()p Fs(S)1275 3525 y Fp(k)r Fj(\000)p Fv(1)1413 3511 y Fu(\))21 b Fs(\000)f Fu(cap)1711 3525 y Fp(B)1766 3535 y Fk(k)1810 3511 y Fu(\()p Fs(S)1900 3525 y Fp(k)r Fj(\000)p Fv(1)2039 3511 y Fu(\))p Fs(j)25 b(\024)g Fr(e)2262 3473 y Fj(\000)p Fp(\016)r(=\017)2427 3511 y Fu(cap)2579 3525 y Fp(B)2634 3535 y Fk(k)2678 3511 y Fu(\()p Fs(S)2768 3525 y Fp(k)r Fj(\000)p Fv(1)2906 3511 y Fu(\))536 b(\(3)p Fr(:)p Fu(98\))50 3750 y(Based)27 b(on)f(\(3.98\))j(and)d(\(3.95\),)k(one)d(can)g(impro)m (v)m(e)f(Prop)s(osition)f(6.1)j(of)34 b([BEGK3])27 b(iterativ)m(ely)g (as)g(ab)s(o)m(v)m(e)50 3895 y(to)k(yield)1134 4046 y Fq(E)1189 4060 y Fp(x)1230 4070 y Fk(k)1280 4046 y Fr(\034)1320 4060 y Fj(S)1365 4070 y Fk(k)1434 4046 y Fu(=)1541 3984 y(cap)1693 3998 y Fp(B)1748 4008 y Fk(k)1792 3984 y Fs(S)1847 3998 y Fp(k)r Fj(\000)p Fv(1)p 1541 4025 444 4 v 1649 4108 a Fs(k)p Fr(h)1746 4122 y Fp(k)1791 4108 y Fs(k)1836 4122 y Fv(2)2011 3945 y Fi(\020)2065 4046 y Fu(1)21 b(+)f Fr(O)s Fu(\()p Fr(e)2371 4008 y Fj(\000)p Fp(\016)r(=\017)2536 4046 y Fu(\))2571 3945 y Fi(\021)3477 4046 y Fu(\(3)p Fr(:)p Fu(99\))50 4277 y(whic)m(h)35 b(implies)e(the)j(\014rst)f (equalit)m(y)h(in)f(Theorem)h(1.1.)58 b(Th)m(us)35 b(all)g(error)h (terms)g(of)g(order)g Fr(\017)3283 4244 y Fp(\013=)p Fv(2)3444 4277 y Fu(can)g(b)s(e)50 4423 y(remo)m(v)m(ed)31 b(in)e(\(3.78\))k(and)d(\(3.75\),)j(completing)c(the)i(pro)s(ofs)e(of)i (Theorems)f(1.1)h(and)f(Theorem)g(1.2.)42 b Fs(}})50 4641 y Ft(Exp)s(onen)m(tial)31 b(distribution)g(of)g(exit)f(times.)39 b Fu(W)-8 b(e)28 b(conclude)e(this)g(c)m(hapter)h(with)f(a)h(result)f (that)h(will)50 4786 y(imply)c(Theorem)i(1.3)h(on)f(the)h(exp)s(onen)m (tial)e(distribution)e(of)j(exit)g(times.)39 b(Let)26 b Fr(L)2835 4753 y Fp(D)2835 4809 y(\017)2926 4786 y Fu(denote)f(the)h(Diric)m(hlet)50 4932 y(op)s(erator)f(with)e(Diric)m (hlet)g(conditions)h(in)f Fr(D)s Fu(.)38 b(T)-8 b(o)25 b(a)m(v)m(oid)h(confusion,)e(w)m(e)h(assume)g(that)g Fr(D)j Fu(=)d Fs(S)3319 4946 y Fp(k)r Fj(\000)p Fv(1)3457 4932 y Fu(.)38 b(Note)50 5077 y(that)26 b(Prop)s(osition)e(3.10)k (\(and)e(its)f(imp)s(o)m(v)m(emen)m(t\))h(also)g(applies)f(to)h(the)g (op)s(erator)h Fr(L)2949 5044 y Fp(D)2949 5100 y(\017)3014 5077 y Fu(,)g(and)f(if)f(w)m(e)h(denote)50 5223 y(b)m(y)180 5199 y(\026)176 5223 y Fr(\025)229 5190 y Fp(i)229 5249 y(k)304 5223 y Fu(the)31 b Fr(i)20 b Fs(\000)g Fr(th)31 b Fu(eigen)m(v)-5 b(alue)30 b(of)h Fr(L)1319 5190 y Fp(D)1319 5245 y(\017)1384 5223 y Fu(,)g(w)m(e)g(see)g(that)g(within)c(our)j (usual)f(errors,)1684 5438 y(\026)1681 5462 y Fr(\025)1734 5424 y Fp(i)1734 5484 y(k)1804 5462 y Fs(\030)c Fr(\025)1953 5476 y Fp(k)r Fv(+)p Fp(i)3432 5462 y Fu(\(3)p Fr(:)p Fu(100\))p eop %%Page: 30 30 30 29 bop 50 295 a Fm(30)50 513 y Fu(for)30 b Fr(i)c Fu(=)e(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(n)20 b Fs(\000)g Fr(k)s Fu(,)30 b(and)g(the)h(corresp)s(onding)d(eigenfunction)2348 489 y(\026)2336 513 y Fr(\036)2390 480 y Fp(i)2390 538 y(k)2465 513 y Fu(satis\014es)810 787 y Fr(\036)864 749 y Fp(i)864 809 y(k)908 787 y Fu(\()p Fr(y)s Fu(\))e(=)1166 725 y Fr(h)1218 739 y Fp(k)r Fv(+)p Fp(i)1344 725 y Fu(\()p Fr(y)s Fu(\))p 1159 766 311 4 v 1159 849 a Fs(k)p Fr(h)1256 863 y Fp(k)r Fv(+)p Fp(i)1383 849 y Fs(k)1428 863 y Fv(2)1480 787 y Fu(\(1)21 b(+)f Fr(O)s Fu(\()p Fr(e)1821 749 y Fj(\000)p Fp(\016)r(=\017)1986 787 y Fu(\)\))h(+)2211 673 y Fp(n)2168 701 y Fi(X)2169 896 y Fp(j)t Fv(=)p Fp(k)2314 787 y Fr(d)2361 801 y Fp(j)2400 787 y Fu(\()p Fr(y)s Fu(\))p Fr(h)2570 801 y Fp(j)2609 787 y Fu(\()p Fr(y)s Fu(\))q Fs(k)p Fr(h)2825 801 y Fp(j)2864 787 y Fs(k)2909 801 y Fv(2)3432 787 y Fu(\(3)p Fr(:)p Fu(101\))50 1097 y(with)g Fs(j)p Fr(d)321 1111 y Fp(j)360 1097 y Fu(\()p Fr(y)s Fu(\))p Fs(j)26 b(\024)f Fr(e)667 1064 y Fj(\000)p Fp(\016)r(=\017)832 1097 y Fu(.)38 b(Let)23 b(us)f(denote)i(henceforth) f(b)m(y)2015 1073 y(\026)2003 1097 y Fr(\036)2057 1064 y Fp(i)2057 1122 y(k)2125 1097 y Fu(the)g(corresp)s(onding)e (normalized)g(eigenfunc-)50 1242 y(tions)30 b(\(e.g.)44 b Fs(k)544 1218 y Fu(\026)532 1242 y Fr(\036)586 1209 y Fp(i)586 1268 y(k)631 1242 y Fs(k)676 1256 y Fv(2)744 1242 y Fu(=)26 b(1\).)43 b(Note)32 b(that)1419 1218 y(\026)1407 1242 y Fr(\036)1461 1209 y Fp(i)1461 1268 y(k)1532 1242 y Fu(=)26 b Fr(\036)1683 1209 y Fp(i)1683 1268 y(k)1728 1242 y Fu(\(1)21 b(+)g Fr(O)s Fu(\()p Fr(e)2070 1209 y Fj(\000)p Fp(\016)r(=\017)2234 1242 y Fu(\)\),)32 b(so)g(in)d(fact)j (they)f(can)g(b)s(e)g(represen)m(ted)50 1388 y(in)e(the)h(same)h(w)m(a) m(y)h(as)e(\(3.101\))j(with)c(rede\014ned)g Fr(d)1787 1402 y Fp(j)1856 1388 y Fu(satisfying)g(the)i(same)g(b)s(ounds.)159 1606 y(Denote)e(b)m(y)e Fr(P)649 1620 y Fp(k)r Fj(\000)p Fp(i)803 1606 y Fu(the)h(pro)5 b(jector)27 b(on)g(the)h(subspace)e (generated)i(b)m(y)f Fr(\036)2586 1573 y Fp(i)2586 1632 y(k)r Fj(\000)p Fv(1)2751 1606 y Fu(and)g(b)m(y)g Fr(P)3106 1620 y Fj(?)3195 1606 y Fu(the)g(pro)5 b(jector)50 1751 y(to)31 b(the)f(subspace)g(orthogonal)h(to)g(span\()1493 1727 y(\026)1481 1751 y Fr(\036)1535 1718 y Fv(1)1535 1777 y Fp(k)r Fj(\000)p Fv(1)1673 1751 y Fr(;)15 b(:)g(:)g(:)h(;)1887 1727 y Fu(\026)1874 1751 y Fr(\036)1928 1712 y Fp(n)p Fj(\000)p Fp(k)1928 1779 y(k)r Fj(\000)p Fv(1)2075 1751 y Fu(\).)42 b(Note)31 b(that)275 1989 y Fq(P)330 2003 y Fp(x)371 2013 y Fk(k)431 1989 y Fu([)p Fr(\034)496 2003 y Fp(D)587 1989 y Fr(>)25 b(T)13 b Fu(])26 b(=)895 1888 y Fi(\020)950 1989 y Fr(\016)990 2003 y Fp(x)1031 2013 y Fk(k)1076 1989 y Fr(;)15 b(e)1158 1952 y Fj(\000)p Fp(T)10 b(L)1318 1924 y Fk(D)1318 1970 y(\017)1379 1989 y Fu(1)-23 b(I)1435 2003 y Fp(D)1496 1985 y Fk(c)1535 1888 y Fi(\021)800 2252 y Fu(=)895 2138 y Fp(n)p Fj(\000)p Fp(k)901 2165 y Fi(X)908 2359 y Fp(i)p Fv(=1)1053 2151 y Fi(\020)1107 2252 y Fr(\016)1147 2266 y Fp(x)1188 2276 y Fk(k)1233 2252 y Fr(;)15 b(e)1315 2214 y Fj(\000)p Fp(T)10 b(L)1475 2187 y Fk(D)1475 2232 y(\017)1536 2252 y Fr(P)1594 2266 y Fp(k)r Fj(\000)p Fp(i)1722 2252 y Fu(1)-23 b(I)1777 2266 y Fp(D)1838 2247 y Fk(c)1877 2151 y Fi(\021)1952 2252 y Fu(+)2043 2151 y Fi(\020)2097 2252 y Fr(\016)2137 2266 y Fp(x)2178 2276 y Fk(k)2223 2252 y Fr(;)15 b(e)2305 2214 y Fj(\000)p Fp(T)10 b(L)2465 2187 y Fk(D)2465 2232 y(\017)2526 2252 y Fr(P)2584 2266 y Fj(?)2646 2252 y Fu(1)-23 b(I)2701 2266 y Fp(D)2762 2247 y Fk(c)2801 2151 y Fi(\021)800 2571 y Fu(=)895 2458 y Fp(n)p Fj(\000)p Fp(k)901 2485 y Fi(X)908 2679 y Fp(i)p Fv(=1)1053 2571 y Fr(e)1095 2534 y Fj(\000)1155 2517 y Fv(\026)1152 2534 y Fp(\025)1195 2506 y Fk(i)1195 2552 y(k)q Fg(\000)p Fl(1)1315 2534 y Fp(T)1384 2547 y Fu(\026)1372 2571 y Fr(\036)1426 2534 y Fp(i)1426 2594 y(k)r Fj(\000)p Fv(1)1564 2571 y Fu(\()p Fr(x)1651 2585 y Fp(k)1696 2571 y Fu(\))1746 2448 y Fi(Z)1797 2654 y Fp(D)1858 2636 y Fk(c)1912 2571 y Fr(dy)19 b(e)2065 2534 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2358 2547 y Fu(\026)2346 2571 y Fr(\036)2400 2534 y Fp(i)2400 2594 y(k)r Fj(\000)p Fv(1)2538 2571 y Fu(\()p Fr(y)s Fu(\))20 b(+)g Fr(O)2854 2470 y Fi(\020)2909 2571 y Fr(e)2951 2534 y Fj(\000)p Fp(T)3064 2517 y Fv(\026)3061 2534 y Fp(\025)3104 2543 y Fk(n)3153 2470 y Fi(\021)3432 2309 y Fu(\(3)p Fr(:)p Fu(102\))50 2849 y(Giv)m(en)30 b(the)h(precise)e(con)m(trol)i(on)f(the) h(eigenfunctions,)e(it)h(is)g(not)g(di\016cult)e(to)k(obtain)d(that)275 3070 y Fq(P)330 3084 y Fp(x)371 3094 y Fk(k)431 3070 y Fu([)p Fr(\034)496 3084 y Fp(D)587 3070 y Fr(>)c(T)13 b Fu(])26 b(=)895 2969 y Fi(\020)950 3070 y Fr(\016)990 3084 y Fp(x)1031 3094 y Fk(k)1076 3070 y Fr(;)15 b(e)1158 3032 y Fj(\000)p Fp(T)10 b(L)1318 3005 y Fk(D)1318 3050 y(\017)1379 3070 y Fu(1)-23 b(I)1435 3084 y Fp(D)1496 3065 y Fk(c)1535 2969 y Fi(\021)800 3332 y Fu(=)895 3218 y Fp(n)p Fj(\000)p Fp(k)901 3246 y Fi(X)908 3439 y Fp(i)p Fv(=1)1053 3332 y Fr(e)1095 3294 y Fj(\000)1155 3278 y Fv(\026)1152 3294 y Fp(\025)1195 3267 y Fk(i)1195 3313 y(k)q Fg(\000)p Fl(1)1315 3294 y Fp(T)1384 3308 y Fu(\026)1372 3332 y Fr(\036)1426 3294 y Fp(i)1426 3354 y(k)r Fj(\000)p Fv(1)1564 3332 y Fu(\()p Fr(x)1651 3346 y Fp(k)1696 3332 y Fu(\))1746 3208 y Fi(Z)1797 3415 y Fp(D)1858 3397 y Fk(c)1912 3332 y Fr(dy)19 b(e)2065 3294 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)2358 3308 y Fu(\026)2346 3332 y Fr(\036)2400 3294 y Fp(i)2400 3354 y(k)r Fj(\000)p Fv(1)2538 3332 y Fu(\()p Fr(y)s Fu(\))20 b(+)g Fr(O)2854 3231 y Fi(\020)2909 3332 y Fr(e)2951 3294 y Fj(\000)p Fp(T)3064 3278 y Fv(\026)3061 3294 y Fp(\025)3104 3303 y Fk(n)3153 3231 y Fi(\021)3432 3229 y Fu(\(3)p Fr(:)p Fu(103\))50 3615 y(No)m(w)31 b(using)e(\(3.79\),)k(\(3.80\),)g(w)m(e)d (get)77 3836 y(\026)65 3860 y Fr(\036)119 3822 y Fp(i)119 3882 y(k)r Fj(\000)p Fv(1)257 3860 y Fu(\()p Fr(x)344 3874 y Fp(k)389 3860 y Fu(\))439 3736 y Fi(Z)490 3942 y Fp(D)551 3924 y Fk(c)605 3860 y Fr(dy)18 b(e)757 3822 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1050 3836 y Fu(\026)1039 3860 y Fr(\036)1093 3822 y Fp(i)1093 3882 y(k)r Fj(\000)p Fv(1)1230 3860 y Fu(\()p Fr(y)s Fu(\))90 4127 y(=)25 b Fr(h)238 4141 y Fp(k)r Fj(\000)p Fv(1+)p Fp(i)458 4127 y Fu(\()p Fr(x)545 4141 y Fp(k)590 4127 y Fu(\))636 3986 y Fi(R)679 4092 y Fp(D)740 4074 y Fk(c)794 4060 y Fr(dy)19 b(e)947 4027 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)1228 4060 y Fr(h)1280 4074 y Fp(k)r Fj(\000)p Fv(1+)p Fp(i)1499 4060 y Fu(\()p Fr(y)s Fu(\))p 636 4106 982 4 v 925 4190 a Fs(k)p Fr(h)1022 4204 y Fp(k)r Fj(\000)p Fv(1+)p Fp(i)1242 4190 y Fs(k)1287 4158 y Fv(2)1287 4214 y(2)1649 4127 y Fu(+)2039 4041 y Fi(X)1740 4240 y Fv(\()p Fp(j;j)1854 4222 y Fg(0)1878 4240 y Fv(\))p Fj(6)p Fv(=\()p Fp(k)r Fj(\000)p Fv(1+)p Fp(i;k)r Fj(\000)p Fv(1+)p Fp(i)p Fv(\))2485 4127 y Fr(d)2532 4141 y Fp(j)2571 4127 y Fr(d)2618 4141 y Fp(j)2652 4123 y Fg(0)2681 4127 y Fr(h)2733 4141 y Fp(j)2772 4127 y Fu(\()p Fr(x)2859 4141 y Fp(k)2904 4127 y Fu(\))2950 3986 y Fi(R)2993 4092 y Fp(D)3054 4074 y Fk(c)3108 4060 y Fr(dy)19 b(e)3261 4027 y Fj(\000)p Fp(F)10 b Fv(\()p Fp(y)r Fv(\))p Fp(=\017)3542 4060 y Fr(h)3594 4074 y Fp(j)3628 4055 y Fg(0)3657 4060 y Fu(\()p Fr(y)s Fu(\))p 2950 4106 826 4 v 3141 4190 a Fs(k)p Fr(h)3238 4204 y Fp(j)3277 4190 y Fs(k)3322 4204 y Fv(2)3363 4190 y Fs(k)p Fr(h)3460 4204 y Fp(j)3499 4190 y Fs(k)3544 4204 y Fv(2)90 4397 y Fu(=)25 b Fr(h)238 4411 y Fp(k)r Fj(\000)p Fv(1+)p Fp(i)458 4397 y Fu(\()p Fr(x)545 4411 y Fp(k)590 4397 y Fu(\)\(1)c(+)f Fr(O)s Fu(\()p Fr(e)966 4360 y Fj(\000)p Fp(\016)r(=\017)1131 4397 y Fu(\)\))h(+)1612 4311 y Fi(X)1313 4510 y Fv(\()p Fp(j;j)1427 4492 y Fg(0)1451 4510 y Fv(\))p Fj(6)p Fv(=\()p Fp(k)r Fj(\000)p Fv(1+)p Fp(i;k)r Fj(\000)p Fv(1+)p Fp(i)p Fv(\))2058 4397 y Fr(d)2105 4411 y Fp(j)2144 4397 y Fr(d)2191 4411 y Fp(j)2225 4393 y Fg(0)2254 4397 y Fr(h)2306 4411 y Fp(j)2345 4397 y Fu(\()p Fr(x)2432 4411 y Fp(k)2477 4397 y Fu(\))p Fr(c)2551 4411 y Fp(j;j)2637 4393 y Fg(0)2666 4397 y Fr(e)2708 4360 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(x)2911 4332 y Fg(0)2911 4378 y Fk(j)2945 4360 y Fv(\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)3155 4369 y Fk(j)3190 4360 y Fv(\)])p Fp(=)p Fv(2)p Fp(\017)3432 4612 y Fu(\(3)p Fr(:)p Fu(104\))50 4758 y(No)m(w)31 b(if)e Fr(j)i Fu(=)25 b Fr(k)s Fu(,)31 b(the)f(term)h(in)e(the)h(last)h(sum)e (is)1189 4992 y Fr(d)1236 5006 y Fp(k)1281 4992 y Fr(d)1328 5006 y Fp(j)1362 4988 y Fg(0)1392 4992 y Fr(c)1431 5006 y Fp(k)r(;j)1528 4988 y Fg(0)1556 4992 y Fr(e)1598 4955 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(x)1801 4927 y Fg(0)1801 4973 y Fk(j)1835 4955 y Fv(\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)2045 4965 y Fk(k)2085 4955 y Fv(\)])p Fp(=)p Fv(2)p Fp(\017)2268 4992 y Fs(\024)25 b Fr(e)2406 4955 y Fj(\000)p Fp(\016)r(=\017)3432 4992 y Fu(\(3)p Fr(:)p Fu(105\))50 5227 y(since)30 b Fr(F)13 b Fu(\()p Fr(x)431 5241 y Fp(j)465 5223 y Fg(0)494 5227 y Fu(\))25 b Fr(>)g(F)13 b Fu(\()p Fr(x)808 5241 y Fp(k)854 5227 y Fu(\))30 b(for)g Fr(j)1100 5194 y Fj(0)q(0)1173 5227 y Fr(>)25 b(k)s Fu(;)30 b(in)f(all)h(other)g(cases,)748 5462 y Fr(h)800 5476 y Fp(j)838 5462 y Fu(\()p Fr(x)925 5476 y Fp(k)970 5462 y Fu(\))c Fs(\031)f Fr(e)1169 5424 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(z)1369 5397 y Fg(\003)1406 5424 y Fv(\()p Fp(x)1475 5433 y Fk(j)1510 5424 y Fp(;)p Fj(M)1617 5434 y Fk(k)q Fl(+)p Fk(i)1728 5424 y Fj(n)p Fp(x)1806 5433 y Fk(j)1840 5424 y Fv(\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)2050 5434 y Fk(k)2091 5424 y Fv(\)\)])p Fp(=\017)2265 5462 y Fr(<)25 b(e)2403 5424 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(x)2606 5433 y Fk(j)2640 5424 y Fv(\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)2850 5434 y Fk(k)2891 5424 y Fv(\)])p Fp(=\017)3432 5462 y Fu(\(3)p Fr(:)p Fu(106\))p eop %%Page: 31 31 31 30 bop 50 295 a Fa(Metastability)27 b(and)g(sp)l(e)l(ctr)l(a)2736 b Fm(31)50 513 y Fu(so)30 b(that)362 753 y Fr(h)414 767 y Fp(j)453 753 y Fu(\()p Fr(x)540 767 y Fp(k)585 753 y Fu(\))p Fr(e)662 716 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(x)865 688 y Fg(0)865 734 y Fk(j)900 716 y Fv(\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)1110 725 y Fk(j)1144 716 y Fv(\)])p Fp(=)p Fv(2)p Fp(\017)1327 753 y Fr(<)25 b(e)1465 716 y Fj(\000)p Fv([)p Fp(F)10 b Fv(\()p Fp(x)1668 725 y Fk(j)1702 716 y Fv(\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)1912 726 y Fk(k)1953 716 y Fv(\)])p Fp(=)p Fv(2)p Fp(\017)2110 753 y Fr(e)2152 716 y Fj(\000)p Fv([)p Fp(F)g Fv(\()p Fp(x)2355 688 y Fg(0)2355 734 y Fk(j)2389 716 y Fv(\))p Fj(\000)p Fp(F)g Fv(\()p Fp(x)2599 726 y Fk(k)2640 716 y Fv(\)])p Fp(=)p Fv(2)p Fp(\017)2822 753 y Fr(<)25 b(\017)2955 716 y Fj(\000)p Fp(\016)r(=\017)3432 753 y Fu(\(3)p Fr(:)p Fu(107\))50 993 y(This)j(sho)m(ws)j(that)227 1264 y Fq(P)282 1278 y Fp(x)323 1288 y Fk(k)383 1264 y Fu([)p Fr(\034)448 1278 y Fp(D)539 1264 y Fr(>)25 b(T)13 b Fu(])25 b(=)g Fr(e)889 1227 y Fj(\000)949 1210 y Fv(\026)946 1227 y Fp(\025)989 1237 y Fk(k)q Fg(\000)p Fl(1)1110 1227 y Fp(T)1167 1264 y Fu(\(1)c(+)f Fr(O)s Fu(\()p Fr(e)1508 1227 y Fj(\000)p Fp(\016)r(=\017)1673 1264 y Fu(\)\))g(+)1854 1151 y Fp(n)p Fj(\000)p Fp(k)1860 1178 y Fi(X)1866 1372 y Fp(i)p Fv(=2)2011 1264 y Fr(e)2053 1227 y Fj(\000)2113 1210 y Fv(\026)2110 1227 y Fp(\025)2153 1199 y Fk(i)2153 1246 y(k)q Fg(\000)p Fl(1)2278 1264 y Fr(O)s Fu(\()p Fr(e)2427 1227 y Fj(\000)p Fp(\016)r(=\017)2592 1264 y Fu(\))h(+)e Fr(O)s Fu(\(1\))p Fr(e)2967 1223 y Fj(\000)p Fp(T)3082 1206 y Fv(\026)3077 1223 y Fp(\025)3120 1193 y Fk(n)p Fg(\000)p Fk(k)3120 1247 y(k)q Fg(\000)p Fl(1)3432 1264 y Fu(\(3)p Fr(:)p Fu(108\))50 1548 y(This)28 b(pro)m(v)m(es)j (Theorem)g(1.3.)41 b Fs(}})50 1796 y Fw(A.)f(App)s(endix)159 2014 y Fu(In)e(this)f(app)s(endix)f(w)m(e)j(pro)m(v)m(e)h(a)e(general)h (p)s(erturbation)d(estimate)k(that)f(is)e(needed)i(in)e(Section)h(5.)50 2160 y(This)28 b(should)h(b)s(e)h(w)m(ell-kno)m(wn,)f(but)h(w)m(e)g(ha) m(v)m(e)i(not)f(b)s(een)e(able)h(to)h(\014nd)e(a)i(precise)f (references.)50 2378 y Fh(Theorem)37 b(7.1:)52 b Fo(L)-5 b(et)34 b Fr(A)f Fo(b)-5 b(e)33 b(a)g(self-adjoint)h(op)-5 b(er)g(ator)36 b(in)d(some)g(\(\014nite-dimensional\))i(Hilb)-5 b(ert)33 b(sp)-5 b(ac)g(e)50 2523 y Fr(L)112 2490 y Fv(2)152 2523 y Fu(\(\012)p Fr(;)15 b(\026)p Fu(\))p Fo(.)45 b(L)-5 b(et)34 b Fr(B)5 b Fu(\()p Fr(\025)p Fu(\))33 b Fo(a)h(c)-5 b(ontinous)34 b(family)h(of)e(b)-5 b(ounde)g(d)35 b(op)-5 b(er)g(ators)37 b(on)d(the)f(same)i(sp)-5 b(ac)g(e)34 b(that)h(satis\014es)50 2669 y(the)27 b(b)-5 b(ound)28 b Fs(k)p Fr(B)5 b Fu(\()p Fr(\025)p Fu(\))p Fs(k)26 b(\024)f Fr(\016)11 b Fu(+)d Fr(\025C)33 b Fo(for)28 b Fu(0)d Fs(\024)g Fr(\016)k Fs(\034)d Fu(1)p Fo(,)i(and)g Fu(0)d Fs(\024)g Fr(C)32 b(<)25 b Fs(1)p Fo(.)40 b(Assume)26 b(that)i Fr(A)f Fo(has)h Fr(k)i Fo(eigenvalues)50 2814 y Fr(\025)103 2828 y Fv(1)144 2814 y Fr(;)15 b(:)g(:)g(:)h(;)f(\025)398 2828 y Fp(k)476 2814 y Fo(in)32 b(an)i(interval)f Fu([0)p Fr(;)15 b(a)p Fu(])34 b Fo(with)f Fr(a)26 b(<)f Fu(1)p Fr(=C)7 b Fo(.)42 b(Then)33 b(the)g(e)-5 b(quatation)1365 3055 y Fu(det\()p Fr(A)21 b Fs(\000)f Fr(\025)p Fu(\(1)-23 b(I)20 b(+)g Fr(B)5 b Fu(\()p Fr(\025)p Fu(\)\)\))26 b(=)f(0)1129 b(\(7)p Fr(:)p Fu(1\))50 3295 y Fo(has)33 b(at)h(most)f Fr(k)j Fo(solutions)e Fr(\025)1059 3262 y Fj(0)1059 3318 y Fv(1)1100 3295 y Fr(;)15 b(:)g(:)g(:)h(;)f(\025)1354 3262 y Fj(0)1354 3321 y Fp(k)1432 3295 y Fo(and)34 b(e)-5 b(ach)33 b(solution)h(satis\014es)g Fs(j)p Fr(\025)2577 3262 y Fj(0)2577 3319 y Fp(i)2627 3295 y Fs(\000)20 b Fr(\025)2771 3309 y Fp(i)2801 3295 y Fs(j)26 b(\024)f Fu(4)p Fr(\016)s(\025)3089 3309 y Fp(i)3121 3295 y Fo(.)50 3513 y Fe(Pro)s(of:)47 b Fu(If)30 b(\(7.1\))i(holds,)d(then)i(there)f (exist)g(a)h(non-zero)g(v)m(ector)h Fr(c)f Fu(suc)m(h)f(that)1503 3754 y(\()p Fr(A)21 b Fs(\000)f Fr(\025)p Fu(\))p Fr(c)26 b Fu(=)f Fr(\025B)5 b Fu(\()p Fr(\025)p Fu(\))p Fr(c)1267 b Fu(\(7)p Fr(:)p Fu(2\))50 3994 y(or)1554 4139 y Fr(c)26 b Fu(=)1815 4078 y Fr(\025)p 1726 4119 233 4 v 1726 4202 a(A)20 b Fs(\000)g Fr(\025)1969 4139 y(B)5 b Fu(\()p Fr(\025)p Fu(\))p Fr(c)1318 b Fu(\(7)p Fr(:)p Fu(3\))50 4342 y(Th)m(us)1232 4488 y Fs(k)p Fr(c)p Fs(k)1361 4502 y Fv(2)1428 4488 y Fs(\024)25 b Fr(\025)p Fs(k)p Fu(\()p Fr(A)c Fs(\000)f Fr(\025)p Fu(\))1925 4450 y Fj(\000)p Fv(1)2023 4488 y Fs(kk)p Fr(B)5 b Fu(\()p Fr(\025)p Fu(\))p Fs(kk)p Fr(c)p Fs(k)2484 4502 y Fv(2)3523 4488 y Fu(\(7)p Fr(:)p Fu(4\))50 4691 y(Since)29 b Fr(c)i Fu(is)e(non-zero,)i(this)f (means)g(that)1380 4931 y Fr(\025)p Fs(k)p Fu(\()p Fr(A)21 b Fs(\000)f Fr(\025)p Fu(\))1781 4893 y Fj(\000)p Fv(1)1879 4931 y Fs(kk)p Fr(B)5 b Fu(\()p Fr(\025)p Fu(\))p Fs(k)27 b(\025)e Fu(1)1144 b(\(7)p Fr(:)p Fu(5\))50 5171 y(No)m(w)31 b(since)f Fr(A)g Fu(is)g(symmetric,)f(w)m(e)i(ha)m(v)m(e)h(that)1154 5413 y Fs(k)p Fu(\()p Fr(A)21 b Fs(\000)f Fr(\025)p Fu(\))1502 5376 y Fj(\000)p Fv(1)1600 5413 y Fs(k)26 b(\024)f Fu(max)1838 5470 y Fp(i)2088 5352 y Fu(1)p 1962 5392 298 4 v 1962 5475 a Fs(j)p Fr(\025)2040 5489 y Fp(i)2091 5475 y Fs(\000)20 b Fr(\025)p Fs(j)2291 5413 y(^)2466 5352 y Fu(1)p 2383 5392 213 4 v 2383 5475 a Fr(a)g Fs(\000)g Fr(\025)3523 5413 y Fu(\(7)p Fr(:)p Fu(6\))p eop %%Page: 32 32 32 31 bop 50 295 a Fm(32)3225 b Fa(R)l(efer)l(enc)l(es)50 513 y Fu(and)30 b(so)1210 594 y Fp(k)1154 673 y Fu(min)1171 730 y Fp(i)p Fv(=1)1321 673 y Fs(j)p Fr(\025)1399 687 y Fp(i)1450 673 y Fs(\000)19 b Fr(\025)p Fs(j)26 b(\024)f Fr(\025)p Fs(k)p Fr(B)5 b Fu(\()p Fr(\025)p Fu(\))p Fs(k)26 b(\024)f Fr(\025)p Fu(\()p Fr(\016)g Fu(+)20 b Fr(C)7 b(\025)p Fu(\))917 b(\(7)p Fr(:)p Fu(7\))50 897 y(whic)m(h)29 b(implies)f(the)i(claimed)f(result.)40 b Fs(})50 1171 y Fw(References)-251 1389 y Fu([BEGK1])47 b(A.)33 b(Bo)m(vier,)h(M.)f (Ec)m(kho\013,)h(V.)f(Ga)m(yrard,)h(and)e(M.)h(Klein,)f(\\Metastabilit) m(y)h(in)f(sto)s(c)m(hastic)h(dynamics)159 1535 y(of)d(disordered)f (mean-\014eld)g(mo)s(dels",)h(Probab.)40 b(Theor.)g(Rel.)h(Fields)28 b Ft(119)p Fu(,)k(99{161\(2001\).)-251 1753 y([BEGK2])47 b(A.)30 b(Bo)m(vier,)h(M.)g(Ec)m(kho\013,)g(V.)f(Ga)m(yrard,)h(and)e (M.)h(Klein)e(\\Metastabilit)m(y)j(and)f(lo)m(w-lying)e(sp)s(ectra)i (in)159 1898 y(rev)m(ersible)f(Mark)m(o)m(v)j(c)m(hains",)e(Comm)m(un.) 40 b(Math.)i(Ph)m(ys.)e Ft(228)p Fu(,)31 b(219-255)j(\(2002\).)-251 2116 y([BEGK3])47 b(A.)37 b(Bo)m(vier,)i(M.)e(Ec)m(kho\013,)i(V.)e(Ga)m (yrard,)h(and)e(M.)h(Klein,)g(\\Metastabilit)m(y)g(in)e(rev)m(ersible)g (di\013usion)159 2262 y(pro)s(cesses)30 b(I.)g(Sharp)f(asymptotics)i (for)f(capacities)h(and)e(exit)i(times,)f(preprin)m(t)e(\(2002\).)-224 2480 y([BluGet])46 b(R.M.)34 b(Blumen)m(thal)e(and)g(R.K.)h(Geto)s(or,) i(\\Mark)m(o)m(v)h(pro)s(cesses)d(and)f(p)s(oten)m(tial)h(theory",)h (Academic)159 2625 y(Press,)c(New)h(Y)-8 b(ork,)31 b(London,)f(1968.) -226 2844 y([BuMa1])47 b(V.A.)32 b(Buslo)m(v)f(and)g(K.A.)h(Mak)-5 b(aro)m(v,)33 b(\\A)f(time-scale)g(hierarc)m(h)m(y)f(with)f(small)g (di\013usion")f(\(Russian\),)159 2989 y(T)-8 b(eoret.)45 b(Mat.)h(Fiz.)e Ft(76)p Fu(,)32 b(219{230)j(\(1988\);)g(translation)c (in)f(Theoret.)44 b(Math.)h(Ph)m(ys.)f Ft(76)p Fu(,)33 b(818{826)159 3135 y(\(1989\).)-226 3353 y([BuMa2])47 b(V.A.)31 b(Buslo)m(v)f(and)g(K.A.)h(Mak)-5 b(aro)m(v,)32 b(\\Life)e(spans)f(and)h(least)g(eigen)m(v)-5 b(alues)30 b(of)h(an)f(op)s(erator)h(of)f(small)159 3498 y(di\013usion")36 b(\(Russian\),)j(Mat.)64 b(Zametki)38 b Ft(51)g Fu(,)i(20{31)g (\(1992\);)k(translation)37 b(in)g(Math.)64 b(Notes)39 b Ft(51)p Fu(,)159 3644 y(14{21)32 b(\(1992\).)-52 3862 y([D1])47 b(E.B.)32 b(Da)m(vies,)h(\\Metastable)h(states)f(of)e (symmetric)g(Mark)m(o)m(v)j(semigroups.)43 b(I.)32 b(Pro)s(c.)44 b(Lond.)g(Math.)159 4007 y(So)s(c.)c(I)s(I)s(I,)30 b(Ser.)40 b Ft(45)p Fu(,)31 b(133{150)i(\(1982\).)-52 4225 y([D2])47 b(E.B.)40 b(Da)m(vies,)j(\\Metastable)e(states)f(of)g(symmetric)e(Mark) m(o)m(v)j(semigroups.)66 b(I)s(I.)39 b(J.)g(Lond.)67 b(Math.)159 4371 y(So)s(c.)40 b(I)s(I,)30 b(Ser.)40 b Ft(26)p Fu(,)31 b(541{556)j(\(1982\).)-82 4589 y([D3])77 b(E.B.)29 b(Da)m(vies,)g(\\Sp)s(ectral)f(prop)s(erties)e(of)j (metastable)g(Mark)m(o)m(v)g(semigroups",)f(J.)g(F)-8 b(unct.)41 b(Anal.)e Ft(52)p Fu(,)159 4735 y(315{329)33 b(\(1983\).)-72 4953 y([D)m(V])47 b(M.D.)32 b(Donsk)m(er)f(and)f (S.R.S.)h(V)-8 b(aradhan,)31 b(\\On)f(the)h(principal)d(eigen)m(v)-5 b(alue)30 b(of)h(second-order)g(elliptic)159 5098 y(di\013eren)m(tial)e (op)s(erators",)i(Comm.)40 b(Pure)30 b(Appl.)39 b(Math.)i Ft(29)p Fu(,)31 b(595{621)j(\(1976\).)-100 5316 y([Do)s(o])47 b(J.L.)26 b(Do)s(ob,)i(\\Classical)d(p)s(oten)m(tial)h(theory)h(and)f (its)f(probabilistic)e(coun)m(terpart",)29 b(Grundlehren)23 b(der)159 5462 y(mathematisc)m(hen)31 b(Wissensc)m(haften)f(262,)i (Springer)c(V)-8 b(erlag,)31 b(Berlin,)e(1984.)p eop %%Page: 33 33 33 32 bop 50 295 a Fa(Metastability)27 b(and)g(sp)l(e)l(ctr)l(a)2736 b Fm(33)-90 513 y Fu([FW])47 b(M.I.)25 b(F)-8 b(reidlin)21 b(and)j(A.D.)h(W)-8 b(en)m(tzell,)27 b(\\Random)d(p)s(erturbations)e (of)i(dynamical)f(systems",)j(Springer,)159 658 y(Berlin-Heidelb)s (erg-New)i(Y)-8 b(ork,)31 b(1984.)-92 876 y([GM])47 b(B.)30 b(Ga)m(v)m(eau)i(and)d(M.)h(Moreau,)h(\\Metastable)g(relaxation)f (times)f(and)g(absorbtion)g(probabilities)d(for)159 1022 y(m)m(ultidimensional)g(sto)s(c)m(hastic)31 b(systems",)g(J.)g(Ph)m (ys.)40 b(A:)31 b(Math.)41 b(Gen.)g Ft(33)p Fu(,)31 b(4837{4850)j (\(2000\).)-59 1240 y([GS])46 b(B.)28 b(Ga)m(v)m(eau)h(and)e(L.S.)g(Sc) m(h)m(ulman,)g(\\Theory)h(of)f(nonequilibrium)c(\014rst-order)j(phase)h (transitions)f(for)159 1385 y(sto)s(c)m(hastic)31 b(dynamics",)f(J.)g (Math.)41 b(Ph)m(ys.)g Ft(39)p Fu(,)31 b(1517{1533)j(\(1998\).)-101 1604 y([HS3])46 b(B.)40 b(Hel\013er,)i(and)d(J.)g(Sj\177)-45 b(ostrand,)42 b(\\Multiple)37 b(w)m(ells)i(in)f(the)i(semiclassical)e (limit.)66 b(I)s(I)s(I.)38 b(In)m(teraction)159 1749 y(through)30 b(nonresonan)m(t)g(w)m(ells",)g(Math.)41 b(Nac)m(hr.)g Ft(124)p Fu(,)32 b(263{313)h(\(1985\).)-126 1967 y([HKS])45 b(R.A.)26 b(Holley)-8 b(,)26 b(S.)f(Kusuok)-5 b(a,)26 b(S.)e(W.)i(Stro)s(o)s(c)m(k,)h(Asymptotics)e(of)g(the)h(sp)s (ectral)e(gap)i(with)e(applications)159 2113 y(to)31 b(the)g(theory)f(of)h(sim)m(ulated)e(annealing.)39 b(J.)31 b(F)-8 b(unct.)41 b(Anal.)f Ft(83)31 b Fu(\(1989\),)i(333{347)-139 2331 y([HMS])46 b(W.)27 b(Huisinga,)e(S.)h(Meyn,)i(and)e(Ch.)f(Sc)m(h)s (\177)-48 b(utte,)28 b(\\Phase)f(transitions)d(and)i(metastabilit)m(y)g (for)g(Mark)m(o)m(v-)159 2476 y(ian)j(and)h(molecular)g(systems",)h (preprin)m(t,)d(FU)j(Berlin)e(\(2002\).)-115 2695 y([Jac)m(k])47 b(J.D.)28 b(Jac)m(kson,)g(\\Classical)e(electro)s(dynamics".)39 b(Second)27 b(edition.)38 b(John)26 b(Wiley)h(&)f(Sons,)i(Inc.,)g(New) 159 2840 y(Y)-8 b(ork-London-Sydney)g(,)30 b(1975.)-53 3058 y([Ka])46 b(T.)41 b(Kato,)k(\\P)m(erturbation)c(theory)h(for)f (linear)f(op)s(erators",)45 b(Second)c(edition.)72 b(Grundlehren)38 b(der)159 3204 y(Mathematisc)m(hen)31 b(Wissensc)m(haften,)g(Band)f (132.)42 b(Springer-V)-8 b(erlag,)29 b(Berlin-New)h(Y)-8 b(ork,)31 b(1976.)-124 3422 y([Kolo])46 b(V.N.)22 b(Kolok)m(oltso)m(v,) j(\\Semiclassical)19 b(analysis)h(for)i(di\013usions)c(and)j(sto)s(c)m (hastic)h(pro)s(cesses",)i(Springer,)159 3567 y(Berlin,)29 b(2000.)-260 3785 y([KoMak])77 b(V.N.)37 b(Kolok)m(oltso)m(v)h(and)e (K.A.)h(Mak)-5 b(aro)m(v,)40 b(\\Asymptotic)d(sp)s(ectral)f(analysis)f (of)h(a)h(small)e(di\013usion)159 3931 y(op)s(erator)24 b(and)f(the)h(life)e(times)h(of)h(the)g(corresp)s(onding)d(di\013usion) g(pro)s(cess",)k(Russian)d(J.)i(Math.)39 b(Ph)m(ys.)159 4076 y Ft(4)p Fu(,)31 b(341{360)i(\(1996\).)-66 4295 y([Ma])47 b(P)-8 b(.)25 b(Mathieu,)h(\\Sp)s(ectra,)g(exit)f(times)f (and)g(long)g(times)h(asymptotics)g(in)e(the)i(zero)g(white)f(noise)g (limit",)159 4440 y(Sto)s(c)m(h.)41 b(Sto)s(c)m(h.)g(Rep.)f Ft(55)p Fu(,)31 b(1{20)h(\(1995\).)-46 4658 y([Mi])46 b(L.)40 b(Miclo,)j(Comp)s(ortemen)m(t)e(de)f(sp)s(ectres)g(d'op)m(\023) -43 b(erateurs)42 b(de)e(Sc)m(hr\177)-45 b(odinger)39 b(\022)-45 b(a)41 b(basse)f(temp)m(\023)-43 b(erature.)159 4804 y(Bull.)39 b(Sci.)h(Math.)h Ft(119)31 b Fu(\(1995\),)i(529{553.) -54 5022 y([RS])45 b(M.)h(Reed)f(and)g(B.)h(Simon,)i(\\Metho)s(ds)e(of) g(mo)s(dern)e(mathematical)i(ph)m(ysics.)84 b(IV.)46 b(Analysis)e(of)159 5167 y(op)s(erators",)31 b(Academic)g(Press,)f(New) h(Y)-8 b(ork-London,)30 b(1978.)12 5385 y([S])46 b(Ch.)27 b(Sc)m(h)s(\177)-48 b(utte,)29 b(\\Conformational)e(dynamics:)38 b(mo)s(delling,)25 b(theory)-8 b(,)30 b(algorithm,)d(and)g(application) f(to)p eop %%Page: 34 34 34 33 bop 50 295 a Fm(34)3225 b Fa(R)l(efer)l(enc)l(es)159 513 y Fu(biomolecules",)29 b(preprin)m(t)g(SC)g(99-18,)k(ZIB-Berlin)c (\(1999\).)-185 731 y([SFHD])47 b(Ch.)27 b(Sc)m(h)s(\177)-48 b(utte,)29 b(A.)g(Fisc)m(her,)f(W.)h(Huisinga,)e(and)h(P)-8 b(.)28 b(Deu\015hard,)g(\\A)g(direct)g(approac)m(h)g(to)h(conforma-)159 876 y(tional)g(dynamics)h(based)g(on)g(h)m(ybrid)e(Mon)m(te)k(Carlo",)e (J.)g(Comput.)40 b(Ph)m(ys.)h Ft(151)p Fu(,)31 b(146{168)j(\(1999\).) -104 1095 y([Szni])45 b(A.-S.)32 b(Sznitman,)g(\\Bro)m(wnian)g(motion,) g(obstacles)h(and)e(random)g(media",)i(Springer)d(Monographs)159 1240 y(in)f(Mathematics.)42 b(Springer,)28 b(Berlin,)h(1998.)-86 1458 y([T)-8 b(a)m(y])47 b(M.E.)24 b(T)-8 b(a)m(ylor,)25 b(\\P)m(artial)f(di\013eren)m(tial)e(equations.)38 b(Basic)24 b(theory",)h(T)-8 b(exts)24 b(in)e(Applied)f(Mathematics,)159 1604 y(Springer,)28 b(Berlin-Heidelb)s(erg-New)g(Y)-8 b(ork)31 b(\(1996\).)-76 1822 y([W1])47 b(A.D.)29 b(W)-8 b(en)m(tzell,)30 b(\\On)e(the)h(asymptotic)f(b)s(eha)m(viour)f(of)i (the)f(greatest)i(eigen)m(v)-5 b(alue)29 b(of)f(a)h(second)f(order)159 1967 y(elliptic)19 b(di\013eren)m(tial)i(op)s(erator)h(with)e(a)i (small)f(parameter)h(in)f(the)h(higher)e(deriv)-5 b(ativ)m(es",)24 b(So)m(viet)e(Math.)159 2113 y(Do)s(cl.)41 b Ft(13)p Fu(,)31 b(13{17)h(\(1972\).)-76 2331 y([W2])47 b(A.D.)29 b(W)-8 b(en)m(tzell,)29 b(\\F)-8 b(orm)m(ulas)28 b(for)f (eigenfunctions)f(and)h(eigenmeasures)h(that)g(are)g(connected)g(with)e (a)159 2476 y(Mark)m(o)m(v)32 b(pro)s(cess",)f(T)-8 b(eor.)41 b(V)-8 b(ero)5 b(jatnost.)42 b(i)30 b(Primenen.)39 b Ft(18)p Fu(,)31 b(329)h(\(1973\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF