This is a multi-part message in MIME format. ---------------0407151502781 Content-Type: text/plain; name="04-215.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-215.keywords" phase separation, Mullins-Sekerka, Hilbert expansion ---------------0407151502781 Content-Type: application/postscript; name="enza.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="enza.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: chFenza.dvi %%Pages: 45 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips chFenza.dvi -o chFenza.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.06.10:1429 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 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b(it)h(will)g(b)r(e)g(this)f (function)h(de\014ned)g(in)g(\(3.8\).)36 b(As)28 b(a)100 4721 y(function)g(from)f Fj(M)g Fq(to)h Fl(C)915 4691 y Fc(1)986 4721 y Fq(\(\012\),)g(it)g(will)g(alw)n(a)n(ys)d(denote)j Fl(m)2007 4733 y Fp(0)2058 4604 y Fk(\022)2129 4665 y Fl(d)p Fq(\()p Fl(\030)t(;)14 b Fq(\000)2333 4677 y Fo(t)2363 4665 y Fq(\))p 2129 4702 266 4 v 2238 4778 a Fl(\025)2405 4604 y Fk(\023)2466 4721 y Fq(,)28 b(the)g(\014rst)f(term)h(on)f(the)h (righ)n(t)f(in)h(\(3.3\).)199 4896 y(The)k(next)g(t)n(w)n(o)f(terms)g (on)h(the)g(righ)n(t)f(side)g(of)h(\(3.3\))g(require)e(more)h (explanation.)48 b(The)32 b(functions)g Fl(h)3479 4908 y Fp(1)3548 4896 y Fq(and)g Fl(\036)3763 4908 y Fp(1)100 5071 y Fq(giv)n(e)h(imp)r(ortan)n(t)h(corrections)e(to)i(the)g(leading) g(term)g Fl(m)1931 5083 y Fp(0)1982 4954 y Fk(\022)2053 5014 y Fl(d)p Fq(\()p Fl(\030)t(;)14 b Fq(\000)2257 5026 y Fo(t)2286 5014 y Fq(\))p 2053 5052 V 2162 5128 a Fl(\025)2329 4954 y Fk(\023)2390 5071 y Fq(.)56 b(Long)33 b(range)g(corrections)f (are)h(giv)n(en)h(b)n(y)100 5240 y Fl(\036)149 5252 y Fp(1)186 5240 y Fq(.)39 b(It)29 b(will)f(b)r(e)h(a)e(smo)r(oth)h (function)h(with)g(deriv)-5 b(ativ)n(es)27 b(of)h(all)g(orders,)f(and)h (will)g(satisfy)g(a)g(Lipsc)n(hitz)g(b)r(ound)g(that)100 5340 y(is)34 b Fn(indep)l(endent)i(of)h Fl(\025)p Fq(.)58 b(Close)33 b(to)h(the)h(surface,)g(more)f(rapidly)f(v)-5 b(arying)33 b(corrections)g(ma)n(y)g(b)r(e)i(required,)g(and)f(if)0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)29 b(14:29)1398 b Fq(9)p eop %%Page: 10 10 10 9 bop 100 83 a Fq(so,)33 b(these)g(are)f(to)h(b)r(e)g(enco)r(ded)g (in)g Fl(h)1294 95 y Fp(1)1331 83 y Fq(.)53 b(W)-7 b(e)33 b(shall)g(deriv)n(e)e(an)i(equation)f(for)h Fl(h)2650 95 y Fp(1)2687 83 y Fq(,)h(and)f(as)f(a)g(consequence)g(of)h(this)100 183 y(equation,)27 b(shall)g(see)g(that)h Fl(h)1018 195 y Fp(1)1083 183 y Fq(is)f(a)g(rapidly)g(deca)n(ying)g(function)h(of)f Fl(z)t Fq(,)g(lik)n(e)h Fl(m)2599 153 y Fc(0)2599 203 y Fp(0)2663 183 y 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y Fo(\025)1795 1426 y Fq(from)g(\(3.1\))g(and)g(this)g(appro)n(ximation.)35 b(First)25 b(recall)f(that)h(the)100 1526 y(Cahn{Hilliard)g(equation)h (is)g(conserv)-5 b(ativ)n(e)25 b(in)i(that)1799 1459 y Fk(R)1838 1555 y Fp(\012)1903 1526 y Fl(m)p Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\)d)p Fl(\030)t Fq(,)28 b(do)r(es)e(not)g(dep)r (end)i(on)e Fl(t)p Fq(.)36 b(If)27 b(this)g(conserv)-5 b(ation)100 1625 y(la)n(w)27 b(is)g(to)g(hold)h(at)f(ev)n(ery)g(order,) f(w)n(e)h(w)n(ould)g(require)1228 1776 y Fk(Z)1274 1965 y Fp(\012)1339 1772 y Fk(\022)1414 1833 y Fq(1)p 1410 1870 49 4 v 1410 1946 a Fl(\025)1469 1889 y(m)1542 1855 y Fc(0)1542 1910 y Fp(0)1592 1772 y Fk(\022)1664 1833 y Fl(d)p Fq(\()p Fl(\030)t(;)14 b Fq(\000)1868 1845 y Fo(t)1897 1833 y Fq(\))p 1664 1870 266 4 v 1772 1946 a Fl(\025)1939 1772 y Fk(\023)2014 1889 y Fl(V)2062 1901 y Fp(0)2100 1889 y Fq(\()p Fl(s)p Fq(\()p Fl(\030)t Fq(\)\))2307 1772 y Fk(\023)2383 1889 y Fq(d)p Fl(\030)28 b Fq(=)22 b(0)27 b Fl(:)1016 b Fq(\(3)p Fl(:)p 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Fq(is)i(giv)n(en)f(b)n(y)838 4047 y Fl(\026)888 4059 y Fp(0)925 4047 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))24 b(=)1207 3934 y Fk(Z)1253 4123 y Fp(\012)1319 4047 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))1583 3930 y Fk(\022)1658 3991 y Fq(1)p 1655 4028 49 4 v 1655 4104 a Fl(\025)1713 4047 y(m)1786 4013 y Fc(0)1786 4068 y Fp(0)1837 3930 y Fk(\022)1908 3991 y Fl(d)p Fq(\()p Fl(\021)s(;)g Fq(\000)2116 4003 y Fo(t)2146 3991 y Fq(\))p 1908 4028 271 4 v 2019 4104 a Fl(\025)2188 3930 y Fk(\023)2263 4047 y Fl(V)2311 4059 y Fp(0)2349 4047 y Fq(\()p Fl(s)p Fq(\()p Fl(\021)s Fq(\))p Fl(;)g(t)p Fq(\))2627 3930 y Fk(\023)2703 4047 y Fq(d)p Fl(\021)22 b Fq(+)c Fl(c)2931 4059 y Fp(0)2968 4047 y Fq(\()p Fl(t)p Fq(\))626 b(\(3)p Fl(:)p Fq(14\))100 4315 y(where)26 b Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))28 b(is)f(the)h(Neumann)f(Greens)g(function)h(for)e (\012)h(\(see)g(the)h(app)r(endix\),)f(and)g Fl(c)2954 4327 y Fp(0)2992 4315 y Fq(\()p Fl(t)p Fq(\))g(is)g(a)g(constan)n(t)f (\(in)i 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y(on)c(\000)267 195 y Fo(t)324 183 y Fq(so)g(that)1663 282 y Fj(k)p Fl(\026)1755 294 y Fp(0)1792 282 y Fj(k)1834 297 y Fp(Lip)o(\(\012\))2056 282 y Fj(\024)c Fl(C)1485 b Fq(\(3)p Fl(:)p Fq(16\))100 431 y(W)-7 b(e)35 b(can)g(no)n(w)g (explain)g(wh)n(y)g(w)n(e)g(ha)n(v)n(e)g(written)g(the)h(\014rst)f (order)f(corrections)g(to)h Fl(m)2864 443 y Fp(0)2936 431 y Fq(as)g(a)g(sum)h(of)f(t)n(w)n(o)g(terms,)100 531 y Fl(\025h)196 543 y Fp(1)244 531 y Fq(+)11 b Fl(\025\036)417 543 y Fp(1)454 531 y Fq(.)36 b(The)24 b(p)r(oin)n(t)g(is)g(that)g Fl(\026)1199 543 y Fp(0)1236 531 y Fq(,)h(b)r(eing)f(an)f(appro)n (ximate)f(single)h(la)n(y)n(er)f(p)r(oten)n(tial,)j(cannot)e(deca)n(y)g (rapidly)g(to)h(a)100 630 y(constan)n(t:)33 b(Single)22 b(la)n(y)n(er)f(p)r(oten)n(tials)h(deca)n(y)f(quite)i(slo)n(wly)-7 b(,)23 b(esp)r(ecially)e(in)i Fl(I)-14 b(R)2515 594 y Fp(2)2552 630 y Fq(.)35 b(Therefore,)23 b(w)n(e)f(split)g(the)h (correction)100 730 y(in)n(to)28 b(t)n(w)n(o)g(pieces:)38 b(One,)29 b(giv)n(en)f(b)n(y)g Fl(\036)1284 742 y Fp(1)1350 730 y Fq(will)h(b)r(e)g(long)f(range,)g(and)g(will)h(ha)n(v)n(e)e(a)i (purely)f(p)r(oten)n(tial)g(theoretic)h(origin)100 830 y(and)i(sp)r(eci\014cation.)47 b(It)31 b(will)h(ho)n(w)n(ev)n(er)d(b)r (e)i(de\014ned)h(in)f(the)h(whole)f(domain,)h(not)f(just)h(in)f(an)g (\\outer)f(la)n(y)n(er".)45 b(The)100 929 y(function)39 b Fl(h)484 941 y Fp(1)560 929 y Fq(will)g(pro)n(vide)e(corrections)g (to)i Fl(m)1652 941 y Fp(0)1715 929 y Fq(+)25 b Fl(\025\036)1902 941 y Fp(1)1979 929 y Fq(that)39 b(are)f(required)f(near)h(\000.)70 b(As)39 b(w)n(e)f(shall)h(see,)i(suc)n(h)100 1029 y(corrections)32 b(are)h Fn(only)i Fq(required)e(v)n(ery)g(near)g(\000;)k(the)d (equation)g(determining)g Fl(h)2692 1041 y Fp(1)2763 1029 y Fq(shall)g(force)f(its)h(supp)r(ort)g(to)g(b)r(e)100 1129 y(exp)r(onen)n(tially)29 b(lo)r(calized)g(near)g(\000.)44 b(W)-7 b(e)30 b(no)n(w)f(use)h(use)g(\(3.5\))f(to)h(determine)g Fl(\036)2616 1141 y Fp(1)2654 1129 y Fq(.)44 b(Assuming)29 b(that)i Fl(\036)3336 1141 y Fp(1)3403 1129 y Fq(enco)r(des)f(all)100 1228 y(\014rst)d(order)f(long)h(range)f(corrections)g(to)h Fl(m)1489 1240 y Fp(0)1527 1228 y Fq(,)g(so)g(that)h Fl(m)1932 1198 y Fc(0)1932 1249 y Fp(0)1997 1228 y Fq(and)f Fl(h)2206 1240 y Fp(1)2271 1228 y Fq(deca)n(y)g(rapidly)g(for)g Fl(\030)32 b Fq(far)27 b(a)n(w)n(a)n(y)e(from)i(\000)3563 1240 y Fo(t)3593 1228 y Fq(,)820 1408 y Fl(m)893 1420 y Fp(1)930 1408 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))23 b Fj(\031)g(\006)p Fq(1)18 b(+)g Fl(\025\036)1517 1420 y Fp(1)1555 1408 y Fq(\()p Fl(\030)t(;)c(t)p Fq(\))166 b(and)g(\001)p Fl(m)2334 1420 y Fp(1)2371 1408 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))24 b Fj(\031)f Fl(\025)p Fq(\001)p Fl(\036)2820 1420 y Fp(1)2858 1408 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))28 b Fl(:)608 b Fq(\(3)p Fl(:)p Fq(17\))100 1587 y(No)n(w)27 b(consider)f(\(3.5\).)37 b(Because)26 b(of)i(\(3.17\),)f(and)g(b)r(ecause)h(of)f(the)h(rapid)f(deca)n(y)g(of) g Fl(h)2814 1599 y Fp(1)2851 1587 y Fq(,)h(for)f Fl(\030)32 b Fq(far)27 b(from)g(\000)3472 1599 y Fo(t)3501 1587 y Fq(,)1410 1830 y Fl(\026)p Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))24 b Fj(\031)1756 1774 y Fq(1)p 1752 1811 49 4 v 1752 1887 a Fl(\025)1811 1830 y(f)9 b Fq(\()p Fj(\006)p Fq(1)17 b(+)h Fl(\025\036)2197 1842 y Fp(1)2235 1830 y Fq(\()p Fl(\030)t(;)c(t)p Fq(\)\))29 b Fl(:)100 2062 y Fq(T)-7 b(o)27 b(leading)g(order)f(in)i Fl(\025)p Fq(,)g Fl(f)9 b Fq(\()p Fj(\006)p Fq(1)17 b(+)i Fl(\025\036)1308 2074 y Fp(1)1346 2062 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\)\))24 b(=)e Fl(\025f)1758 2032 y Fc(0)1782 2062 y Fq(\(1\))p Fl(\036)1937 2074 y Fp(1)1974 2062 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\).)38 b(Hence)28 b(w)n(e)f(m)n(ust)h(ha)n(v)n (e)1523 2306 y Fl(\036)1572 2318 y Fp(1)1610 2306 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))23 b(=)1970 2250 y(1)p 1902 2287 179 4 v 1902 2363 a Fl(f)1952 2339 y Fc(0)1975 2363 y Fq(\(1\))2091 2306 y Fl(\026)2141 2318 y Fp(0)2178 2306 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))1339 b(\(3)p Fl(:)p 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b(then)i(follo)n(ws)d(immediately)i(from)f(\(3.16\))g(that)1585 3104 y Fj(k)p Fl(\036)1676 3116 y Fp(1)1713 3104 y Fj(k)1755 3119 y Fp(Lip)o(\(\012\))1978 3104 y Fj(\024)2132 3048 y Fl(C)p 2075 3085 V 2075 3161 a(f)2125 3137 y Fc(0)2148 3161 y Fq(\(1\))2292 3104 y Fl(:)1373 b Fq(\(3)p Fl(:)p Fq(19\))100 3352 y(Moreo)n(v)n(er,)25 b(with)j(this)g(de\014nition)1046 3602 y Fl(\025)p Fq(\001\()p Fl(\025\036)1292 3614 y Fp(1)1332 3602 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\)\))23 b(=)1721 3546 y Fl(\025)p 1656 3583 V 1656 3659 a(f)1706 3635 y Fc(0)1729 3659 y Fq(\(1\))1845 3602 y Fl(m)1918 3568 y Fc(0)1918 3622 y Fp(0)1969 3485 y Fk(\022)2040 3546 y Fl(d)p Fq(\()p Fl(\030)t(;)14 b Fq(\000)2244 3558 y Fo(t)2274 3546 y Fq(\))p 2040 3583 266 4 v 2149 3659 a Fl(\025)2316 3485 y Fk(\023)2391 3602 y Fl(V)2439 3614 y Fp(0)2477 3602 y Fq(\()p Fl(s)p Fq(\()p Fl(\030)t Fq(\))p Fl(;)g Fq(\000)2741 3614 y Fo(t)2770 3602 y Fq(\))28 b Fl(:)835 b Fq(\(3)p Fl(:)p Fq(20\))100 3856 y(No)n(w)30 b(w)n(e)h(examine)g(\(3.2\))g(close)f(to)h(\000)1308 3868 y Fo(t)1369 3856 y Fq(to)g(deduce)g(an)g(equation)f(for)h Fl(h)2394 3868 y Fp(1)2462 3856 y Fq(and)g(the)h(motion)f(of)g(\000) 3209 3868 y Fo(t)3238 3856 y Fq(.)48 b(Since)31 b(in)h(\(3.2\))100 3956 y(time)e(en)n(ters)e(simply)i(as)f(a)g(parameter)f(w)n(e)h(a)n(v)n (oid)f(writing)h(it)h(in)g(the)g(follo)n(wing,)f(when)h(no)f(confusion) g(arises.)41 b(W)-7 b(e)100 4055 y(need)27 b(to)h(express)e(the)i (Laplacian)f(in)g(the)h(\()p Fl(z)t(;)14 b(s)p Fq(\))28 b(co)r(ordinate)e(system.)37 b(This)27 b(is)h(easily)f(w)n(ork)n(ed)f (out)h(to)h(b)r(e)904 4234 y Fl(\025)952 4200 y Fp(2)989 4234 y Fq(\001)p Fl(f)k Fq(=)23 b(\()p Fl(f)1292 4246 y Fo(z)r(z)1383 4234 y Fq(+)18 b Fl(\025)1514 4200 y Fp(2)1551 4234 y Fl(f)1592 4246 y Fo(ss)1659 4234 y Fq(\))h Fj(\000)f Fl(\025K)6 b Fq(\()p Fl(s)p Fq(\))p Fl(f)2062 4246 y Fo(z)2118 4234 y Fj(\000)18 b Fl(\025)2249 4200 y Fp(2)2287 4234 y Fl(K)2364 4200 y Fp(2)2401 4234 y Fq(\()p Fl(s)p Fq(\))p 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y(\014)2494 5255 y(\014)2494 5305 y(\014)2521 5359 y Fo(z)r Fp(=)p Fo(d)p Fp(\()p Fo(\030)r(;)p Fp(\000\))p Fo(=\025)2882 5276 y Fq(+)18 b Fj(O)r Fq(\()p Fl(\025)p Fq(\))29 b Fl(;)0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)g(14:29)1377 b Fq(11)p eop %%Page: 12 12 12 11 bop 100 83 a Fq(In)37 b(what)h(follo)n(ws,)h(w)n(e)e(shall)h(use) f(primes)g(to)h(denote)g(deriv)-5 b(ativ)n(es)36 b(with)i(resp)r(ect)g (to)f Fl(z)t Fq(.)66 b(W)-7 b(e)38 b(obtain)g(from)f(the)100 183 y(calculations)26 b(ab)r(o)n(v)n(e)g(and)i(\(3.2\))f(that)261 433 y Fl(\026)311 445 y Fp(0)349 433 y Fq(\()p Fl(\025z)t(;)14 b(s)p Fq(\))23 b(=)704 377 y(1)p 700 414 49 4 v 700 490 a Fl(\025)772 433 y Fq([)q Fj(\000)p Fl(m)934 399 y Fc(00)934 454 y Fp(0)975 433 y Fq(\()p Fl(z)t Fq(\))c(+)f Fl(f)9 b Fq(\()p Fl(m)1339 445 y Fp(0)1376 433 y Fq(\()p Fl(z)t Fq(\)\)])18 b(+)g([)p Fl(h)1710 399 y Fc(00)1710 454 y Fp(1)1753 433 y Fq(\()p Fl(z)t(;)c(s)p Fq(\))k(+)g Fl(K)6 b Fq(\()p Fl(s)p Fq(\))p Fl(m)2290 399 y Fc(0)2290 454 y Fp(0)2327 433 y Fq(\()p Fl(z)t Fq(\))18 b(+)g Fl(f)2585 399 y Fc(0)2608 433 y Fq(\()p Fl(m)2713 445 y Fp(0)2751 433 y Fq(\)\()p Fl(\036)2864 445 y Fp(1)2920 433 y Fq(+)g Fl(h)3051 445 y Fp(1)3088 433 y Fq(\)])h(+)f Fj(O)r Fq(\()p Fl(\025)p Fq(\))263 b(\(3)p Fl(:)p Fq(22\))100 678 y(First,)27 b(the)h(term)g(prop)r(ortional)e(to)h Fl(\025)1294 648 y Fc(\000)p Fp(1)1411 678 y Fq(in)h(\(3.22\))f(m)n(ust)h(v)-5 b(anish,)27 b(and)h(so)f Fl(m)2570 690 y Fp(0)2634 678 y Fq(m)n(ust)h(satisfy)1489 864 y Fj(\000)p Fl(m)1627 830 y Fc(00)1627 885 y Fp(0)1669 864 y Fq(\()p Fl(z)t Fq(\))18 b(+)g Fl(f)9 b Fq(\()p Fl(m)2032 876 y Fp(0)2069 864 y Fq(\()p Fl(z)t Fq(\)\))23 b(=)g(0)k Fl(:)100 1051 y Fq(This)f(equation)g(is)g(satis\014ed)g(b)n(y)g(the)h(free)f(energy)f (minimizing)i(pro\014le)41 b(\026)-57 b Fl(m)p Fq(,)26 b(and)h(this)f(forces)g Fl(m)3134 1063 y Fp(0)3197 1051 y Fq(to)h(b)r(e)f(equal)g(to)42 b(\026)-58 b Fl(m)100 1151 y Fq({)27 b(up)h(to)f(corrections)f(that)i(are)e(exp)r(onen)n 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y(are)f(Lipsc)n(hitz,)i(w)n(e)f(\014nally)g(ha)n(v)n(e)1029 4769 y Fj(L)p Fl(h)1134 4781 y Fp(1)1171 4769 y Fq(\()p Fl(z)t(;)14 b(s)p Fq(\))23 b(=)1464 4652 y Fk(\022)1570 4713 y Fq(3)p 1536 4750 111 4 v 1536 4767 a Fj(p)p 1605 4767 42 4 v 68 x Fq(2)1656 4769 y Fl(\026)1706 4781 y Fp(0)p Fo(;)p Fp(0)1796 4769 y Fq(\()p Fl(s;)14 b Fq(0\))19 b Fj(\000)f Fl(K)6 b Fq(\()p Fl(s)p Fq(\))2260 4652 y Fk(\023)2350 4769 y Fq(\026)-57 b Fl(m)2408 4735 y Fc(0)2431 4769 y Fq(\()p Fl(z)t Fq(\))18 b(+)g Fj(O)r Fq(\()p Fl(\025)p Fq(\))29 b Fl(:)817 b Fq(\(3)p Fl(:)p Fq(27\))100 5036 y(The)24 b(op)r(erator)f Fj(L)i Fq(is)g(self)g(adjoin)n(t)f(on)h Fl(L)1353 5005 y Fp(2)1390 5036 y Fq(\()p Fl(I)-14 b(R)q Fq(\),)26 b(and)e(has)g(a)h(n)n(ull)f(space)g(spanned)h(b)n(y)40 b(\026)-58 b Fl(m)2844 5005 y Fc(0)2867 5036 y Fq(.)36 b(Therefore,)25 b(the)g(condition)100 5135 y(for)i(solv)-5 b(abilit)n(y)27 b(of)g Fj(L)p Fl(h)821 5147 y Fp(1)881 5135 y Fq(=)c Fl(g)30 b Fq(is)1564 5185 y Fk(Z)1610 5373 y Fo(I)-16 b(R)1692 5298 y Fl(g)s Fq(\()p Fl(z)t Fq(\))15 b(\026)-57 b Fl(m)1915 5263 y Fc(0)1938 5298 y Fq(\()p Fl(z)t Fq(\)d)p Fl(z)26 b Fq(=)c(0)28 b Fl(:)1352 b Fq(\(3)p Fl(:)p Fq(28\))0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)29 b(14:29)1377 b Fq(12)p eop %%Page: 13 13 13 12 bop 100 83 a Fq(Eviden)n(tly)27 b(this)h(is)f(p)r(ossible)g(in)h (the)g(case)f(at)g(hand)h(if)g(and)f(only)h(if)924 368 y Fl(\026)974 380 y Fp(0)p Fo(;)p Fp(0)1064 368 y Fq(\()p Fl(s;)14 b Fq(0\))23 b(=)g Fl(S)5 b(K)h Fq(\()p Fl(s)p Fq(\))p Fl(;)345 b(S)28 b Fq(=)2138 312 y(1)p 2138 349 42 4 v 2138 425 a(4)2203 255 y Fk(Z)2249 444 y Fo(I)-16 b(R)2331 368 y Fq(\()16 b(\026)-58 b Fl(m)2436 334 y Fc(0)2459 368 y Fq(\()p Fl(z)t Fq(\)\))2598 322 y Fp(2)2649 368 y Fl(dz)27 b Fq(=)2855 243 y Fj(p)p 2925 243 V 2925 312 a Fq(2)p 2855 349 111 4 v 2890 425 a(3)3688 368 y(\(3)p Fl(:)p Fq(29\))100 643 y(in)d(whic)n(h)h(case)f(the)h(righ)n(t)e(hand)i (side)f(of)h(\(3.27\))f(v)-5 b(anishes)24 b(up)g(to)h Fj(O)r Fq(\()p Fl(\025)p Fq(\),)i(and)d(so)g(w)n(e)g(tak)n(e)g Fl(h)3004 655 y Fp(1)3064 643 y Fj(\021)f Fq(0.)35 b(In)25 b(other)f(w)n(ords,)100 743 y(the)k(compatibilit)n(y)g(condition)g (\(3.28\))f(forces)g(\(3.29\))h(and)f(allo)n(ws)g(us)h(to)g(tak)n(e)g Fl(h)2678 755 y Fp(1)2738 743 y Fq(=)c(0,)k(so)f(that)i(there)e(is)h (no)g(short)100 843 y(range)e(correction)g(at)h(the)h(\014rst)g(order)e (*.)36 b(\(Short)28 b(range)e(corrections)g(will)h(b)r(e)h(required)f (at)h(higher)f(orders\).)199 945 y(Next,)33 b(w)n(e)e(iden)n(tify)h Fl(V)910 957 y Fp(0)948 945 y Fq(:)45 b(It)31 b(is)h(clear)e(from)h (\(3.29\))g(that)h Fl(\026)2077 957 y Fp(0)p Fo(;)p Fp(0)2198 945 y Fq(is)g(the)g(Diric)n(hlet)f(extension)g(of)h Fl(S)5 b(K)36 b Fq(on)31 b(\000,)i(with)100 1077 y Fl(S)h Fq(=)279 1008 y Fj(p)p 348 1008 42 4 v 69 x Fq(2)o Fl(=)p Fq(3.)48 b(By)31 b(standard)g(elemen)n(ts)g(of)g(the)h(theory)f(of)g(single)g (la)n(y)n(er)f(p)r(oten)n(tials)h(\(see)g(the)h(the)g(app)r(endix\))g (and)100 1177 y(their)27 b(Diric)n(hlet)h(data,)217 1443 y Fl(\026)267 1455 y Fp(0)p Fo(;)p Fp(0)357 1443 y Fq(\()p Fl(\030)t(;)14 b Fq(\000\))19 b Fj(\000)690 1387 y Fq(1)p 662 1424 99 4 v 662 1500 a Fj(j)p Fq(\000)p Fj(j)783 1330 y Fk(Z)830 1519 y Fp(\000)889 1443 y Fl(\026)939 1455 y Fp(0)p Fo(;)p Fp(0)1029 1443 y Fq(\()p Fl(\021)s Fq(\)d)p Fl(\021)27 b Fq(=)1338 1330 y Fk(Z)1385 1519 y Fp(\000)1444 1443 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))p Fl(V)1742 1455 y Fp(0)1780 1443 y Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)1985 1455 y Fo(\021)2045 1443 y Fj(\000)2166 1387 y Fq(1)p 2138 1424 V 2138 1500 a Fj(j)p Fq(\000)p Fj(j)2260 1330 y Fk(Z)2306 1519 y Fp(\000)2365 1330 y Fk(Z)2411 1519 y Fp(\000)2470 1443 y Fl(G)p Fq(\()p Fl(\030)t(;)g(\021) s Fq(\))p Fl(V)2768 1455 y Fp(0)2807 1443 y Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)3012 1455 y Fo(\021)3053 1443 y Fq(d)p Fl(S)3150 1455 y Fo(\030)3270 1443 y Fl(\030)27 b Fj(2)c Fq(\012)217 b(\(3)p Fl(:)p Fq(30\))100 1716 y(where,)27 b(with)h Fj(T)597 1728 y Fp(\000)670 1716 y Fq(denoting)f(the)h(Diric)n (hlet{Neumann)g(op)r(erator)e(for)h(\000,)1190 1990 y Fl(V)1238 2002 y Fp(0)1276 1990 y Fq(\()p Fl(\030)t Fq(\))c(=)g Fl(S)5 b Fj(T)1592 2002 y Fp(\000)1651 1873 y Fk(\022)1712 1990 y Fl(K)h Fq(\()p Fj(\001)p Fq(\))19 b Fj(\000)2016 1934 y Fq(1)p 1988 1971 V 1988 2047 a Fj(j)p Fq(\000)p Fj(j)2109 1877 y Fk(Z)2156 2066 y Fp(\000)2215 1990 y Fl(K)6 b Fq(\()p Fl(s)p Fq(\)d)p Fl(s)2480 1873 y Fk(\023)2555 1990 y Fq(\()p Fl(\030)t Fq(\))28 b Fl(:)978 b Fq(\(3)p Fl(:)p Fq(31\))100 2269 y(Comparing)43 b(with)j(\(3.15\),)i(w)n(e)d (see)g(that)g(\(3.31\))f(determines)h(the)g(v)n(elo)r(cit)n(y)f (\014eld,)50 b(and)45 b(w)n(e)f(recognize)g(it)h(as)100 2368 y(the)e(Mullins{Sek)n(erk)-5 b(a)42 b(\015o)n(w.)82 b(Next,)48 b(since)42 b(for)h(an)n(y)f(simple)h(closed)g(curv)n(e)f (\000,)2812 2301 y Fk(R)2851 2398 y Fp(\000)2910 2368 y Fl(K)6 b Fq(\()p Fl(s)p Fq(\)d)p Fl(s)49 b Fq(=)f(2)p Fl(\031)s Fq(,)f(w)n(e)c(ha)n(v)n(e)100 2431 y Fk(Z)146 2620 y Fp(\000)205 2544 y 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Fq(Since)i(w)n(e)f(require)f(that)i Fl(\026)1051 3157 y Fp(0)1117 3145 y Fq(simply)f(b)r(e)h(a)f (\\smeared")f(v)n(ersion)g(of)h Fl(\026)2403 3157 y Fp(0)p Fo(;)p Fp(0)2493 3145 y Fq(,)h(w)n(e)f(m)n(ust)h(use)f(this)h(same)e (constan)n(t)h(as)100 3245 y(the)g(constan)n(t)e Fl(c)613 3257 y Fp(0)651 3245 y Fq(\()p Fl(t)p Fq(\))i(in)g(\(3.14\).)36 b(W)-7 b(e)28 b(\014nally)f(ha)n(v)n(e)g(that)886 3524 y Fl(\026)936 3536 y Fp(0)973 3524 y Fq(\()p Fl(\030)t(;)14 b Fq(\000\))24 b(=)1277 3411 y Fk(Z)1324 3599 y Fp(\012)1389 3524 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))1653 3407 y Fk(\022)1728 3467 y Fq(1)p 1725 3504 49 4 v 1725 3581 a Fl(\025)1783 3524 y(m)1856 3489 y Fc(0)1856 3544 y Fp(0)1907 3407 y Fk(\022)1978 3467 y Fl(d)p Fq(\()p Fl(\021)s(;)g Fq(\000)2186 3479 y Fo(t)2216 3467 y Fq(\))p 1978 3504 271 4 v 2089 3581 a Fl(\025)2258 3407 y Fk(\023)2333 3524 y Fl(V)2381 3536 y Fp(0)2419 3524 y Fq(\()p Fl(s)p Fq(\()p Fl(\021)s Fq(\))p Fl(;)g(t)p Fq(\))2697 3407 y Fk(\023)2773 3524 y 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Fp(2)p Fo(\022)r Fp(\()p Fo(x;t)p Fp(\))100 815 y Fq(In)28 b(our)g(problem,)g(the)g(function)45 b(\026)-58 b Fl(m)1256 697 y Fk(\022)1327 758 y Fl(d)p Fq(\()p Fl(\030)t(;)14 b Fq(\000)1531 770 y Fo(t)1561 758 y Fq(\))p 1327 795 266 4 v 1436 871 a Fl(\025)1603 697 y Fk(\023)1692 815 y Fq(pla)n(ys)28 b(the)g(role)g(of)g(a)g(lo)r(cal)g(Maxw)n(ellian.)38 b(In)28 b(the)h(kinetic)f(theory)100 984 y(problem,)20 b(to)f(determine)g(the)g(ev)n(olution)f(of)h(the)g(lo)r(cal)f(Maxw)n (ellian,)i(one)e(just)i(needs)e(to)h(determine)g(the)g(ev)n(olution)f (of)100 1084 y(the)27 b(\\h)n(ydro)r(dynamic)f(momen)n(ts")h Fl(\032)p Fq(\()p Fl(x;)14 b(t)p Fq(\),)28 b Fl(u)p Fq(\()p Fl(x;)14 b(t)p Fq(\))28 b(and)g Fl(\022)r Fq(\()p Fl(x;)14 b(t)p Fq(\).)38 b(The)27 b(functions)h(\\cen)n(ter")e(the)i(lo)r(cal)f (Maxw)n(ellian)100 1258 y(in)40 b(exactly)f(the)h(same)f(w)n(a)n(y)g (that)h(\000)1307 1270 y Fo(t)1376 1258 y Fq(cen)n(ters)f(the)h(fron)n (t)f Fl(m)2111 1270 y Fp(0)2162 1141 y Fk(\022)2233 1202 y Fl(d)p Fq(\()p Fl(\030)t(;)14 b Fq(\000)2437 1214 y Fo(t)2467 1202 y Fq(\))p 2233 1239 V 2342 1315 a Fl(\025)2509 1141 y Fk(\023)2610 1258 y Fq(in)40 b(our)f(problem.)73 b(In)40 b(the)g(Hilb)r(ert)100 1428 y(expansion,)26 b(to)i(leading)f (order,)f(one)h(writes)1673 1531 y Fl(f)32 b Fq(=)22 b Fl(M)9 b Fq(\(1)18 b(+)g Fl(\025h)p Fq(\))100 1693 y(and)24 b(seeks)f(a)h(solution)f(of)h(the)h(equation)e(p)r(o)n(w)n (ers)g(of)h Fl(\025)h Fq(just)f(as)g(w)n(e)f(did)i(here.)35 b(The)24 b(F)-7 b(redholm)24 b(criterion)f(pro)n(vides)g(a)100 1792 y(compatibilit)n(y)29 b(condition)g(for)f(solving)g(an)h(equation) g(in)n(v)n(olving)e(the)j(linearized)e(Boltzmann)h(op)r(erator,)f(and)h (this)100 1892 y(pro)n(vides)f(the)j(equations)e(of)h(motion)g(for)g Fl(\032)p Fq(\()p Fl(x;)14 b(t)p Fq(\),)31 b Fl(u)p Fq(\()p Fl(x;)14 b(t)p Fq(\))31 b(and)f Fl(\022)r Fq(\()p Fl(x;)14 b(t)p Fq(\))31 b(just)g(as)f(the)g(compatibilit)n(y)g(condition)g(for) 100 1992 y(solving)d(an)i(equation)f(in)n(v)n(olving)f(our)h(op)r (erator)f Fj(L)i Fq(led)g(to)f(the)h(conclusion)f(that)h(\000)2781 2004 y Fo(t)2839 1992 y Fq(ev)n(olv)n(es)e(under)h(the)h(Mullins-)100 2091 y(Sek)n(erk)-5 b(a)26 b(\015o)n(w.)199 2193 y(If)g(one)f(con)n (tin)n(ues)f(the)i(Hilb)r(ert)g(expansion)e(to)h(higher)g(order,)f(one) h(obtains)g(further)g(re\014nemen)n(ts)g(to)g(the)h(ev)n(olu-)100 2292 y(tion)c(equations)g(for)g(the)h(h)n(ydro)r(dynamical)d(momen)n (ts:)35 b(Next)22 b(come)g(the)h(Na)n(vier{Stok)n(es)d(equations,)j (and)f(then)h(the)100 2392 y(Burnett)30 b(equations.)45 b(Con)n(tin)n(uing)29 b(it)i(still)g(further,)g(one)f(can)g(construct)g (high)g(order)f(appro)n(ximate)g(solutions)g(of)100 2492 y(the)24 b(Boltzmann)g(equation.)35 b(These)24 b(in)h(turn,)g(as)f(w)n (as)f(sho)n(wn)h(b)n(y)g(Ca\015isc)n(h)f([4],)i(can)f(b)r(e)h(used)f (to)g(pro)r(duce)g(solutions)100 2591 y(of)k(the)g(Boltzmann)g (equation.)37 b(This)28 b(has)g(recen)n(tly)f(b)r(een)h(extended)h(to)f 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b Fq(\000)1437 1878 y Fp(\(1\))1437 1942 y Fo(t)1526 1922 y Fq(\))24 b(=)30 b(lim)1669 1973 y Fo(s)p Fc(!)p Fp(0)1824 1865 y Fq(1)p 1824 1903 42 4 v 1825 1979 a Fl(s)1889 1829 y Fk(\020)1939 1922 y Fl(\036)1988 1934 y Fp(1)2025 1922 y Fq(\()p Fl(\030)t(;)2140 1901 y Fq(~)2134 1922 y(\000)2186 1934 y Fo(t)p Fp(+)p Fo(s)2298 1922 y Fq(\))19 b Fj(\000)f Fl(\036)2481 1934 y Fp(1)2518 1922 y Fq(\()p Fl(\030)t(;)2633 1901 y Fq(~)2627 1922 y(\000)2679 1934 y Fo(t)2709 1922 y Fq(\))2741 1829 y Fk(\021)2832 1922 y Fl(:)833 b Fq(\(3)p Fl(:)p Fq(37\))100 2161 y(This)27 b(w)n(a)n(y)-7 b(,)1306 2268 y Fl(@)p 1291 2305 79 4 v 1291 2381 a(@)5 b(t)1380 2324 y(\036)1429 2336 y Fp(1)1466 2324 y Fq(\()p Fl(\030)t(;)14 b Fq(\000)1627 2281 y Fp(\(1\))1627 2345 y Fo(t)1717 2324 y Fq(\))23 b(=)g Fl(D)1929 2336 y Fo(V)1968 2344 y Fh(0)2004 2324 y Fl(\036)p Fq(\()p Fl(\030)t(;)14 b Fq(\000)2214 2281 y Fp(\(1\))2214 2345 y Fo(t)2304 2324 y Fq(\))19 b(+)f Fj(O)r Fq(\()p Fl(\025)p Fq(\))100 2554 y(since)27 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Fp(\000)2375 87 y Fq(as)27 b(in)i(\(2.8\).)39 b(The)28 b(part)g Fj(h)p Fl(V)3240 99 y Fp(1)3278 87 y Fj(i)3310 99 y Fp(\000)3384 87 y Fq(denotes)g(the)100 186 y(a)n(v)n(erage)c(of)k Fl(V)540 198 y Fp(1)605 186 y Fq(o)n(v)n(er)e(\000.)37 b(It's)28 b(constan)n(t)f(v)-5 b(alue)27 b(is)g(determined)h(b)n(y)g(\(3.40\):)1252 401 y Fj(h)p Fl(V)1332 413 y Fp(1)1370 401 y Fj(i)1402 413 y Fp(\000)1471 401 y Fq(=)22 b Fj(\000)1727 345 y Fq(1)p 1633 382 229 4 v 1633 479 a(2)p Fj(j)p Fq(\000)1750 436 y Fp(\(1\))1750 499 y Fo(t)1839 479 y Fj(j)1885 288 y Fk(Z)1932 477 y Fp(\012)1997 401 y Fl(D)2066 413 y Fo(V)2105 421 y Fh(0)2141 401 y Fl(\036)2190 413 y Fp(1)2228 401 y Fq(\()p Fl(\030)t(;)14 b Fq(\000)2389 358 y Fp(\(1\))2389 422 y Fo(t)2478 401 y Fq(\)d)p Fl(\030)33 b(:)1040 b Fq(\(3)p Fl(:)p Fq(41\))100 643 y(where)38 b(the)h Fl(\036)554 655 y Fp(1)631 643 y Fq(is)g(giv)n(en)f(in)h(\(3.18\).)70 b(\(This)39 b(is)g(somewhat)f(di\013eren)n(t)h(from)g(the)g(expression) f(for)g Fj(h)p Fl(V)3426 655 y Fp(1)3503 643 y Fq(giv)n(en)g(in)100 743 y(Theorem)26 b(1.2,)h(but)h(as)f(w)n(e)h(shall)f(see,)g(the)h (di\013erence)g(is)f Fj(O)r Fq(\()p Fl(\025)p Fq(\)\).)199 884 y(The)22 b(part)g Fl(V)606 841 y Fp(\(0\))587 906 y(1)717 884 y Fq(will)g(b)r(e)h(orthogonal)d(to)i(the)g(constan)n(ts.) 34 b(It)22 b(will)h(b)r(e)f(determined)g(b)n(y)g(a)g(compatibilit)n(y)g (condition)100 984 y(that)36 b(arises)e(when)i(w)n(e)f(solv)n(e)g(for)g Fl(h)1273 996 y Fp(2)1310 984 y Fq(.)62 b(The)36 b(pattern)g(is)f(the)h (same)g(in)g(all)f(higher)g(orders:)52 b(The)36 b(constan)n(t)f(part) 100 1083 y(of)k(the)g Fl(k)s Fq(th)g(order)f(v)n(elo)r(cit)n(y)g (\014eld,)k Fj(h)p Fl(V)1367 1095 y Fo(k)1409 1083 y Fj(i)1441 1095 y Fp(\000)1525 1083 y Fq(will)d(b)r(e)h(determined)f(b)n (y)g(the)g(compatibilit)n(y)g(condition)f(needed)i(to)100 1224 y(solv)n(e)29 b(Laplace's)g(equation)h(for)g Fl(\026)1192 1236 y Fo(k)1233 1224 y Fq(.)46 b(The)30 b(non-constan)n(t)g(part)g Fl(V)2224 1181 y Fp(\(0\))2206 1249 y Fo(k)2344 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(the)g(curv)n(e)f(and)h(shap)r(e)f(of)h(the)g(in)n(terface)f(matter.)37 b(With)27 b(this)h(c)n(hoice)d(of)100 1767 y Fj(h)p Fl(V)180 1779 y Fp(1)218 1767 y Fj(i)250 1779 y Fp(\000)295 1767 y Fq(,)f(the)g(compatibilit)n(y)e(condition)h(\(3.39\))f(is)h (satis\014ed)g(and)g(w)n(e)g(can)f(no)n(w)h(solv)n(e)e(\(3.38\))i(for)f Fl(\026)3169 1779 y Fp(0)3216 1767 y Fq(+)9 b Fl(\025\026)3388 1779 y Fp(1)3425 1767 y Fq(.)36 b(First,)24 b(w)n(e)100 1867 y(use)h(the)g(full)h(ev)n(olution,)e(under)h(\(3.35\),)g(to)g (di\013eren)n(tiate)g(the)g(\014rst)g(term)g(in)h(\(3.38\).)35 b(Then)25 b(w)n(e)g(apply)g(the)g(Green's)100 1967 y(function)j(to)f (eac)n(h)g(of)h(the)g(pieces.)36 b(T)-7 b(aking)27 b(in)h(accoun)n(t)f (\(3.13\))f(w)n(e)i(obtain)f(that)749 2214 y Fl(\026)799 2226 y Fp(1)836 2214 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))24 b(=)1132 2158 y(1)p 1129 2195 49 4 v 1129 2271 a Fl(\025)1201 2101 y Fk(Z)1247 2289 y Fp(\012)1312 2214 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))p Fl(m)1635 2180 y Fc(0)1635 2234 y Fp(0)1687 2072 y Fk( )1763 2158 y Fl(d)p Fq(\()p Fl(\030)t(;)g Fq(\000)1967 2115 y Fp(\(1\))1967 2178 y Fo(t)2057 2158 y Fq(\))p 1763 2195 326 4 v 1902 2271 a Fl(\025)2099 2072 y Fk(!)2179 2214 y Fl(V)2227 2226 y Fp(1)2264 2214 y Fq(\()p Fl(s)p Fq(\()p Fl(\021)s Fq(\)\)d)p Fl(\021)23 b Fq(+)18 b Fl(p)p Fq(\()p Fl(\030)t(;)c(t)p Fq(\))19 b(+)f Fl(c)3019 2226 y Fp(1)3056 2214 y Fq(\()p Fl(t)p Fq(\))538 b(\(3)p Fl(:)p Fq(42\))100 2456 y(where)27 b(w)n(e)g(set)1226 2603 y Fl(p)p Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))24 b(=)1550 2490 y Fk(Z)1596 2678 y Fp(\012)1662 2603 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))1926 2510 y Fk(h)1966 2603 y Fl(D)2035 2615 y Fo(V)2074 2623 y Fh(0)2110 2603 y Fl(\036)2159 2615 y Fp(1)2197 2603 y Fq(\()p Fl(\030)t(;)g Fq(\000)2358 2559 y Fp(\(1\))2358 2623 y Fo(t)2447 2603 y Fq(\))2479 2510 y Fk(i)2533 2603 y Fq(d)p Fl(\021)31 b(;)1014 b Fq(\(3)p Fl(:)p Fq(43\))100 2809 y(and)26 b Fl(c)296 2821 y Fp(1)333 2809 y Fq(\()p Fl(t)p Fq(\))h(is)f(a)f(constan)n(t)h(\(in)g Fl(\030)t Fq(\))h(to)f(b)r(e)h(determined.)36 b(\(Again,)26 b(this)h(is)f (somewhat)f(di\013eren)n(t)i(from)e(the)i(expression)100 2908 y(for)k Fl(p)p Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))31 b(giv)n(en)g(in)h(Theorem)e(1.2,)i(but)g(as)f(w)n(e)g(shall)g(see,)h (the)g(di\013erence)f(is)g Fj(O)r Fq(\()p Fl(\025)p Fq(\)\).)51 b(In)31 b(the)h(next)g(section,)g(w)n(e)100 3008 y(shall)22 b(deriv)n(e)g(a)g(more)g(explicit)h(form)n(ula,)g(at)g(least)f(in)h (terms)f(of)h(p)r(oten)n(tial)g(theory)-7 b(,)23 b(for)f Fl(p)p Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\).)36 b(F)-7 b(or)22 b(the)h(time)g(b)r(eing,)100 3108 y(it)29 b(is)g(con)n(v)n (enien)n(t)e(to)i(w)n(ork)f(with)h(this)g(compact)g(form.)40 b(The)29 b(\014rst)g(term)f(on)h(the)g(righ)n(t)f(is)h(an)g(appro)n (ximate)e(single)100 3244 y(la)n(y)n(er)e(p)r(oten)n(tial)j(and)f(it)h (is)g(still)g(to)f(b)r(e)h(fully)g(determined,)g(since)g(w)n(e)f(do)g (not)h(kno)n(w)f Fl(V)2873 3201 y Fp(\(0\))2855 3266 y(1)2990 3244 y Fq(y)n(et.)199 3387 y(T)-7 b(o)n(w)n(ard)24 b(this)h(end,)h(w)n(e)f(\014rst)g(determine)g Fl(\036)1555 3399 y Fp(2)1593 3387 y Fq(.)36 b(As)25 b(b)r(efore,)g(consider)g Fl(\030)k Fq(far)c(from)g(\000)2800 3343 y Fp(\(1\))2800 3407 y Fo(t)2914 3387 y Fq(where)f Fl(m)3224 3356 y Fp(2)3224 3407 y(0)3275 3387 y Fj(\000)14 b Fq(1)24 b(and)h Fl(h)3627 3399 y Fp(2)3689 3387 y Fq(are)100 3486 y(negligible.)36 b(W)-7 b(e)28 b(then)g(ha)n(v)n(e)754 3702 y Fl(\026)804 3714 y Fp(0)860 3702 y Fq(+)18 b Fl(\025\026)1041 3714 y Fp(1)1102 3702 y Fj(\031)1203 3646 y Fq(1)p 1199 3683 49 4 v 1199 3759 a Fl(\025)1258 3702 y(f)1321 3635 y Fk(\000)1359 3702 y Fq(1)g Fj(\006)g Fl(\025\036)1599 3714 y Fp(1)1656 3702 y Fq(+)g Fl(\025)1787 3668 y Fp(2)1824 3702 y Fl(\036)1873 3714 y Fp(2)1911 3635 y Fk(\001)1972 3702 y Fj(\031)23 b Fl(f)2110 3668 y Fc(0)2133 3702 y Fq(\(1\))14 b(\()p Fl(\036)2334 3714 y Fp(1)2390 3702 y Fq(+)k Fl(\025\036)2570 3714 y Fp(2)2608 3702 y Fq(\))h(+)2752 3646 y Fl(f)2802 3616 y Fc(00)2844 3646 y Fq(\(1\))p 2752 3683 199 4 v 2830 3759 a(2)2960 3702 y Fl(\025\036)3057 3668 y Fp(2)3057 3723 y(1)3123 3702 y Fl(:)100 3960 y Fq(Since)31 b Fl(\026)370 3972 y Fp(0)437 3960 y Fq(=)e Fl(f)581 3930 y Fc(0)603 3960 y Fq(\(1\))p Fl(\036)758 3972 y Fp(1)796 3960 y Fq(,)k(w)n(e)e(de\014ne)g Fl(\036)1270 3972 y Fp(2)1339 3960 y Fq(b)n(y)g Fl(\036)1507 3972 y Fp(2)1568 3960 y Fq(=)1735 3904 y(1)p 1666 3941 179 4 v 1666 4017 a Fl(f)1716 3993 y Fc(0)1739 4017 y Fq(\(1\))1869 3843 y Fk(\024)1912 3960 y Fl(\026)1962 3972 y Fp(1)2018 3960 y Fj(\000)2173 3904 y Fl(f)2223 3874 y Fc(00)2265 3904 y Fq(\(1\))p 2111 3941 323 4 v 2111 4017 a(2\()p Fl(f)2235 3993 y Fc(0)2258 4017 y Fq(\(1\)\))2396 3993 y Fp(2)2443 3960 y Fl(\026)2493 3926 y Fp(2)2493 3981 y(0)2531 3843 y Fk(\025)2597 3960 y Fq(=)2695 3904 y(1)p 2695 3941 42 4 v 2695 4017 a(2)2747 3960 y Fl(\026)2797 3972 y Fp(1)2852 3960 y Fq(+)2945 3904 y(3)p 2945 3941 V 2945 4017 a(8)2997 3960 y Fl(\026)3047 3926 y Fp(2)3047 3981 y(0)3084 3960 y Fq(.)48 b(As)32 b(b)r(efore,)g(w)n(e)f(use)100 4130 y(this)h(de\014nition)h(globally)f(in)g(\012,)i(and)e Fl(\036)1390 4142 y Fp(2)1461 4130 y Fq(is)g(Lipsc)n(hitz)g(with)i(a)e (norm)g(b)r(ounded)g(indep)r(enden)n(tly)i(of)e Fl(\025)p Fq(.)52 b(W)-7 b(e)33 b(no)n(w)100 4230 y(return)24 b(to)h(\(3.2\),)g (and)g(replace)e Fl(\026)1156 4200 y Fo(\025)1225 4230 y Fq(b)n(y)h Fl(\026)1387 4242 y Fp(0)1437 4230 y Fq(+)13 b Fl(\025\026)1613 4242 y Fp(1)1676 4230 y Fq(and)24 b Fl(m)1907 4200 y Fo(\025)1975 4230 y Fq(b)n(y)h(\(3.34\).)35 b(F)-7 b(rom)25 b(\(3.21\),)f(and)h(the)g(p)r(oten)n(tial)g(theoretic) 100 4329 y(de\014nition)j(of)f Fl(\036)612 4341 y Fp(1)650 4329 y Fq(,)838 4498 y Fj(\000)18 b Fl(\025)p Fq(\001\()p Fl(m)1143 4510 y Fp(0)1200 4498 y Fq(+)g Fl(\025\036)1380 4510 y Fp(1)1437 4498 y Fq(+)g Fl(\025)1568 4464 y Fp(2)1605 4498 y Fl(h)1653 4510 y Fp(2)1709 4498 y Fq(+)g Fl(\025)1840 4464 y Fp(2)1877 4498 y Fl(\036)1926 4510 y Fp(2)1964 4498 y Fq(\))23 b(=)838 4706 y Fj(\000)935 4650 y Fq(1)p 931 4687 49 4 v 931 4763 a Fl(\025)990 4706 y Fq([)p Fl(m)1086 4672 y Fc(00)1086 4726 y Fp(0)1128 4706 y Fq(])c(+)f Fl(K)6 b(m)1403 4672 y Fc(0)1403 4726 y Fp(0)1458 4706 y Fq(+)18 b Fl(\025)1603 4589 y Fk(\024)1647 4706 y Fj(\000)p Fl(h)1760 4672 y Fc(00)1760 4726 y Fp(2)1820 4706 y Fq(+)1982 4650 y(1)p 1913 4687 179 4 v 1913 4763 a Fl(f)1963 4739 y Fc(0)1986 4763 y Fq(\(1\))2102 4706 y Fl(m)2175 4672 y Fc(0)2175 4726 y Fp(0)2212 4706 y Fl(V)2260 4718 y Fp(0)2316 4706 y Fq(+)g Fl(K)2476 4672 y Fp(2)2513 4706 y Fl(z)t(m)2629 4672 y Fc(0)2629 4726 y Fp(0)2665 4589 y Fk(\025)2728 4706 y Fq(+)g Fj(O)r Fq(\()p Fl(\025)2959 4672 y Fp(2)2997 4706 y Fq(\))28 b Fl(:)100 4924 y Fq(Clearly)-7 b(,)954 5026 y(1)p 950 5064 49 4 v 950 5140 a Fl(\025)1009 5083 y(f)9 b Fq(\()p Fl(m)1164 5095 y Fp(0)1219 5083 y Fq(+)18 b Fl(\025\036)1399 5095 y Fp(1)1456 5083 y Fq(+)g Fl(\025)1587 5048 y Fp(2)1624 5083 y Fl(h)1672 5095 y Fp(2)1728 5083 y Fq(+)g Fl(\025)1859 5048 y Fp(2)1897 5083 y Fl(\036)1946 5095 y Fp(2)1983 5083 y Fq(\))24 b(=)954 5260 y(1)p 950 5297 V 950 5373 a Fl(\025)1009 5316 y(f)9 b Fq(\()p Fl(m)1164 5328 y Fp(0)1201 5316 y Fq(\))18 b(+)g Fl(f)1384 5282 y Fc(0)1407 5316 y Fq(\()p Fl(m)1512 5328 y Fp(0)1550 5316 y Fq(\))p Fl(\036)1631 5328 y Fp(1)1687 5316 y Fq(+)g Fl(\025f)1868 5282 y Fc(0)1892 5316 y Fq(\()p Fl(m)1997 5328 y Fp(0)2034 5316 y Fq(\)[)p Fl(h)2137 5328 y Fp(2)2193 5316 y Fq(+)g Fl(\036)2325 5328 y Fp(2)2363 5316 y Fq(])g(+)g Fl(\025)2545 5260 y(f)2595 5230 y Fc(00)2638 5260 y Fq(\()p Fl(m)2743 5272 y Fp(0)2780 5260 y Fq(\))p 2545 5297 267 4 v 2658 5373 a(2)2822 5316 y Fl(\036)2871 5282 y Fp(2)2871 5337 y(1)2937 5316 y Fl(:)0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)29 b(14:29)1377 b Fq(16)p eop %%Page: 17 17 17 16 bop 100 83 a Fq(Hence)27 b(from)h(\(3.2\),)f(since)43 b(\026)-58 b Fl(m)28 b Fq(solv)n(es)e(\(3.6\),)i(and)f(since)g Fl(m)1963 95 y Fp(0)2024 83 y Fq(=)38 b(\026)-58 b Fl(m)28 b Fq(up)g(to)f(exp)r(onen)n(tially)g(small)g(corrections,)498 303 y Fl(\026)548 315 y Fp(0)603 303 y Fq(+)18 b Fl(\025\026)784 315 y Fp(1)845 303 y Fq(=)23 b([)p Fl(K)6 b(m)1106 269 y Fc(0)1106 324 y Fp(0)1161 303 y Fq(+)18 b Fl(f)1294 269 y Fc(0)1317 303 y Fq(\()p Fl(m)1422 315 y Fp(0)1459 303 y Fq(\))p Fl(\036)1540 315 y Fp(1)1578 303 y Fq(])840 511 y(+)g Fl(\025)985 394 y Fk(\024)1030 511 y Fj(L)p Fl(h)1135 523 y Fp(2)1190 511 y Fq(+)g Fl(K)1350 477 y Fp(2)1387 511 y Fl(z)t(m)1503 477 y Fc(0)1503 531 y Fp(0)1558 511 y Fq(+)1720 455 y(1)p 1651 492 179 4 v 1651 568 a Fl(f)1701 544 y Fc(0)1724 568 y Fq(\(1\))1840 511 y Fl(m)1913 477 y Fc(0)1913 531 y Fp(0)1950 511 y Fl(V)1998 523 y Fp(0)2054 511 y Fq(+)g Fl(f)2187 477 y Fc(0)2210 511 y Fq(\()p Fl(m)2315 523 y Fp(0)2352 511 y Fq(\))p Fl(\036)2433 523 y Fp(2)2490 511 y Fq(+)2583 455 y(1)p 2583 492 42 4 v 2583 568 a(2)2634 511 y Fl(f)2684 477 y Fc(00)2726 511 y Fq(\()p Fl(m)2831 523 y Fp(0)2869 511 y Fq(\))p Fl(\036)2950 477 y Fp(2)2950 531 y(1)2988 394 y Fk(\025)3050 511 y Fq(+)g Fj(O)r Fq(\()p Fl(\025)3281 477 y Fp(2)3319 511 y Fq(\))28 b Fl(:)199 795 y Fq(Therefore,)20 b(since)f Fl(\036)836 807 y Fp(1)897 795 y Fq(=)k(\(1)p Fl(=f)1151 765 y Fc(0)1173 795 y Fq(\(1\)\))p Fl(\026)1361 807 y Fp(0)1421 795 y Fq(=)g(\(1)p Fl(=)p Fq(2\))p Fl(\026)1749 807 y Fp(0)1785 795 y Fq(,)f(w)n(e)d(can)f(use)i(the)f(iden)n(tit)n(y)h (\(1)r Fj(\000)r Fl(f)2849 765 y Fc(0)2871 795 y Fq(\()p Fl(m)2976 807 y Fp(0)3013 795 y Fq(\))p Fl(=f)3137 765 y Fc(0)3160 795 y Fq(\(1\)\))j(=)g(\(3)p Fl(=)3525 726 y Fj(p)p 3593 726 V 3593 795 a Fq(2\))p Fl(m)3740 765 y Fc(0)3740 815 y Fp(0)3777 795 y Fq(,)100 894 y(and)k(ha)n(v)n(e)666 1048 y Fl(\026)716 1060 y Fp(1)776 1048 y Fq(=)877 992 y(1)p 874 1029 49 4 v 874 1105 a Fl(\025)946 931 y Fk(\022)1007 1048 y Fl(K)d Fj(\000)1230 992 y Fq(3)p 1195 1029 111 4 v 1195 1046 a Fj(p)p 1264 1046 42 4 v 68 x Fq(2)1316 1048 y Fl(\026)1366 1060 y Fp(0)1403 931 y Fk(\023)1478 1048 y Fl(m)1551 1014 y Fc(0)1551 1069 y Fp(0)772 1305 y Fq(+)855 1188 y Fk(\024)898 1305 y Fj(L)p Fl(h)1003 1317 y Fp(2)1059 1305 y Fq(+)18 b Fl(K)1219 1271 y Fp(2)1256 1305 y Fl(z)t(m)1372 1271 y Fc(0)1372 1326 y Fp(0)1427 1305 y Fq(+)1589 1249 y(1)p 1520 1286 179 4 v 1520 1362 a Fl(f)1570 1338 y Fc(0)1593 1362 y Fq(\(1\))1709 1305 y Fl(m)1782 1271 y Fc(0)1782 1326 y Fp(0)1819 1305 y Fl(V)1867 1317 y Fp(0)1923 1305 y Fq(+)g Fl(f)2056 1271 y Fc(0)2079 1305 y Fq(\()p Fl(m)2184 1317 y Fp(0)2221 1305 y Fq(\))p Fl(\036)2302 1317 y Fp(2)2359 1305 y Fq(+)2452 1249 y(1)p 2452 1286 42 4 v 2452 1362 a(2)2503 1305 y Fl(f)2553 1271 y Fc(00)2595 1305 y Fq(\()p Fl(m)2700 1317 y Fp(0)2738 1305 y Fq(\))p Fl(\036)2819 1271 y Fp(2)2819 1326 y(1)2857 1188 y Fk(\025)2919 1305 y Fq(+)g Fj(O)r Fq(\()p Fl(\025)p Fq(\))29 b Fl(:)3688 1177 y Fq(\(3)p Fl(:)p Fq(44\))100 1532 y(Notice)41 b(that)g(the)h(\014rst)f(term)g(on) g(the)g(righ)n(t)g(is)g(b)r(ounded)g(uniformly)g(in)h Fl(\025)f Fq(b)r(ecause)g Fl(\026)3033 1544 y Fp(0)3111 1532 y Fq(is)g(Lipsc)n(hitz,)k Fl(\026)3653 1544 y Fp(0)3736 1532 y Fq(=)100 1631 y Fl(\026)150 1643 y Fp(0)p Fo(;)p Fp(0)258 1631 y Fq(+)18 b Fj(O)r Fq(\()p Fl(\025)p Fq(\),)29 b(and)f Fl(m)808 1601 y Fc(0)808 1652 y Fp(0)873 1631 y Fq(deca)n(ys)e(rapidly)-7 b(.)36 b(F)-7 b(rom)27 b(\(3.44\))g(w)n(e)g (obtain)845 1904 y Fj(L)p Fl(h)950 1916 y Fp(2)1011 1904 y Fq(=)22 b Fl(\026)1148 1916 y Fp(1)1204 1904 y Fj(\000)1300 1848 y Fq(1)p 1297 1885 49 4 v 1297 1961 a Fl(\025)1369 1787 y Fk(\022)1430 1904 y Fl(K)i Fj(\000)1653 1848 y Fq(3)p 1618 1885 111 4 v 1618 1902 a Fj(p)p 1687 1902 42 4 v 69 x Fq(2)1739 1904 y Fl(\026)1789 1916 y Fp(0)1826 1787 y Fk(\023)1901 1904 y Fl(m)1974 1870 y Fc(0)1974 1925 y Fp(0)864 2162 y Fj(\000)947 2045 y Fk(\024)991 2162 y Fl(K)1068 2127 y Fp(2)1104 2162 y Fl(z)t(m)1220 2127 y Fc(0)1220 2182 y Fp(0)1275 2162 y Fq(+)1437 2106 y(1)p 1368 2143 179 4 v 1368 2219 a Fl(f)1418 2195 y Fc(0)1441 2219 y Fq(\(1\))1557 2162 y Fl(m)1630 2127 y Fc(0)1630 2182 y Fp(0)1667 2162 y Fl(V)1715 2174 y Fp(0)1771 2162 y Fq(+)18 b Fl(f)1904 2127 y Fc(0)1927 2162 y Fq(\()p Fl(m)2032 2174 y Fp(0)2070 2162 y Fq(\))p Fl(\036)2151 2174 y Fp(2)2207 2162 y Fq(+)2300 2106 y(1)p 2300 2143 42 4 v 2300 2219 a(2)2352 2162 y Fl(f)2402 2127 y Fc(00)2444 2162 y Fq(\()p Fl(m)2549 2174 y Fp(0)2586 2162 y Fq(\))p Fl(\036)2667 2127 y Fp(2)2667 2182 y(1)2705 2045 y Fk(\025)2767 2162 y Fq(+)g Fj(O)r Fq(\()p Fl(\025)p Fq(\))p Fl(:)3688 2033 y Fq(\(3)p Fl(:)p Fq(45\))100 2432 y(T)-7 b(o)29 b(solv)n(e)f(\(3.45\),)h(the)g(compatibilit)n(y)g (condition)g(\(3.28\))g(need)g(to)g(b)r(e)h(satis\014ed.)42 b(This)29 b(condition)g(will)g(determine)100 2572 y Fl(V)167 2529 y Fp(\(0\))148 2595 y(1)283 2572 y Fq(and)f(therefore)e Fl(\026)843 2584 y Fp(1)908 2572 y Fq(will)i(b)r(e)g(fully)g (determined.)37 b(Denote)733 2844 y Fl(g)773 2856 y Fp(1)810 2844 y Fq(\()p Fl(s)p Fq(\))24 b(=)1025 2731 y Fk(Z)1071 2920 y Fo(I)-16 b(R)1152 2727 y Fk(\024)1196 2844 y Fl(K)1273 2810 y Fp(2)1310 2844 y Fl(z)t(m)1426 2810 y Fc(0)1426 2864 y Fp(0)1480 2844 y Fq(+)1642 2788 y(1)p 1573 2825 179 4 v 1573 2901 a Fl(f)1623 2877 y Fc(0)1646 2901 y Fq(\(1\))1762 2844 y Fl(m)1835 2810 y Fc(0)1835 2864 y Fp(0)1873 2844 y Fl(V)1921 2856 y Fp(0)1977 2844 y Fq(+)18 b Fl(f)2110 2810 y Fc(0)2133 2844 y Fq(\()p Fl(m)2238 2856 y Fp(0)2275 2844 y Fq(\))p Fl(\036)2356 2856 y Fp(2)2412 2844 y Fq(+)2505 2788 y(1)p 2505 2825 42 4 v 2505 2901 a(2)2557 2844 y Fl(f)2607 2810 y Fc(00)2649 2844 y Fq(\()p Fl(m)2754 2856 y Fp(0)2791 2844 y Fq(\))p Fl(\036)2872 2810 y Fp(2)2872 2864 y(1)2910 2727 y Fk(\025)2968 2844 y Fl(m)3041 2810 y Fc(0)3041 2864 y Fp(0)3078 2844 y Fq(d)p Fl(z)100 3114 y Fq(Since)27 b Fl(m)389 3084 y Fc(0)389 3135 y Fp(0)427 3114 y Fq(\()p Fl(z)t Fq(\))g(is)h(ev)n(en,)f Fl(f)907 3084 y Fc(00)949 3114 y Fq(\()p Fl(m)1054 3126 y Fp(0)1091 3114 y Fq(\))c(=)g(6)p Fl(m)1349 3126 y Fp(0)1386 3114 y Fq(,)28 b Fl(\036)1486 3126 y Fp(1)1551 3114 y Fq(and)f Fl(\036)1761 3126 y Fp(2)1827 3114 y Fq(ha)n(v)n(e)f(a)h (Lipsc)n(hitz)h(b)r(ound)g(indep)r(enden)n(tly)g(on)f Fl(\025)i Fq(w)n(e)e(obtain)594 3384 y Fl(g)634 3396 y Fp(1)671 3384 y Fq(\()p Fl(s)p Fq(\))c(=)885 3271 y Fk(Z)931 3460 y Fo(I)-16 b(R)1013 3267 y Fk(\024)1135 3328 y Fq(1)p 1066 3365 179 4 v 1066 3441 a Fl(f)1116 3417 y Fc(0)1139 3441 y Fq(\(1\))1255 3384 y Fl(m)1328 3350 y Fc(0)1328 3405 y Fp(0)1365 3384 y Fl(V)1413 3396 y Fp(0)1470 3384 y Fq(+)18 b Fl(f)1603 3350 y Fc(0)1625 3384 y Fq(\()p Fl(m)1730 3396 y Fp(0)1768 3384 y Fq(\))p Fl(\036)1849 3396 y Fp(2)1887 3384 y Fq(\(0)p Fl(;)c(s)p Fq(\))k(+)2180 3328 y(1)p 2180 3365 42 4 v 2180 3441 a(2)2232 3384 y Fl(f)2282 3350 y Fc(00)2324 3384 y Fq(\()p Fl(m)2429 3396 y Fp(0)2466 3384 y Fq(\))p Fl(\036)2547 3350 y Fp(2)2547 3405 y(1)2585 3384 y Fq(\(0)p Fl(;)c(s)p Fq(\))2767 3267 y Fk(\025)2825 3384 y Fl(m)2898 3350 y Fc(0)2898 3405 y Fp(0)2935 3384 y Fq(d)p Fl(z)22 b Fq(+)c Fj(O)r Fq(\()p Fl(\025)p Fq(\))797 3592 y(=)23 b(2)p Fl(S)5 b(V)1031 3604 y Fp(0)1068 3592 y Fq(\()p Fl(s)p Fq(\))19 b(+)f Fj(O)r Fq(\()p Fl(\025)p Fq(\))p Fl(:)3688 3463 y Fq(\(3)p Fl(:)p Fq(46\))100 3812 y(where)27 b Fl(S)32 b Fq(is)c(the)g(surface)e(tension)i(de\014ned)g(in)g (\(3.29\).)36 b(W)-7 b(e)28 b(next)f(in)n(v)n(estigate)1230 4082 y Fl(g)1270 4094 y Fp(2)1307 4082 y Fq(\()p Fl(s)p Fq(\))c(=)1535 4026 y(1)p 1531 4063 49 4 v 1531 4139 a Fl(\025)1603 3969 y Fk(Z)1649 4158 y Fo(I)-16 b(R)1731 3965 y Fk(\022)1792 4082 y Fl(K)24 b Fj(\000)2014 4026 y Fq(3)p 1980 4063 111 4 v 1980 4080 a Fj(p)p 2049 4080 42 4 v 68 x Fq(2)2101 4082 y Fl(\026)2151 4094 y Fp(0)2188 3965 y Fk(\023)2263 4082 y Fq(\()p Fl(m)2368 4048 y Fc(0)2368 4103 y Fp(0)2405 4082 y Fq(\))2437 4048 y Fp(2)2475 4082 y Fq(\()p Fl(z)t Fq(\)d)p Fl(z)100 4348 y Fq(Then)560 4501 y Fl(g)600 4513 y Fp(2)637 4501 y Fq(\()p Fl(s)p Fq(\))f(=)865 4445 y(1)p 861 4482 49 4 v 861 4558 a Fl(\025)933 4388 y Fk(Z)979 4577 y Fo(I)-16 b(R)1061 4384 y Fk(\022)1122 4501 y Fl(K)24 b Fj(\000)1345 4445 y Fq(3)p 1310 4482 111 4 v 1310 4499 a Fj(p)p 1379 4499 42 4 v 69 x Fq(2)1431 4501 y Fl(\026)1481 4513 y Fp(0)p Fo(;)p Fp(0)1571 4384 y Fk(\023)1646 4501 y Fq(\()p Fl(m)1751 4467 y Fc(0)1751 4522 y Fp(0)1788 4501 y Fq(\))1820 4467 y Fp(2)1858 4501 y Fq(\()p Fl(z)t Fq(\)d)p Fl(z)d Fj(\000)2223 4445 y Fq(3)p 2164 4482 160 4 v 2164 4499 a Fj(p)p 2233 4499 42 4 v 69 x Fq(2)p Fl(\025)2347 4388 y Fk(Z)2393 4577 y Fo(I)-16 b(R)2461 4501 y Fq(\()p Fl(\026)2543 4513 y Fp(0)2599 4501 y Fj(\000)18 b Fl(\026)2732 4513 y Fp(0)p Fo(;)p Fp(0)2822 4501 y Fq(\))c(\()p Fl(m)2973 4467 y Fc(0)2973 4522 y Fp(0)3011 4501 y Fq(\()p Fl(z)t Fq(\)\))3150 4455 y Fp(2)3201 4501 y Fq(d)p Fl(z)31 b(:)100 4728 y Fq(The)g(second)f(term,)i(in)n(v)n(olving)e(the)h(di\013erence)g(b)r (et)n(w)n(een)g(the)h(\\smeared")d(and)h(exact)h(single)g(la)n(y)n(er)e (p)r(oten)n(tials)i(is)100 4827 y(easily)h(seen)h(to)g(b)r(e)h Fj(O)r Fq(\()p Fl(\025)894 4797 y Fp(2)932 4827 y Fq(\).)55 b(As)33 b(for)g(the)h(\014rst)f(one,)h(note)f(that)h(w)n(e)f(ha)n(v)n (e)f Fl(\026)2562 4839 y Fp(0)p Fo(;)p Fp(0)2652 4827 y Fq(\()p Fl(s;)14 b(z)t Fq(\))32 b(=)g Fl(S)5 b(K)28 b Fq(+)21 b Fl(a\025z)26 b Fq(+)c Fj(O)r Fq(\()p Fl(\025)3597 4797 y Fp(2)3635 4827 y Fq(\))34 b(for)100 4927 y Fl(z)26 b(>)d Fq(0)j(and)h Fl(\026)532 4939 y Fp(0)p Fo(;)p Fp(0)622 4927 y Fq(\()p Fl(s;)14 b(z)t Fq(\))23 b(=)f Fl(S)5 b(K)23 b Fq(+)17 b Fl(b\025z)j Fq(+)d Fj(O)r Fq(\()p Fl(\025)1520 4897 y Fp(2)1558 4927 y Fq(\))28 b(for)e Fl(z)g(>)d Fq(0.)36 b(The)27 b(quan)n(tit)n(y)g Fl(b)17 b Fj(\000)f Fl(a)27 b Fq(is)g(just)h(the)f(jump)h(in)f(the)g(normal)100 5026 y(deriv)-5 b(ativ)n(e)26 b(of)i Fl(\026)625 5038 y Fp(0)p Fo(;)p Fp(0)743 5026 y Fq(across)d(the)j(in)n(terface)f(at)h Fl(s)p Fq(,)f(whic)n(h)h(is)f Fl(V)2028 5038 y Fp(0)2066 5026 y Fq(\()p Fl(s)p Fq(\).)37 b(Hence)28 b(with)g Fl(C)34 b Fq(de\014ned)28 b(b)n(y)1326 5298 y Fl(C)i Fq(=)1502 5185 y Fk(Z)1548 5373 y Fo(I)-16 b(R)1630 5298 y Fj(j)p Fl(z)t Fj(j)p Fq(\()p Fl(m)1824 5263 y Fc(0)1824 5318 y Fp(0)1861 5298 y Fq(\))1893 5263 y Fp(2)1930 5298 y Fq(d)p Fl(z)27 b Fq(=)2139 5241 y(4)14 b(ln\(2\))k Fj(\000)g Fq(1)p 2139 5278 374 4 v 2305 5355 a(6)2551 5298 y Fl(;)1114 b Fq(\(3)p Fl(:)p Fq(47\))0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)29 b(14:29)1377 b Fq(17)p eop %%Page: 18 18 18 17 bop 100 83 a Fq(w)n(e)27 b(ha)n(v)n(e)1510 183 y Fl(g)1550 195 y Fp(2)1587 183 y Fq(\()p Fl(s)p Fq(\))d(=)e Fl(C)6 b(V)1914 195 y Fp(0)1952 183 y Fq(\()p Fl(s)p Fq(\))19 b(+)f Fj(O)r Fq(\()p Fl(\025)p Fq(\))29 b Fl(:)1299 b Fq(\(3)p Fl(:)p Fq(48\))100 339 y(The)27 b(compatibilit)n(y)h (condition)f(that)h(w)n(e)f(m)n(ust)h(ha)n(v)n(e)e(in)i(order)e(to)i (solv)n(e)e(for)h Fl(h)2635 351 y Fp(2)2700 339 y Fq(is)h(that)1278 486 y Fk(Z)1375 599 y Fl(\026)1425 611 y Fp(1)1462 599 y Fq(\()p Fl(\025z)t(;)14 b(s)p Fq(\))p Fl(m)1766 565 y Fc(0)1766 620 y Fp(0)1803 599 y Fq(\()p Fl(z)t Fq(\)d)p Fl(z)27 b Fq(=)22 b Fl(g)2149 611 y Fp(1)2186 599 y Fq(\()p Fl(s)p Fq(\))d(+)f Fl(g)2431 611 y Fp(2)2468 599 y Fq(\()p Fl(s)p Fq(\))28 b Fl(;)1066 b Fq(\(3)p Fl(:)p Fq(49\))100 863 y(where)26 b Fl(\026)389 875 y Fp(1)453 863 y Fq(is)g(giv)n(en)g (in)h(\(3.42\).)36 b(It)27 b(is)f(con)n(v)n(enien)n(t)g(to)g(single)g (out)h(from)f Fl(\026)2422 875 y Fp(1)2486 863 y Fq(the)h(part)f(still) h(unkno)n(wn)g(whic)n(h)f(will)h(b)r(e)100 963 y(determined)g(so)g (that)h(\(3.49\))f(holds.)37 b(W)-7 b(e)28 b(denote)883 1235 y Fl(\026)933 1247 y Fp(1)p Fo(;)p Fp(0)1023 1235 y Fq(\()p Fl(\030)t(;)14 b Fq(\000\))24 b(=)1341 1179 y(1)p 1337 1216 49 4 v 1337 1292 a Fl(\025)1410 1122 y Fk(Z)1456 1310 y Fp(\012)1521 1235 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))p Fl(m)1844 1201 y Fc(0)1844 1255 y Fp(0)1896 1118 y Fk(\022)1967 1179 y Fl(d)p Fq(\()p Fl(\030)t(;)g Fq(\000\))p 1967 1216 237 4 v 2061 1292 a Fl(\025)2214 1118 y Fk(\023)2289 1235 y Fl(V)2356 1192 y Fp(\(0\))2337 1257 y(1)2445 1235 y Fq(\()p Fl(s)p Fq(\()p Fl(\021)s Fq(\)\)d)p Fl(\021)23 b Fq(+)18 b Fl(c)2885 1247 y Fp(1)2922 1235 y Fq(\()p Fl(t)p Fq(\))672 b(\(3)p Fl(:)p Fq(50\))100 1502 y(and)925 1656 y(~)-49 b Fl(\026)968 1668 y Fp(1)1006 1656 y Fq(\()p Fl(\030)t(;)14 b Fq(\000\))23 b(=)p Fl(<)f(V)1422 1668 y Fp(1)1483 1656 y Fl(>)1584 1599 y Fq(1)p 1581 1636 49 4 v 1581 1712 a Fl(\025)1653 1542 y Fk(Z)1699 1731 y Fp(\012)1764 1656 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))p Fl(m)2087 1621 y Fc(0)2087 1676 y Fp(0)2139 1538 y Fk(\022)2211 1599 y Fl(d)p Fq(\()p Fl(\021)s(;)g Fq(\000)p 2211 1636 209 4 v 2291 1712 a Fl(\025)2429 1538 y Fk(\023)2504 1656 y Fq(d)p Fl(\021)22 b Fq(+)c Fl(p)p Fq(\()p Fl(\030)t(;)c Fq(\000\))28 b Fl(:)706 b Fq(\(3)p Fl(:)p Fq(51\))100 1887 y(T)-7 b(aking)26 b(in)i(accoun)n(t)f(\(3.46\))g(and)g(\(3.48\))g(w)n(e)g(then)h(write)g (\(3.49\))f(as)280 2038 y Fk(Z)377 2151 y Fl(\026)427 2163 y Fp(1)p Fo(;)p Fp(0)517 2151 y Fq(\()p Fl(\025z)t(;)14 b(s)p Fq(\))p Fl(m)821 2117 y Fc(0)821 2172 y Fp(0)858 2151 y Fq(\()p Fl(z)t Fq(\)d)p Fl(z)26 b Fq(=)d(\(2)p Fl(S)g Fq(+)18 b Fl(C)6 b Fq(\))p Fl(V)1540 2163 y Fp(0)1597 2151 y Fj(\000)1680 2038 y Fk(Z)1783 2151 y Fq(~)-49 b Fl(\026)1826 2163 y Fp(1)1864 2151 y Fq(\()p Fl(\025z)t(;)14 b(s)p Fq(\))p Fl(m)2168 2117 y Fc(0)2168 2172 y Fp(0)2205 2151 y Fq(\()p Fl(z)t Fq(\)d)p Fl(z)26 b Fq(=)d(\(2)p Fl(S)g Fq(+)18 b Fl(C)6 b Fq(\))p Fl(V)2887 2163 y Fp(0)2944 2151 y Fj(\000)18 b Fq(2)7 b(~)-49 b Fl(\026)3119 2163 y Fp(1)3155 2151 y Fq(\(0)p Fl(;)14 b(s)p Fq(\))19 b(+)f Fj(O)r Fq(\()p Fl(\025)p Fq(\))100 2411 y(ha)n(ving)33 b(~)-49 b Fl(\026)417 2423 y Fp(1)483 2411 y Fq(a)27 b(Lipsc)n(hitz)h(b)r(ound)g(indep)r(enden)n(t)h(on)e Fl(\025)p Fq(.)39 b(As)28 b(in)g(the)g(previous)f(section,)g(w)n(e)h (ma)n(y)f(substitute)i Fl(\026)3595 2423 y Fp(1)p Fo(;)p Fp(0)3713 2411 y Fq(b)n(y)100 2511 y(the)f(corresp)r(onding)d(single)i (la)n(y)n(er)f(p)r(oten)n(tial)1160 2770 y Fl(\026)1210 2782 y Fp(1)p Fo(;)p Fp(0)p Fo(;)p Fp(0)1353 2770 y Fq(\()p Fl(\030)t(;)14 b Fq(\000\))23 b(=)g(2)1713 2657 y Fk(Z)1758 2846 y Fp(\000)1818 2770 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))p Fl(V)2135 2727 y Fp(\(0\))2116 2793 y(1)2225 2770 y Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)2430 2782 y Fo(\021)2489 2770 y Fq(+)k Fl(c)2608 2782 y Fp(1)2646 2770 y Fq(\()p Fl(t)p Fq(\))948 b(\(3)p Fl(:)p Fq(52\))100 3034 y(and)27 b(further)h(restrict)e(to)i Fl(z)e Fq(=)d(0,)k(obtaining) 836 3181 y Fk(Z)933 3294 y Fl(\026)983 3306 y Fp(1)p Fo(;)p Fp(0)p Fo(;)p Fp(0)1126 3294 y Fq(\(0)p Fl(;)14 b(s)p Fq(\))p Fl(m)1381 3260 y Fc(0)1381 3315 y Fp(0)1418 3294 y Fq(\()p Fl(z)t Fq(\)d)p Fl(z)26 b Fq(=)d(2)p Fl(\026)1816 3306 y Fp(1)p Fo(;)p Fp(0)p Fo(;)p Fp(0)1958 3294 y Fq(\(0)p Fl(;)14 b(s)p Fq(\))23 b(=)g(\(2)p Fl(S)g Fq(+)18 b Fl(C)6 b Fq(\))p Fl(V)2627 3306 y Fp(0)2684 3294 y Fj(\000)18 b Fq(2)7 b(~)-49 b Fl(\026)2859 3306 y Fp(1)2896 3294 y Fq(\(0)p Fl(;)14 b(s)p Fq(\))846 3537 y(=)22 b(\(2)p Fl(S)h Fq(+)18 b Fl(C)6 b Fq(\))p Fl(V)1309 3549 y Fp(0)1366 3537 y Fq(+)18 b(4)p Fj(h)p Fl(V)1571 3549 y Fp(1)1608 3537 y Fj(i)1640 3549 y Fp(\000)1700 3424 y Fk(Z)1746 3612 y Fp(\000)1805 3537 y Fl(G)p Fq(\()p Fl(s;)c(\021)s Fq(\)d)p Fl(S)2151 3549 y Fo(\021)2211 3537 y Fq(+)k(2)p Fl(p)p Fq(\()p Fl(s)p Fq(\))g(+)g Fj(O)r Fq(\()p Fl(\025)p Fq(\))p Fl(:)3688 3417 y Fq(\(3)p Fl(:)p Fq(53\))100 3805 y(Inserting)27 b(\(3.52\))f(in)n(to)i(\(3.53\))f(and)g(in)n (tegrating)f(o)n(v)n(er)g(\000)i(w)n(e)f(obtain)g(that)707 4077 y(2)p Fl(c)785 4089 y Fp(1)822 4077 y Fq(\()p Fl(t)p Fq(\))d(=)e Fj(\000)1130 4021 y Fq(4)p 1102 4058 99 4 v 1102 4134 a Fj(j)p Fq(\000)p Fj(j)1224 3960 y Fk(\032)1286 3964 y(Z)1332 4152 y Fp(\000)1391 3964 y Fk(Z)1437 4152 y Fp(\000)1496 4077 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))p Fl(V)1814 4034 y Fp(\(0\))1794 4099 y(1)1903 4077 y Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)2108 4089 y Fo(\021)2150 4077 y Fq(d)p Fl(S)2247 4089 y Fo(\030)2301 4077 y Fq(+)2384 3964 y Fk(Z)2430 4152 y Fp(\000)2490 4077 y Fq(d)p Fl(S)2587 4089 y Fo(\030)2637 3964 y Fk(Z)2683 4152 y Fp(\000)2742 4077 y Fl(G)p Fq(\()p Fl(\030)t(;)g(\021)s Fq(\)d)p Fl(S)3089 4089 y Fo(\021)3130 3960 y Fk(\033)726 4327 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y(Z)1939 5113 y Fp(\000)1998 5037 y Fl(G)p Fq(\()p Fl(\030)t(;)c(\021)s Fq(\)d)p Fl(S)2345 5049 y Fo(\021)2405 5037 y Fj(\000)2526 4981 y Fq(1)p 2498 5018 99 4 v 2498 5094 a Fj(j)p Fq(\000)p Fj(j)2619 4924 y Fk(Z)2666 5113 y Fp(\000)2725 5037 y Fq(d)p Fl(S)2822 5049 y Fo(\030)2872 4924 y Fk(Z)2918 5113 y Fp(\000)2977 5037 y Fl(G)p Fq(\()p Fl(\030)t(;)g(\021)s Fq(\)d)p Fl(S)3324 5049 y Fo(\021)3366 4920 y Fk(\025)509 5294 y Fj(\000)602 5238 y Fq(1)p 602 5275 42 4 v 602 5351 a(2)667 5177 y Fk(\024)711 5294 y Fl(p)p Fq(\()p Fl(\030)t Fq(\))19 b Fj(\000)997 5238 y Fq(1)p 969 5275 99 4 v 969 5351 a Fj(j)p Fq(\000)p Fj(j)1091 5181 y Fk(Z)1137 5370 y Fp(\000)1196 5294 y Fl(p)p Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)1443 5306 y Fo(\021)1484 5177 y Fk(\025)1569 5294 y Fl(;)0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)29 b(14:29)1377 b Fq(18)p eop %%Page: 19 19 19 18 bop 100 83 a Fq(where)22 b Fj(S)385 95 y Fp(\000)453 83 y Fq(is)g(the)h(op)r(erator)e(de\014ned)i(in)g(\(10.9\).)34 b(Finally)23 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Fk(\022)1654 4352 y Fl(@)p 1639 4389 79 4 v 1639 4465 a(@)5 b(t)1727 4408 y(V)19 b Fq(\()p Fl(s;)14 b Fq(0\))19 b(+)f Fl(W)12 b Fq(\()p Fl(s)p Fq(\))p Fl(K)6 b Fq(\()p Fl(s)p Fq(\))p Fl(V)19 b Fq(\()p Fl(s;)14 b Fq(0\))2700 4291 y Fk(\023)2802 4408 y Fl(:)100 4631 y Fq(Applying)27 b(the)h(Diric)n(hlet{Neumann)g (op)r(erator,)e(see)h(\(10.8\),)g(w)n(e)g(ha)n(v)n(e)f(the)i(result.)p 2847 4586 57 4 v 2847 4636 4 50 v 2900 4636 V 2847 4639 57 4 v 127 4868 a Fr(4.2)j(Application)h(to)f Fl(D)983 4887 y Fo(V)1022 4907 y Fh(0)1074 4868 y Fl(\026)1124 4880 y Fp(0)p Fo(;)p Fp(0)199 5049 y Fq(Let)d(\000)g(ev)n(olv)n(e)e (under)h(\(4.2\),)g(from)h(\(4.3\),)f(using)g(\(4.6\))h(again,)e(w)n(e) h(obtain)1035 5239 y Fl(@)p 1020 5276 79 4 v 1020 5352 a(@)5 b(t)1109 5296 y(f)k Fq(\()p Fl(s;)14 b Fq(0\))22 b(=)h Fl(S)1531 5239 y(@)p 1516 5276 V 1516 5352 a(@)5 b(t)1605 5296 y(K)h Fq(\()p Fl(s;)14 b(t)p Fq(\))1852 5200 y Fk(\014)1852 5250 y(\014)1852 5300 y(\014)1880 5354 y Fo(t)p Fp(=0)2011 5296 y Fq(+)2104 5239 y(2)p Fl(\031)s(S)p 2104 5276 148 4 v 2110 5352 a Fj(j)p Fq(\000)p Fj(j)2208 5328 y Fp(2)2276 5182 y Fk(Z)2322 5371 y Fp(\000)2381 5296 y Fl(K)6 b Fq(\()p Fl(s)p Fq(\))p Fl(W)12 b Fq(\()p Fl(s)p Fq(\)d)p Fl(s)28 b(:)0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)h(14:29)1377 b Fq(21)p eop %%Page: 22 22 22 21 bop 199 83 a Fq(A)28 b(w)n(ell)g(kno)n(wn)e(computation)i(yields) f(the)h(result)f(that)1152 256 y Fl(@)p 1137 293 79 4 v 1137 369 a(@)5 b(t)1226 312 y(K)h Fq(\()p Fl(s;)14 b(t)p Fq(\))1473 217 y Fk(\014)1473 267 y(\014)1473 316 y(\014)1500 370 y Fo(t)p Fp(=0)1637 312 y Fq(=)22 b Fj(\000)1803 195 y Fk(\022)1893 256 y Fq(d)1939 226 y Fp(2)p 1874 293 123 4 v 1874 369 a Fq(d)p Fl(s)1959 345 y Fp(2)2006 312 y Fl(W)12 b Fq(\()p Fl(s)p Fq(\))19 b(+)f Fl(K)6 b Fq(\()p Fl(s)p Fq(\))2481 278 y Fp(2)2518 312 y Fl(W)12 b Fq(\()p Fl(s)p Fq(\))2711 195 y Fk(\023)3729 312 y Fq(\(4)p Fl(:)p Fq(7\))100 551 y(Hence)27 b(in)h(this)g(case,)f(since)g (from)h(\(4.3\))f Fl(S)5 b(K)h Fq(\()p 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Fq(d)p Fl(S)2729 5308 y Fo(\030)2788 5296 y Fq(=)23 b Fl( )2930 5308 y Fo(V)2969 5316 y Fh(0)3006 5296 y Fq(\()p Fl(\030)t(;)14 b Fq(\000\))28 b Fl(:)3688 5174 y Fq(\(4)p Fl(:)p Fq(11\))0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)h(14:29)1377 b Fq(22)p eop %%Page: 23 23 23 22 bop 100 83 a Fq(Then)27 b(\(4.10\),)g(when)h Fl(V)42 b Fq(=)23 b Fl(V)1022 95 y Fp(0)1059 83 y Fq(,)28 b(is)g(equal)f(to)955 366 y Fj(\000)p Fl(S)1090 249 y Fk(\022)1189 310 y Fq(1)p 1161 347 99 4 v 1161 423 a Fj(j)p Fq(\000)p Fj(j)1282 253 y Fk(Z)1329 441 y Fp(\000)1388 366 y Fl(K)1465 332 y Fp(2)1501 366 y Fq(\()p Fl(\030)t Fq(\))p Fl(W)12 b Fq(\()p Fl(\030)t Fq(\)d)p Fl(S)1896 378 y Fo(\030)1952 366 y Fj(\000)2067 310 y Fq(2)p Fl(\031)p 2045 347 136 4 v 2045 423 a Fj(j)p Fq(\000)p Fj(j)2143 399 y Fp(2)2205 253 y Fk(Z)2251 441 y Fp(\000)2310 366 y Fl(K)6 b Fq(\()p Fl(\030)t Fq(\))p Fl(W)12 b Fq(\()p Fl(\030)t Fq(\)d)p Fl(S)2782 378 y Fo(\030)2819 249 y Fk(\023)2922 366 y Fl(:)100 649 y Fq(No)n(w)27 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b(b)r(e)i(the)f (sequence)f(of)h(v)n(ector)e(\014elds)i(in)n(tro)r(duced)g(in)g(the)g (ansatz.)58 b(W)-7 b(e)35 b(split)g(them,)100 1474 y(according)25 b(to)j(\(2.7\))f(and)h(\(2.8\),)f(as)1291 1729 y Fl(V)1339 1741 y Fo(j)1397 1729 y Fq(=)c Fl(V)1552 1686 y Fp(\(0\))1533 1752 y Fo(j)1659 1729 y Fq(+)18 b Fj(h)p Fl(V)1822 1741 y Fo(j)1858 1729 y Fj(i)166 b Fl(j)28 b Fq(=)23 b(0)p Fl(;)14 b(::;)g(N)27 b Fj(\000)18 b Fq(1)p Fl(:)1119 b Fq(\(5)p Fl(:)p Fq(4\))100 1997 y(The)25 b Fl(V)335 1954 y Fp(\(0\))316 2020 y Fo(j)449 1997 y Fq(will)g(b)r(e)h (determined)f(applying)f(the)i(Diric)n(hlet-Neuman)f(op)r(erator,)e(b)n (y)i(p)r(oten)n(tial)g(theory)-7 b(,)25 b(in)g(Theorem)100 2097 y(5.2.)36 b(The)27 b Fj(h)p Fl(V)516 2109 y Fo(j)552 2097 y Fj(i)p Fq(,)h(the)g(part)f(constan)n(t)g(on)g(\000,)h(will)g(b)r (e)g(determined)g(in)f(Theorem)g(5.1,)g(stated)h(next.)132 2306 y Fr(Theorem)j(5.1)121 b Fn(Fix)30 b Fl(N)j(>)23 b Fq(1)p Fn(.)40 b(L)l(et)30 b Fq(\000)1421 2262 y Fp(\()p Fo(N)6 b 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Fl(\026)1923 2878 y Fo(i)1951 2866 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))200 b Fn(in)30 b Fq(\012)18 b Fj(\002)g Fq([0)p Fl(;)c(T)e Fq(])p Fl(;)935 b Fq(\(5)p Fl(:)p Fq(5\))100 3181 y Fn(with)30 b Fl(\026)330 3193 y Fo(i)358 3181 y Fn(,)g(for)g Fl(i)23 b Fq(=)g(0)p Fl(;)14 b Fq(1)p Fl(;)g(::N)26 b Fj(\000)18 b Fq(1)29 b Fn(given)h(in)g(\(6.13\),)i(\(6.17\))g(and)e(\(6.27\),)i(satisfying) 872 3407 y Fl(@)p 857 3444 79 4 v 857 3520 a(@)5 b(t)945 3463 y(m)1018 3429 y Fp(\()p Fo(N)h Fp(\))1133 3463 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))24 b(=)e(\001)p Fl(\026)1534 3429 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))1734 3463 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))19 b(+)f Fl(R)2070 3475 y Fp(1)2108 3463 y Fq(\()p Fl(\030)t(;)c(t;)g(\025)p Fq(\))200 b Fn(in)30 b Fq(\012)18 b Fj(\002)g Fq(\(0)p Fl(;)c(T)e Fq(\))p Fl(;)675 b Fq(\(5)p Fl(:)p Fq(6\))100 3741 y Fn(with)34 b Fl(R)347 3753 y Fp(1)417 3741 y Fn(given)g(in)f (\(6.9\).)51 b(F)-6 b(urther)33 b Fl(\026)1349 3711 y Fp(\()p Fo(N)6 b Fc(\000)p 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Fn(wher)l(e)30 b Fl(C)6 b Fq(\()p Fl(T)12 b Fq(\))30 b Fn(is)g(a)g(c)l(onstant)f(indep)l (endent)h(on)g Fl(\025)p Fn(.)100 5003 y Fq(The)39 b(pro)r(of)g(of)g (Theorems)f(5.1)g(is)h(deferred)g(to)g(Section)h(6.)71 b(The)39 b(next)h(theorem)e(assures)g(the)i(existence)f(and)100 5103 y(\(essen)n(tial\))27 b(uniqueness)h(of)g(the)g(functions)g Fl(m)1583 5115 y Fo(j)1618 5103 y Fq(,)g Fl(j)h Fq(=)23 b(0)p Fl(;)14 b(::N)9 b Fq(,)28 b(ha)n(ving)e(the)j(prop)r(erties)e (requiremed)g(in)h(the)g(ansatz.)100 5203 y(Existence)34 b(and)i(unicit)n(y)f(are)g(obtained)g(pro)n(vided)f(a)h(compatibilit)n (y)h(condition)f(is)g(satis\014ed.)60 b(This)36 b(determines)100 5339 y Fl(V)167 5296 y Fp(\(0\))148 5362 y Fo(j)256 5339 y Fq(,)27 b(the)h(orthogonal)e(part)h(of)g(the)h(v)n(elo)r(cit)n(y)f (\014elds.)0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)i(14:29)1377 b Fq(24)p eop %%Page: 25 25 25 24 bop 135 83 a Fr(Theorem)34 b(5.2)131 b Fn(L)l(et)32 b Fl(T)44 b Fn(b)l(e)32 b(the)h(lifetime)h(of)g(the)e(solution)h(of)h (\(1.5\))f(in)g Fj(M)p Fn(.)47 b(L)l(et)32 b Fl(\026)2926 53 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))3126 83 y Fq(\()p Fj(\001)p Fl(;)14 b(t)p Fq(\))p Fn(,)34 b Fl(t)28 b Fj(2)h Fq([0)p Fl(;)14 b(T)e Fq(])p Fn(,)32 b(as)100 224 y(in)e(The)l(or)l(em) i(5.1.)43 b(Then,)32 b(pr)l(ovide)l(d)g(the)f Fl(V)1502 181 y Fp(\(0\))1483 247 y Fo(j)1591 224 y Fn(,)g Fl(j)f Fq(=)24 b(0)p Fl(;)14 b(::)p Fq(\()p Fl(N)28 b Fj(\000)18 b Fq(1\))31 b Fn(ar)l(e)g(chosen)g(as)g(in)g(\(8.18\),)i(\(8.25\))f (and)f(\(8.37\),)100 323 y(ther)l(e)e(exist)h Fl(m)573 335 y Fo(j)608 323 y Fn(,)g Fl(j)e Fq(=)23 b(0)p Fl(;)14 b(::;)g(N)37 b Fn(having)32 b(the)d(r)l(e)l(quir)l(ement)g(state)l(d)h (in)g(the)g(ansatz)f(such)h(that)522 582 y Fl(\026)572 547 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))772 582 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))24 b(=)f Fj(\000)p Fl(\025)p Fq(\001)p Fl(m)1310 547 y Fp(\()p Fo(N)6 b Fp(\))1424 582 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))20 b(+)1711 525 y(1)p 1707 563 49 4 v 1707 639 a Fl(\025)1766 582 y(f)9 b Fq(\()p Fl(m)1921 547 y Fp(\()p Fo(N)d Fp(\))2036 582 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\)\))19 b(+)f Fl(R)2404 594 y Fp(2)2441 582 y Fq(\()p Fl(\030)t(;)c(t;)g(\025)p Fq(\))200 b Fn(in)30 b Fq(\012)19 b Fj(\002)f Fq(\(0)p Fl(;)c(T)e Fq(])p Fl(;)350 b Fq(\(5)p Fl(:)p Fq(9\))100 837 y Fn(with)31 b Fl(R)344 849 y Fp(2)413 837 y Fn(given)h(in)f (\(8.45\).)44 b(F)-6 b(urther)31 b Fl(m)1397 807 y Fp(\()p Fo(N)6 b Fp(\))1512 837 y Fq(\()p Fj(\001)p Fl(;)14 b(t)p Fq(\))p Fn(,)32 b(for)g Fl(t)26 b Fj(2)g Fq([0)p Fl(;)14 b(T)e Fq(])p Fn(,)30 b(is)i(a)f Fl(C)2464 807 y Fc(1)2535 837 y Fq(\(\012\))h Fn(function)f(that)g(satis\014es)g(homo)l(ge-)100 937 y(ne)l(ous)e(Neumann)g(b)l(oundary)h(c)l(onditions)h(and)1413 1128 y Fq(sup)1413 1198 y Fo(\030)r Fc(2)p Fp(\012)1593 1128 y Fq(sup)1551 1202 y Fo(t)p Fc(2)p Fp([0)p Fo(;T)9 b Fp(])1774 1128 y Fj(j)p Fl(R)1860 1140 y Fp(2)1897 1128 y Fq(\()p Fl(\030)t(;)14 b(t;)g(\025)p Fq(\))p Fj(j)24 b(\024)f Fl(C)6 b(\025)2401 1094 y Fo(N)2464 1128 y Fl(:)1201 b Fq(\(5)p Fl(:)p Fq(10\))199 1538 y(The)24 b(pro)r(of)f(of)h(this)g (result)f(is)h(giv)n(en)f(in)h(Section)f(8.)35 b(Theorem)23 b(5.1)g(and)h(Theorem)f(5.2)f(pro)n(vide)h(the)h(t)n(w)n(o)f(steps)h (to)100 1638 y(construct)e(the)h(appro)n(ximate)e(solution)h(to)h (\(1.2\).)35 b(F)-7 b(rom)22 b(Theorem)g(5.1)g(and)h(5.2)f(one)g (obtains)g(easily)g(the)i(follo)n(wing)100 1737 y(comprehensiv)n(e)i (result,)h(whic)n(h)h(ampli\014es)f(Theorem)g(1.1.)36 b(Its)28 b(pro)r(of)f(is)g(giv)n(en)g(in)h(Section)f(9.)100 1886 y Fr(Theorem)e(5.3)48 b Fn(F)-6 b(or)26 b(al)t(l)g Fl(N)32 b(>)23 b Fq(1)p Fn(,)j(ther)l(e)f(ar)l(e)h(uniquely)g(de\014ne) l(d)f(se)l(quenc)l(es)g(of)i(ve)l(ctor)e(\014elds)h Fl(V)3112 1898 y Fo(j)3148 1886 y Fn(,)g Fl(j)i Fq(=)23 b(0)p Fl(;)14 b(::;)g Fq(\()p Fl(N)j Fj(\000)9 b Fq(1\))p Fn(,)100 1985 y(on)27 b Fj(M)f Fn(and)i(functions)f Fl(m)932 1997 y Fo(j)967 1985 y Fn(,)h Fl(j)g Fq(=)22 b(0)p Fl(;)14 b(::;)g(N)9 b Fn(,)27 b(fr)l(om)h Fj(M)e Fn(to)h Fl(C)1941 1955 y Fc(1)2012 1985 y Fq(\(\012\))g Fn(as)h(in)f(the)g(ansatz)g(such) g(that)g(the)g(fol)t(lowing)i(holds.)100 2085 y(F)-6 b(or)22 b(any)h Fq(\000)451 2097 y Fp(0)511 2085 y Fj(2)h(M)p Fn(,)g(cho)l(ose)g Fl(k)1038 2097 y Fp(0)1098 2085 y Fj(\025)f Fl(\024)p Fq(\(\000)1318 2097 y Fp(0)1355 2085 y Fq(\))p Fn(,)i(set)d Fl(\025)1607 2097 y Fp(0)1668 2085 y Fq(=)1799 2052 y Fp(1)p 1765 2066 101 4 v 1765 2114 a(2)p Fo(k)1833 2122 y Fh(0)1899 2085 y Fn(and)h(let)f Fl(T)34 b Fn(b)l(e)22 b(the)h(lifetime)h(of)f(the)g(solution)g(of)g (\(1.5\))h(in)e Fj(M)p Fn(,)100 2229 y(ac)l(c)l(or)l(ding)32 b(to)f(\(2.6\).)43 b(Then)32 b(for)g(al)t(l)g Fl(t)25 b(<)g(T)12 b Fn(,)30 b(for)i(al)t(l)g Fl(\025)26 b Fj(2)f Fq(\(0)p Fl(;)14 b(\025)2113 2241 y Fp(0)2151 2229 y Fq(])31 b Fn(we)g(c)l(an)g(c)l(onstruct)f Fq(\()16 b(~)-58 b Fl(m)2949 2199 y Fp(\()p Fo(N)6 b Fp(\))3064 2229 y Fl(;)20 b Fq(~)-48 b Fl(\026)3151 2199 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))3351 2229 y Fq(\))25 b Fj(2)h Fl(C)3554 2199 y Fc(1)3624 2229 y Fq(\(\012)20 b Fj(\002)100 2328 y Fq([0)p Fl(;)14 b(T)e Fq(]\))30 b Fn(wher)l(e)48 b Fq(~)-58 b Fl(m)657 2298 y Fp(\()p Fo(N)6 b Fp(\))803 2328 y Fn(is)32 b(a)g Fl(\025)1016 2298 y Fo(N)1111 2328 y Fn(mo)l(di\014c)l(ation)g(of)h Fl(m)1754 2298 y Fp(\()p Fo(N)6 b Fp(\))1868 2328 y Fn(,)33 b(i.e.)45 b Fq(sup)2209 2349 y Fp(\()p Fo(\030)r(;t)p Fp(\))p Fc(2)p Fp(\012)p Fc(\002)p Fp([0)p Fo(;T)9 b Fp(])2638 2328 y Fj(j)16 b Fq(~)-58 b Fl(m)2734 2298 y Fp(\()p Fo(N)6 b Fp(\))2849 2328 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))20 b Fj(\000)f Fl(m)3197 2298 y Fp(\()p Fo(N)6 b Fp(\))3312 2328 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))p Fj(j)27 b(\024)f Fl(C)6 b(\025)3737 2298 y Fo(N)100 2470 y Fn(and)31 b Fq(~)-49 b Fl(\026)305 2440 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))529 2470 y Fn(is)25 b(a)g Fl(\025)728 2440 y Fo(N)6 b Fc(\000)p Fp(1)900 2470 y Fn(mo)l(di\014c)l(ation)26 b(of)f Fl(\026)1506 2440 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))1705 2470 y Fn(,)26 b(i.e.)38 b Fq(sup)2032 2490 y Fp(\()p Fo(\030)r(;t)p Fp(\))p Fc(2)p Fp(\012)p Fc(\002)p Fp([0)p Fo(;T)9 b Fp(])2462 2470 y Fj(j)e Fq(~)-49 b Fl(\026)2535 2440 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))2734 2470 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))6 b Fj(\000)g Fl(\026)3032 2440 y Fp(\()p Fo(N)g Fc(\000)p Fp(1\))3233 2470 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))p Fj(j)24 b(\024)f Fl(C)6 b(\025)3652 2440 y Fo(N)g Fc(\000)p Fp(1)100 2570 y Fn(satisfying)561 2761 y Fl(@)p 546 2798 79 4 v 546 2874 a(@)f(t)650 2817 y Fq(~)-57 b Fl(m)708 2783 y Fp(\()p Fo(N)6 b Fp(\))822 2817 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))24 b(=)f(\001)7 b(~)-49 b Fl(\026)1224 2783 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))1424 2817 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))200 b Fn(in)30 b Fq(\012)18 b Fj(\002)g Fq(\(0)p Fl(;)c(T)e Fq(\))543 3042 y(~)-49 b Fl(\026)586 3008 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))786 3042 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))24 b(=)e Fj(\000)p Fl(\025)p Fq(\001)16 b(~)-58 b Fl(m)1323 3008 y Fp(\()p Fo(N)6 b Fp(\))1438 3042 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))19 b(+)1725 2986 y(1)p 1721 3023 49 4 v 1721 3099 a Fl(\025)1780 3042 y(f)9 b Fq(\()15 b(~)-57 b Fl(m)1935 3008 y Fp(\()p Fo(N)6 b Fp(\))2049 3042 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\)\))20 b(+)e Fl(R)q Fq(\()p Fl(\030)t(;)c(t;)g(\025)p Fq(\))200 b Fn(in)29 b Fq(\012)19 b Fj(\002)f Fq(\(0)p Fl(;)c(T)e Fq(\))p Fl(:)3688 2922 y Fq(\(5)p Fl(:)p Fq(11\))100 3244 y Fn(wher)l(e)1374 3344 y Fq(sup)1374 3414 y Fo(\030)r Fc(2)p Fp(\012)1554 3344 y Fq(sup)1512 3418 y Fo(t)p Fc(2)p Fp([0)p Fo(;T)d Fp(])1735 3344 y Fj(j)p Fl(R)q Fq(\()p Fl(\030)t(;)14 b(t;)g(\025)p Fq(\))p Fj(j)23 b(\024)g Fl(C)6 b(\025)2325 3310 y Fo(N)g Fc(\000)p Fp(1)2503 3344 y Fl(:)100 3602 y Fn(F)-6 b(urther,)38 b Fq(~)-49 b Fl(\026)477 3572 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))677 3602 y Fq(\()p Fj(\001)p Fl(;)14 b(t)p Fq(\))31 b Fn(and)48 b Fq(~)-58 b Fl(m)1098 3572 y Fp(\()p Fo(N)6 b Fp(\))1213 3602 y Fq(\()p Fj(\001)p Fl(;)14 b(t)p Fq(\))p Fn(,)32 b(for)g Fl(t)25 b Fj(2)h Fq([0)p Fl(;)14 b(T)e Fq(])p Fn(,)30 b(satisfy)j(Neumann)d(homo)l(gene)l(ous)i(b)l(oundary)f(c)l (onditions)100 3702 y(on)e(the)h(b)l(oundary)h(of)f Fq(\012)p Fn(.)39 b(In)29 b(addition)1247 3893 y Fq(sup)1205 3966 y Fo(t)p Fc(2)p Fp([0)p Fo(;T)9 b Fp(])1428 3893 y Fq(sup)1428 3963 y Fo(\030)r Fc(2)p Fp(\012)1566 3893 y Fj(j)e Fq(~)-49 b Fl(\026)1639 3859 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))1839 3893 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))19 b Fj(\000)f Fl(\026)2162 3905 y Fp(0)p Fo(;)p Fp(0)2252 3893 y Fq(\()p Fl(\030)t(;)c(t)p Fq(\))p Fj(j)24 b(\024)f Fl(C)6 b(\025;)994 b Fq(\(5)p Fl(:)p Fq(12\))100 4176 y Fn(wher)l(e)30 b Fl(\026)384 4188 y Fp(0)p Fo(;)p Fp(0)504 4176 y Fn(is)g(the)g (solution)g(of)g(\(1.8\),)i(\(1.9\),)971 4470 y Fq(sup)929 4544 y Fo(t)p Fc(2)p Fp([0)p Fo(;T)9 b Fp(])1300 4470 y Fq(sup)1151 4562 y Fo(\030)r Fc(2N)g Fp(\()p Fo(\025)1356 4570 y Fh(0)1389 4562 y Fo(;)p Fp(\000)1450 4531 y Fh(\()p Fg(N)c Fh(\))1450 4581 y Fg(t)1548 4562 y Fp(\))1588 4325 y 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Fp(\))2126 4814 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))19 b Fj(\007)f Fq(1)2441 4719 y Fk(\014)2441 4769 y(\014)2441 4818 y(\014)2491 4814 y Fj(\024)23 b Fl(C)6 b(\025)30 b(:)943 b Fq(\(5)p Fl(:)p Fq(14\))199 5240 y(Once)25 b(the)g(appro)n(ximate)f(solution)g (to)h(\(1.2\))g(is)f(constructed)h(it)g(remains)g(to)f(sho)n(w)g(that)i (it)f(is)g(indeed)g(\\close")f(to)100 5340 y(the)h(solution)g(of)g (\(1.2\).)36 b(This)25 b(can)g(b)r(e)h(ac)n(hiev)n(ed)e(arguing)g(as)h (in)g([1].)36 b(One)25 b(needs)g(only)g(to)g(v)n(erify)g(that)g(the)h (sp)r(ectral)0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)j(14:29)1377 b Fq(25)p eop %%Page: 26 26 26 25 bop 100 83 a Fq(estimate)29 b(used)g(there)f(still)i(holds)e(for) h(the)g(appro)n(ximated)f(solution)g(w)n(e)h(constructed.)41 b(This)29 b(is)g(indeed)g(the)g(case,)100 183 y(as)e(can)g(b)r(e)h (easily)f(v)n(eri\014ed.)0 329 y Fm(6.)50 b(Construction)36 b(of)i(the)f(appro)m(ximate)g(c)m(hemical)f(p)s(oten)m(tial)199 524 y Fq(In)f(this)f(section)g(w)n(e)g(apply)g(classical)e(p)r(oten)n (tial)i(theory)g(to)g(pro)n(v)n(e)e(Theorem)i(5.1.)55 b(W)-7 b(e)35 b(lo)r(ok)e(for)h(a)g(function)100 624 y Fl(\026)150 594 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))377 624 y Fq(from)27 b Fj(M)h Fq(to)f Fl(C)867 594 y Fc(1)938 624 y Fq(\(\012\))h(ha)n(ving)f(the)h(form)1211 917 y Fl(\026)1261 883 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))1461 917 y Fq(\()p Fl(\030)t(;)14 b Fq(\000\))24 b(=)1765 814 y Fo(N)6 b Fc(\000)p Fp(1)1777 838 y Fk(X)1784 1015 y Fo(i)p Fp(=0)1923 917 y Fl(\025)1971 883 y Fo(i)1999 917 y Fl(\026)2049 929 y Fo(i)2077 917 y Fq(\()p Fl(\030)t(;)14 b Fq(\000\))166 b Fl(\030)28 b Fj(2)23 b Fq(\012)28 b Fl(;)1040 b Fq(\(6)p Fl(:)p Fq(1\))100 1218 y(where)22 b Fl(\026)385 1230 y Fo(i)413 1218 y Fq(,)i Fl(i)e Fq(=)h(0)p Fl(;)14 b(::;)g(N)j Fj(\000)9 b Fq(1,)24 b(are)e(functions)h(to)g(b)r (e)g(determined.)36 b(W)-7 b(e)23 b(insert)g Fl(m)2603 1188 y Fp(\()p Fo(N)6 b Fp(\))2718 1218 y Fq(,)24 b(as)e(in)h(the)h (ansatz,)f(and)f Fl(\026)3577 1188 y Fp(\()p Fo(N)6 b Fc(\000)p Fp(1\))3777 1218 y Fq(,)100 1354 y(as)29 b(in)g(\(6.1\),)h(b) r(oth)g(ev)-5 b(aluated)29 b(at)h(\000)1253 1311 y Fp(\()p Fo(N)6 b Fp(\))1253 1375 y Fo(t)1397 1354 y Fq(where)29 b(\000)1691 1311 y Fp(\()p Fo(N)6 b Fp(\))1691 1375 y Fo(t)1836 1354 y Fq(is)29 b(the)h(solution)f(of)h(\(1.5\),)f(in)n(to)h (\(3.1\).)42 b(W)-7 b(e)30 b(obtain)f(\()p Fl(N)g Fj(\000)20 b Fq(1\))100 1495 y(Laplace)30 b(equations)g(for)g Fl(\026)964 1507 y Fo(i)992 1495 y Fq(\()p Fj(\001)p Fl(;)14 b Fq(\000)1136 1452 y Fp(\()p Fo(N)6 b Fp(\))1136 1515 y Fo(t)1251 1495 y Fq(\),)32 b Fl(i)c Fq(=)h(1)p Fl(;)14 b(::)p Fq(\()p Fl(N)29 b Fj(\000)20 b Fq(1\).)47 b(The)31 b(compatibilit)n(y)g (condition)g(needed)g(to)g(solv)n(e)e(these)100 1636 y(Laplace)d(equations)h(determines)g Fj(h)p Fl(V)1277 1648 y Fo(j)1313 1636 y Fj(i)p Fq(\(\000)1429 1592 y Fp(\()p Fo(N)6 b Fp(\))1429 1656 y Fo(t)1544 1636 y Fq(\))28 b(for)f Fl(j)h Fq(=)23 b(0)p Fl(;)14 b(::;)g Fq(\()p Fl(N)27 b Fj(\000)18 b Fq(1\).)100 1898 y(When)29 b(di\013eren)n (tiating)g Fl(m)953 1868 y Fp(\()p Fo(N)6 b Fp(\))1068 1898 y Fq(\()p Fj(\001)p Fl(;)14 b Fq(\000)1212 1855 y Fp(\()p Fo(N)6 b Fp(\))1212 1919 y Fo(t)1327 1898 y Fq(\))29 b(with)h(resp)r(ect)f(to)g Fl(t)g Fq(w)n(e)g(need)g(to)g(tak)n (e)g(in)n(to)f(accoun)n(t)h(that)g Fl(m)3363 1868 y Fp(\()p Fo(N)6 b Fp(\))3507 1898 y Fq(dep)r(ends)100 1998 y(on)27 b(\000)267 2010 y Fo(t)296 1998 y Fq(,)h(trough)f(a)g(fast)g(and)h(slo) n(w)e(scale,)h(the)h(fast)g(scale)f(brings)f(a)i(factor)e Fl(\025)2515 1968 y Fc(\000)p Fp(1)2605 1998 y Fq(.)100 2216 y Fr(Notation)66 b Fn(L)l(et)32 b Fl(m)h Fn(b)l(e)g(a)g(function)g (fr)l(om)h Fj(M)e Fn(to)h Fl(C)1799 2186 y Fc(1)1870 2216 y Fq(\(\012\))g Fn(of)h(the)f(typ)l(e)g(\(2.9\).)49 b(L)l(et)33 b Fl(V)51 b Fn(b)l(e)33 b(a)h(ve)l(ctor)f(\014eld)g(on)g Fj(M)p Fn(.)100 2316 y(We)c(denote)i(by)1169 2484 y Fl(D)1238 2496 y Fo(V)1296 2484 y Fl(m)p Fq(\()p Fl(\030)t(;)14 b Fq(\000\))23 b(=)1686 2428 y(1)p 1683 2465 49 4 v 1683 2541 a Fl(\025)1741 2484 y(h)1789 2449 y Fc(0)1812 2484 y Fq(\()1854 2428 y Fl(d)p Fq(\()p Fl(\030)t(;)14 b Fq(\000\))p 1854 2465 237 4 v 1949 2541 a Fl(\025)2101 2484 y(;)g(s)p Fq(\()p Fl(\030)t(;)g Fq(\000\)\))p Fl(V)20 b Fq(\()p Fl(s)p Fq(\()p Fl(\030)t Fq(\)\))31 b Fl(;)998 b Fq(\(6)p Fl(:)p Fq(2\))100 2711 y Fn(wher)l(e)36 b(we)f(indic)l(ate)i(with)f (prime)g(the)f(derivative)j(of)e Fl(h)f Fn(with)h(r)l(esp)l(e)l(ct)f (to)g(the)h(\014rst)e(variable)j Fl(z)g Fq(=)3320 2670 y Fo(d)p Fp(\()p Fo(\030)r(;)p Fp(\000\))p 3320 2691 180 4 v 3390 2739 a Fo(\025)3510 2711 y Fn(.)55 b(When)100 2854 y Fl(W)178 2866 y Fo(N)264 2854 y Fq(=)352 2792 y Fk(P)439 2812 y Fo(N)6 b Fc(\000)p Fp(1)439 2879 y Fo(j)s Fp(=0)601 2854 y Fl(\025)649 2824 y Fo(j)685 2854 y Fl(V)733 2866 y Fo(j)768 2854 y Fn(,)30 b(with)g Fl(V)1051 2866 y Fp(0)1089 2854 y Fn(,...)p Fl(V)1237 2866 y Fo(N)6 b Fc(\000)p Fp(1)1417 2854 y Fn(a)30 b(ve)l(ctor)g(\014elds)g(on)g Fj(M)1315 3163 y Fl(D)1384 3175 y Fo(W)1446 3183 y Fg(N)1504 3163 y Fl(m)p Fq(\()p Fl(\030)t(;)14 b Fq(\000\))24 b(=)1881 3059 y Fo(N)6 b Fc(\000)p Fp(1)1893 3084 y Fk(X)1896 3261 y Fo(j)s Fp(=0)2039 3163 y Fl(\025)2087 3129 y Fo(j)2122 3163 y Fl(D)2191 3175 y Fo(V)2230 3183 y Fg(j)2265 3163 y Fl(m)p Fq(\()p Fl(\030)t(;)14 b Fq(\000\))30 b Fl(:)100 3573 y Fq(Note)24 b(that)g(b)n(y)f(the)i(orthogonalit)n(y)c(of)j Fj(r)1392 3585 y Fo(\030)1428 3573 y Fl(d)h Fq(with)f(resp)r(ect)g(to)f (the)i(surface)e(there)g(is)h(no)g(con)n(tribution)f(in)h(\(6.2\))g (from)100 3672 y Fl(s)p Fq(\()p Fl(\030)t(;)14 b Fq(\000\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(then)570 3888 y Fl(@)5 b(m)692 3857 y Fp(\()p Fo(N)h Fp(\))p 570 3925 237 4 v 649 4001 a Fl(@)f(t)816 3944 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))24 b(=)e Fl(D)1167 3956 y Fo(W)1229 3964 y Fg(N)1287 3944 y Fq(\()p Fl(m)1392 3909 y Fp(\()p Fo(N)6 b Fp(\))1507 3944 y Fq(\))23 b(=)g Fl(D)1719 3956 y Fo(V)1758 3964 y Fh(0)1795 3944 y Fl(m)1868 3956 y Fp(0)1923 3944 y 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Fp(1)1455 2539 y Fq(\()p Fl(\030)t(;)14 b(t;)g(\025)p Fq(\))24 b(=)f Fl(R)1886 2551 y Fo(N)1949 2539 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))19 b(+)f Fl(E)5 b Fq(\()p Fl(\030)t(;)14 b(t;)g(\025)p Fq(\))1185 b(\(6)p Fl(:)p Fq(9\))100 2742 y(It)28 b(can)f(b)r(e)h(easily)f(estimated)1448 2846 y(sup)1305 2920 y Fp(\()p Fo(\030)r(;t)p Fp(\))p Fc(2)p Fp(\012)p Fc(\002)p Fp([0)p Fo(;T)9 b Fp(])1730 2846 y Fj(j)p Fl(R)1816 2858 y Fp(1)1853 2846 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))p Fj(j)24 b(\024)e Fl(C)6 b Fq(\()p Fl(T)12 b Fq(\))p Fl(\025)2396 2812 y Fo(N)6 b Fc(\000)p Fp(1)2572 2846 y Fl(:)1093 b Fq(\(6)p Fl(:)p Fq(10\))100 3108 y(F)-7 b(urther,)22 b(one)f(gains)g(an)g(extra)g(p)r(o)n(w)n(er)f(of)i Fl(\025)g Fq(when)g(in)n(tegrating)e Fl(R)2175 3120 y Fp(1)2212 3108 y Fq(,)j(since)e(the)h(terms)g(of)f(order)f Fl(\025)3165 3078 y Fo(N)6 b Fc(\000)p Fp(1)3335 3108 y Fq(ha)n(v)n(e)21 b(supp)r(ort)100 3208 y(in)27 b Fj(N)12 b Fq(\()p Fl(\025)356 3220 y Fp(0)395 3208 y Fq(\),)1391 3379 y(sup)1350 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y(more)22 b(systematic.)35 b(Finally)23 b(in)h(Lemma)f(6.3)f(w)n(e)h(outline)h(the)f(pro)r(of)g(for)g(solving)f (the)i(equation)e(to)i(a)e(generic)h(order.)100 4251 y Fr(Remark:)83 b Fq(In)30 b(the)f(next)g(lemmas,)g(as)f(w)n(ell)h(in)g (all)g(the)g(pap)r(er,)g Fl(G)h Fq(stands)e(for)h(the)g(Neumann)g (Green's)g(function)100 4351 y(on)e(\012,)h(see)f(\(10.1\).)100 4571 y Fr(Lemma)i(6.1)118 b Fn(Ther)l(e)31 b(exists)e(an)h(unique)f (\(up)h(to)g(c)l(onstant)f(in)h Fl(\030)t Fn(\))f(solution)h(of)h (\(6.6\))g(pr)l(ovide)l(d)1313 4729 y Fk(Z)1359 4917 y Fp(\000)1400 4925 y Fg(t)1446 4842 y Fl(V)1494 4854 y Fp(0)1532 4842 y Fq(\()p Fl(\021)s(;)14 b Fq(\000)1697 4854 y Fo(t)1726 4842 y Fq(\)d)p Fl(S)1855 4854 y Fo(\021)1919 4842 y Fq(=)23 b(0)169 b Fl(t)23 b Fj(2)g Fq([0)p Fl(;)14 b(T)e Fq(])29 b Fl(:)1101 b Fq(\(6)p Fl(:)p Fq(12\))100 5120 y Fn(It)29 b(is)h(given)g(by)786 5294 y Fl(\026)836 5306 y Fp(0)873 5294 y Fq(\()p Fl(\030)t(;)14 b Fq(\000)1034 5306 y Fo(t)1063 5294 y Fq(\))24 b(=)1206 5181 y Fk(Z)1253 5370 y Fp(\012)1318 5294 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))1582 5177 y Fk(\022)1657 5238 y Fq(1)p 1654 5275 V 1654 5351 a Fl(\025)1712 5294 y(m)1785 5260 y Fc(0)1785 5315 y Fp(0)1836 5177 y Fk(\022)1907 5238 y Fl(d)p Fq(\()p Fl(\021)s(;)g Fq(\000)2115 5250 y Fo(t)2145 5238 y Fq(\))p 1907 5275 271 4 v 2018 5351 a Fl(\025)2187 5177 y Fk(\023)2262 5294 y Fl(V)2310 5306 y Fp(0)2348 5294 y Fq(\()p Fl(s)p Fq(\()p Fl(\021)s Fq(\))p Fl(;)g(t)p Fq(\))2626 5177 y Fk(\023)2702 5294 y Fq(d)p Fl(\021)22 b Fq(+)c Fl(c)2930 5306 y Fp(0)2967 5294 y Fq(\()p Fl(t)p Fq(\))30 b Fl(;)574 b Fq(\(6)p Fl(:)p Fq(13\))0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)29 b(14:29)1377 b Fq(27)p eop %%Page: 28 28 28 27 bop 100 83 a Fn(wher)l(e)30 b Fl(c)370 95 y Fp(0)407 83 y Fq(\()p Fl(t)p Fq(\))g Fn(is)h(a)f(c)l(onstant)f(\(in)h Fl(\030)t Fn(\))f(to)h(b)l(e)g(determine)l(d.)39 b(It)30 b(is)g(a)g Fl(C)2242 53 y Fc(1)2312 83 y Fq(\(\012\))h Fn(function)e(for)i Fl(t)23 b Fj(2)g Fq([0)p Fl(;)14 b(T)e Fq(])p Fn(.)100 302 y Fb(Pro)s(of:)41 b Fq(The)28 b(solv)-5 b(abilit)n(y)27 b(of)g(\(6.6\))h(requires)e(that)i(for)f(all) g Fl(t)c Fj(2)h Fq([0)p Fl(;)14 b(T)e Fq(])1213 443 y Fk(Z)1260 632 y Fp(\012)1325 439 y Fk(\022)1399 500 y Fq(1)p 1396 537 49 4 v 1396 613 a Fl(\025)1454 556 y(m)1527 522 y Fc(0)1527 577 y Fp(0)1578 439 y Fk(\022)1649 500 y Fl(d)p Fq(\()p Fl(\021)s(;)i Fq(\000)1857 512 y Fo(t)1887 500 y Fq(\))p 1649 537 271 4 v 1760 613 a Fl(\025)1929 439 y Fk(\023)2004 556 y Fl(V)2052 568 y Fp(0)2090 556 y Fq(\()p Fl(s)p Fq(\()p Fl(\021)s Fq(\))p Fl(;)g(t)p Fq(\))2368 439 y Fk(\023)2444 556 y Fq(d)p Fl(\021)26 b Fq(=)d(0)1001 b(\(6)p Fl(:)p Fq(14\))100 811 y(This)27 b(forces)g(to)g(tak)n(e)g Fl(V)852 823 y Fp(0)918 811 y Fq(suc)n(h)g(that)h(\(6.12\))e(holds.)37 b(No)n(w)27 b(since)1499 990 y(d)p Fl(\021)g Fq(=)22 b Fl(\025)p Fq(\(1)d Fj(\000)f Fl(\025z)t(K)6 b Fq(\()p Fl(s)p Fq(\)\)d)p Fl(s)p Fq(d)p Fl(z)1291 b Fq(\(6)p Fl(:)p Fq(15\))100 1170 y(and)27 b Fl(m)334 1140 y Fc(0)334 1191 y Fp(0)399 1170 y Fq(is)g(ev)n(en)g(w)n(e)h(ha)n(v)n(e)e(that)i(\(6.14\))f(is)g (satis\014ed.)p 1877 1125 57 4 v 1877 1174 4 50 v 1930 1174 V 1877 1177 57 4 v 199 1430 a Fr(Remark)d Fq(In)h(dimension)g Fl(d)e Fq(=)f(2,)j(the)g(v)n(elo)r(cit)n(y)f(\014eld)h Fl(V)1958 1442 y Fp(0)2021 1430 y Fq(coincides)f(with)h Fl(V)2622 1387 y Fp(\(0\))2603 1452 y(0)2711 1430 y Fq(,)g(the)h (constan)n(t)d(part)i(b)r(eing)f(zero.)100 1530 y(If)f(w)n(e)g(w)n(ere) f(w)n(orking)f(in)i(three)g(or)f(more)g(dimensions,)i(the)f(in)n (tegral)f(\(6.14\))g(w)n(ould)h(not)g(ha)n(v)n(e)f(v)-5 b(anished)23 b(iden)n(tically)-7 b(,)100 1629 y(but)25 b(w)n(ould)f(ha)n(v)n(e)g(b)r(een)h(a)f(term)g(of)h Fj(O)r Fq(\()p Fl(\025)1371 1599 y Fp(2)1409 1629 y Fq(\).)37 b(This)24 b(w)n(ould)g(ha)n(v)n(e)g(caused)g(only)g(a)g(sligh)n(t)h (complication,)f(and)h(w)n(e)f(shall)100 1729 y(explain)j(ho)n(w)g(to)g (deal)h(with)g(suc)n(h)f(problems)g(in)h(Lemma)f(6.2)g(when)h(w)n(e)f 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y Fl(\026)1479 2915 y Fp(1)1516 2903 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))24 b(=)f Fl(\026)1849 2915 y Fp(1)p Fo(;)p Fp(0)1939 2903 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))19 b(+)24 b(~)-48 b Fl(\026)2262 2915 y Fp(1)2299 2903 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))1218 b(\(6)p Fl(:)p Fq(17\))100 3083 y Fn(wher)l(e)37 b Fq(~)-49 b Fl(\026)384 3095 y Fp(1)451 3083 y Fn(is)30 b(given)h(in)e(\(6.26\),)765 3337 y Fl(\026)815 3349 y Fp(1)p Fo(;)p Fp(0)905 3337 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))23 b(=)1187 3224 y Fk(Z)1233 3413 y Fp(\012)1298 3337 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))1562 3220 y Fk(\022)1638 3281 y Fq(1)p 1634 3318 49 4 v 1634 3394 a Fl(\025)1693 3337 y(m)1766 3303 y Fc(0)1766 3358 y Fp(0)1817 3220 y Fk(\022)1888 3281 y Fl(d)p Fq(\()p Fl(\021)s(;)g Fq(\000)2096 3293 y Fo(t)2126 3281 y Fq(\))p 1888 3318 271 4 v 1999 3394 a Fl(\025)2168 3220 y Fk(\023)2243 3337 y Fl(V)2310 3294 y Fp(\(0\))2291 3359 y(1)2399 3337 y Fq(\()p Fl(s)p Fq(\()p Fl(\021)s Fq(\))p Fl(;)g(t)p 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Fp(1)1867 4309 y Fq(,)g Fl(m)1991 4321 y Fp(0)2056 4309 y Fq(and)f Fl(V)2265 4321 y Fp(0)2331 4309 y Fq(are)f(already)g (determined)i(and)g(so)e(w)n(e)i(de\014ne)1559 4556 y Fl(b)1595 4568 y Fp(1)1632 4556 y Fq(\()p Fl(t)p Fq(\))c(=)1837 4443 y Fk(Z)1884 4632 y Fp(\012)1949 4556 y Fl(D)2018 4568 y Fo(V)2057 4576 y Fh(0)2093 4556 y Fl(m)2166 4568 y Fp(1)2204 4556 y Fq(d)p Fl(\030)32 b(:)1347 b Fq(\(6)p Fl(:)p Fq(20\))100 4802 y(Set)1369 4901 y Fl(V)1417 4913 y Fp(1)1455 4901 y Fq(\(\000)1539 4913 y Fo(t)1569 4901 y Fq(\))23 b Fj(\021)g Fl(V)1778 4858 y Fp(\(0\))1760 4923 y(1)1868 4901 y Fq(\(\000)1952 4913 y Fo(t)1981 4901 y Fq(\)+)g Fl(<)f(V)2236 4913 y Fp(1)2297 4901 y Fl(>)h Fq(\(\000)2469 4913 y Fo(t)2498 4901 y Fq(\))1158 b(\(6)p Fl(:)p Fq(21\))100 5050 y(Require)27 b(\(6.16\).)36 b(Then)28 b(w)n(e)f(obtain)1332 5185 y Fk(Z)1378 5373 y Fp(\012)1444 5298 y Fl(D)1513 5310 y Fo(V)1552 5318 y Fh(1)1588 5298 y Fl(m)1661 5310 y Fp(0)1698 5298 y Fq(d)p Fl(\030)h Fq(=)22 b(2)p Fj(j)p Fq(\000)2012 5310 y Fo(t)2041 5298 y Fj(j)h Fl(<)g(V)2223 5310 y Fp(1)2284 5298 y Fl(>)f Fq(\(\000)2455 5310 y Fo(t)2485 5298 y Fq(\))28 b Fl(:)1120 b Fq(\(6)p Fl(:)p Fq(22\))0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)29 b(14:29)1377 b Fq(28)p eop %%Page: 29 29 29 28 bop 100 83 a Fq(Hence,)27 b(to)h(satisfy)f(\(6.19\))g(w)n(e)g(m)n (ust)h(tak)n(e)1467 347 y Fl(<)23 b(V)1603 359 y Fp(1)1640 347 y Fq(\(\000)1724 359 y Fo(t)1754 347 y Fq(\))g Fl(>)p Fq(=)f Fj(\000)2100 291 y Fq(1)p 2036 328 169 4 v 2036 404 a(2)p Fj(j)p Fq(\000)2153 416 y Fo(t)2182 404 y Fj(j)2215 347 y Fl(b)2251 359 y Fp(1)2288 347 y Fq(\()p Fl(t)p Fq(\))28 b Fl(:)1255 b Fq(\(6)p Fl(:)p Fq(23\))100 621 y(This)26 b(determines)h Fl(<)c(V)844 633 y Fp(1)881 621 y Fq(\(\000)965 633 y Fo(t)995 621 y Fq(\))g Fl(>)p Fq(,)k(the)g(pro)5 b(jection)26 b(of)g Fl(V)1842 633 y Fp(1)1880 621 y Fq(\(\000)1964 633 y Fo(t)1994 621 y Fq(\))h(on)n(to)f(the)h(constan)n(ts.)36 b(It)27 b(still)g(remains)f (to)g(determine)100 761 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Fp(2)2120 263 y Fl(;)g(::;)g(\036)2289 275 y Fo(n)p Fc(\000)p Fp(1)2420 263 y Fq(\)\()p Fl(\030)t Fq(\))416 b Fl(\030)27 b Fj(2)d Fq(\012)18 b Fj(n)g(N)12 b Fq(\()p Fl(\025)3412 275 y Fp(0)3450 263 y Fq(\))206 b(\(7)p Fl(:)p Fq(15\))100 562 y Fr(The)32 b(Remainder:)100 780 y Fq(The)27 b(remainder)g(term)g (is)h(giv)n(en)f(b)n(y)488 960 y Fl(\025)554 939 y Fq(~)536 960 y Fl(R)599 972 y Fp(2)636 960 y Fq(\()p Fl(\030)t(;)14 b(t;)g(\025)p Fq(\))24 b(=)f Fl(\025)1052 926 y Fo(N)6 b Fp(+1)1199 960 y Fl(A)p Fq(\()p Fl(\030)t(;)14 b(t;)g(\025)p Fq(\))20 b(+)e Fl(\025)1668 926 y Fo(N)6 b Fp(+2)1815 960 y Fq(\001)p Fl(\036)1933 972 y Fo(N)1997 960 y Fq(\()p Fl(\030)t(;)14 b(t)p Fq(\))19 b(+)f Fl(E)2331 972 y Fp(1)2369 960 y Fq(\()p Fl(\030)t(;)c(t;)g(\025)p Fq(\))19 b(+)f Fl(\025)2775 926 y Fo(N)6 b Fp(+1)2922 960 y Fl(B)2985 972 y Fo(N)g Fp(+1)3132 960 y Fq(\()p Fl(\030)t(;)14 b(t;)g(\025)p Fq(\))p Fl(:)277 b Fq(\(7)p Fl(:)p Fq(16\))100 1140 y(>F)-7 b(rom)27 b(\(7.3\),)g(\(7.5\),)h(\(7.7\),)f(w)n(e)g(ha)n (v)n(e)1510 1320 y(sup)1367 1394 y Fp(\()p Fo(\030)r(;t)p Fp(\))p Fc(2)p Fp(\012)p Fc(\002)p Fp([0)p Fo(;T)9 b Fp(])1792 1320 y Fj(j)1833 1299 y Fq(~)1815 1320 y Fl(R)1878 1332 y Fp(2)1915 1320 y Fq(\()p Fl(\030)t(;)14 b(t;)g(\025)p Fq(\))p Fj(j)24 b(\024)f Fl(C)6 b(\025)2419 1286 y Fo(N)2510 1320 y Fl(:)1155 b Fq(\(7)p Fl(:)p Fq(17\))0 1645 y Fm(8.)50 b(Pro)s(of)37 b(of)h(Theorem)f(5.2:)50 b(Analysis)37 b(of)g(compatibilit)m(y)d(conditions)199 1836 y Fq(In)23 b(this)f(section)g(w)n(e)g(analyze)f(the)i(equations)e(obtained)h(in)h (Section)f(7.)35 b(As)22 b(in)h(Section)f(3,)h(the)g(strategy)e(is)h (to)g(\014nd)100 1972 y(at)k(eac)n(h)g(order)g(in)h Fl(\025)p Fq(,)g(\014rst,)g(the)g(slo)n(wly)f(v)-5 b(arying)25 b(part,)i(the)g Fl(\036)2070 1984 y Fo(i)2098 1972 y Fq(,)g(solving)f(the)h(equations)f(for)g Fl(\030)i Fj(2)23 b Fq(\012)17 b Fj(n)f(N)c Fq(\()p Fl(\025)3503 1984 y Fp(0)3541 1972 y Fl(;)i Fq(\000)3630 1929 y Fp(\()p Fo(N)6 b Fp(\))3630 1992 y Fo(t)3745 1972 y Fq(\).)100 2072 y(Then)29 b(w)n(e)g(extend)g Fl(\036)762 2084 y Fo(i)820 2072 y Fq(globally)f(in)h(\012)g(and)g(determine)h(the)f(rapidly)g (deca)n(ying)f(part)g Fl(h)2875 2084 y Fo(i)2932 2072 y Fq(solving)g(the)i(equations)e(in)100 2208 y Fl(\030)f Fj(2)d(N)12 b Fq(\()364 2174 y Fo(\025)403 2182 y Fh(0)p 365 2189 72 4 v 384 2236 a Fp(2)446 2208 y Fl(;)i Fq(\000)535 2165 y Fp(\()p Fo(N)6 b Fp(\))535 2228 y Fo(t)650 2208 y Fq(\).)38 b(Ho)n(w)n(ev)n(er)26 b(here,)h(in)i(order)d(to)i(con)n (tin)n(ue)f(to)h(arbitrary)e(order,)g(it)j(is)e(con)n(v)n(enien)n(t)g (to)h(mo)r(dify)g(the)100 2307 y(w)n(a)n(y)c(w)n(e)h(extract)g(the)i (compatibilit)n(y)e(condition)h(required)e(to)i(solv)n(e)f(the)h (equation)f(for)g(the)h Fl(h)3095 2319 y Fo(i)3123 2307 y Fq(.)36 b(The)26 b(mo)r(di\014cation)100 2407 y(is)k(to)g(add)h(and)f (subtract)g(to)g(eac)n(h)g(order)f(a)h(term)g(of)h(lo)n(w)n(er)e(order) g Fl(\025)2323 2377 y Fo(i)p Fp(+1)2435 2407 y Fl(\013)2488 2419 y Fo(i)2516 2407 y Fq(\()p Fl(s;)14 b Fq(\000\))i(\026)-58 b Fl(m)2781 2377 y Fc(0)2804 2407 y Fq(\()p Fl(z)t Fq(\),)31 b(with)g Fl(\013)3210 2419 y Fo(i)3238 2407 y Fq(\()p Fj(\001)p Fl(;)14 b Fq(\000\))28 b Fj(2)g Fl(C)3590 2377 y Fc(1)3661 2407 y Fq(\(\000\).)100 2507 y(Adding)20 b(and)g(subtracting)f(terms)h(do)r(es)g(not)g(c)n(hange,)h(of)f (course,)g(the)h(total)e(quan)n(tit)n(y)h(but)h(it)f(mo)r(di\014es)h (the)f(equation)100 2606 y(w)n(e)30 b(obtain)h(at)g(eac)n(h)f(single)g (order.)46 b(W)-7 b(e)31 b(obtain)g(at)g(an)n(y)f Fl(i)e Fj(\025)h Fq(1)h(order)g(in)h Fl(\025)g Fq(the)h(corresp)r(onding)d Fl(m)3329 2618 y Fo(i)3387 2606 y Fq(split)j(in)f(one)100 2706 y(part,)g(the)g(function)h Fl(\036)831 2718 y Fo(i)890 2706 y Fq(de\014ned)g(globally)d(in)j(\012,)f(satisfying)g(Neumann)g (condition)g(on)g(the)g(b)r(oundary)f(of)h(\012,)h(the)100 2842 y(other)g(part,)i(the)f Fl(h)728 2854 y Fo(i)755 2842 y Fq(,)i(is)d(di\013eren)n(t)h(from)g(zero)e(only)i(in)g(a)f 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b(\(8)p Fl(:)p Fq(23\))100 4071 y(exp)r(onen)n(tially)27 b(fast.)39 b(Namely)-7 b(,)28 b(since)g(the)h(exp)r(onen)n(tial)f(con)n(v)n (ergence)d(of)44 b(\026)-57 b Fl(m)o Fq(\()p Fj(\001)p Fq(\))29 b(to)f Fj(\006)p Fq(1)g(and)g(the)h(one)e(of)44 b(\026)-57 b Fl(m)3528 4041 y Fc(0)3551 4071 y Fq(\()p Fj(\001)p Fq(\))29 b(and)100 4171 y Fl(h)148 4141 y Fc(0)148 4192 y Fp(1)185 4171 y Fq(\()p Fj(\001)p Fl(;)14 b(s)p Fq(\),)28 b(for)f Fl(s)c Fj(2)g Fq(\000,)28 b(to)f(0)h(w)n(e)f(need)h (only)f(to)g(v)n(erify)g(that)802 4428 y(lim)757 4485 y Fc(j)p Fo(z)r Fc(j!1)976 4311 y Fk(\024)1020 4428 y Fl(\025)p Fq(\001)p Fl(\036)1186 4440 y Fp(1)1224 4428 y Fq(\()p Fl(\025z)t(;)14 b(s)p Fq(\))19 b Fj(\000)f Fl(f)1598 4440 y Fp(2)1635 4428 y Fq(\()e(\026)-58 b Fl(m;)14 b(m)1850 4440 y Fp(1)1887 4428 y Fq(\)\()p Fl(\025z)t(;)g(s)p Fq(\))19 b(+)2262 4371 y Fl(f)2312 4341 y Fc(0)2335 4371 y Fq(\()d(\026)-58 b Fl(m)p Fq(\()p Fl(z)t Fq(\)\))p 2262 4408 V 2331 4485 a Fl(f)2381 4461 y Fc(0)2404 4485 y Fq(\(1\))2589 4428 y Fl(f)2630 4440 y Fp(2)2667 4428 y Fq(\(1)p Fl(;)14 b(\036)2827 4440 y Fp(1)2864 4428 y Fq(\))2896 4311 y Fk(\025)2964 4428 y Fq(=)22 b(0)27 b Fl(:)545 b Fq(\(8)p Fl(:)p Fq(24\))100 4710 y(>F)-7 b(rom)28 b(\(8.7\))g(and)g(\(3.14\))g(w)n(e)g(ha)n(v)n(e)f(that)i Fl(\025)p Fq(\001)p Fl(\036)1622 4722 y Fp(1)1660 4710 y Fq(\()p Fl(\025z)t(;)14 b(s)p Fq(\))25 b(=)2072 4677 y Fp(1)p 2015 4691 147 4 v 2015 4739 a Fo(f)2054 4722 y Fa(0)2076 4739 y Fp(\(1\))2171 4710 y Fl(V)2238 4667 y Fp(\(0\))2219 4732 y(0)2327 4710 y Fq(\()p Fl(s)p Fq(\))17 b(\026)-59 b Fl(m)2503 4680 y Fc(0)2527 4710 y Fq(\()p Fl(z)t Fq(\).)39 b(Then)29 b(as)f Fj(j)p Fl(z)t Fj(j)c(!)g(1)p Fq(,)29 b(it)g(con)n(v)n(erges)100 4849 y(exp)r(onen)n(tially)h(fast)h (to)h(0.)47 b(F)-7 b(urther)31 b(since)g(lim)1623 4864 y Fc(j)p Fo(z)r Fc(j!1)1847 4849 y Fl(f)1897 4819 y Fc(0)1920 4849 y Fq(\()16 b(\026)-58 b Fl(m)p Fq(\()p Fl(z)t Fq(\)\))30 b(=)e Fl(f)2337 4819 y Fc(0)2360 4849 y Fq(\(1\))h(=)g Fl(f)2639 4819 y Fc(0)2662 4849 y Fq(\()p Fj(\000)p Fq(1\))i(and)g(lim) 3144 4864 y Fc(j)p Fo(z)r Fc(j!1)3368 4849 y Fl(h)3416 4861 y Fp(1)3453 4849 y Fq(\()p Fl(z)t(;)14 b(s)p Fq(\))29 b(=)g(0)100 4948 y(exp)r(onen)n(tially)e(fast,)g(w)n(e)g(ha)n(v)n(e)g (\(8.24\))g(and)g(therefore)g(\(8.23\).)36 b(As)28 b(done)f(b)r(efore,) g(w)n(e)h(extend)f(\(8.21\))g(in)h Fl(I)-14 b(R)q Fq(.)127 5141 y Fr(Lemma)30 b(8.3)90 b Fn(Set)967 5294 y Fl(V)1033 5251 y Fp(\(0\))1015 5316 y(1)1123 5294 y Fq(\()p Fl(\030)t(;)14 b Fq(\000\))23 b(=)g Fj(T)1472 5306 y Fp(\000)1531 5177 y Fk(\024)1585 5238 y Fq(1)p 1585 5275 42 4 v 1585 5351 a(4)1636 5294 y Fl(B)1699 5306 y Fp(1)1736 5294 y Fq(\()p Fj(\001)p Fq(\))d Fj(\000)1985 5238 y Fq(1)p 1936 5275 140 4 v 1936 5351 a(4)p Fj(j)p Fq(\000)p Fj(j)2099 5181 y Fk(Z)2145 5370 y Fp(\000)2204 5294 y Fl(B)2267 5306 y Fp(1)2304 5294 y Fq(\()p Fl(s)p Fq(\)d)p Fl(s)2492 5177 y Fk(\025)2550 5294 y Fq(\()p Fl(\030)t Fq(\))86 b Fl(\030)27 b Fj(2)d Fq(\000)754 b(\(8)p Fl(:)p Fq(25\))0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)29 b(14:29)1377 b Fq(36)p eop %%Page: 37 37 37 36 bop 100 83 a Fn(wher)l(e)38 b Fl(B)405 95 y Fp(1)442 83 y Fq(\()p Fl(s)p Fq(\))g Fn(is)g(de\014ne)l(d)g(in)f(\(8.28\))i(and) f Fj(T)1549 95 y Fp(\000)1632 83 y Fn(is)g(the)f(op)l(er)l(ator)i (de\014ne)l(d)f(in)f(\(10.8\).)64 b(Then)38 b(ther)l(e)f(ar)l(e)h (uniquely)100 183 y(determine)l(d)i Fl(\013)590 195 y Fp(2)628 183 y Fq(\()p Fj(\001)p Fl(;)14 b Fq(\000\))42 b Fj(2)f Fl(C)1007 153 y Fc(1)1078 183 y Fq(\(\000\))f Fn(and)h Fl(h)1454 195 y Fp(2)1491 183 y Fq(\()p Fj(\001)p Fl(;)14 b(s)p Fq(\))42 b Fj(2)g Fq(\003)1851 153 y Fc(1)1921 183 y Fq(\()p Fl(I)-15 b(R)r Fq(\))p Fn(,)43 b Fl(h)2187 195 y Fp(2)2224 183 y Fq(\(0)p Fl(;)14 b(s)p Fq(\))41 b(=)g(0)f Fn(with)g Fl(s)i Fj(2)g Fq(\000)p Fn(,)g(solution)e(of)h (\(8.21\).)100 282 y(Mor)l(e)l(over)31 b Fl(h)513 294 y Fp(2)550 282 y Fq(\()p Fj(\001)p Fl(;)14 b(s)p Fq(\))30 b Fn(and)g(its)g(derivatives)i(with)e(r)l(esp)l(e)l(ct)g(to)f Fl(z)k 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b(of)i Fl(\036)2469 2053 y Fo(h)2540 2041 y Fq(across)d(\000,)j(and)f(w)n(e)h(ha)n(v)n(e)e (that)1587 2308 y Fl(h)p Fq(\()p Fl(\030)t Fq(\))e(=)1850 2191 y Fk(\024)1929 2252 y Fl(@)p 1904 2289 99 4 v 1904 2365 a(@)5 b(n)2013 2308 y(\036)2062 2320 y Fo(h)2105 2191 y Fk(\025)2149 2391 y Fp(\000)2208 2308 y Fq(\()p Fl(\030)t Fq(\))1376 b(\(10)p Fl(:)p Fq(5\))100 2579 y(where)27 b(the)h(righ)n(t)f(hand)g(side)h(is)f(the)h(di\013erence)g (in)f(the)h(normal)f(deriv)-5 b(ativ)n(es)27 b(at)g Fl(\030)g Fj(2)d Fq(\000:)1086 2729 y Fk(\024)1165 2790 y Fl(@)p 1140 2827 V 1140 2903 a(@)5 b(n)1248 2846 y(\036)1297 2858 y Fo(h)1341 2729 y Fk(\025)1385 2929 y Fp(\000)1444 2846 y Fq(\()p Fl(\030)t Fq(\))23 b(=)1659 2729 y Fk(\022)1730 2790 y Fl(@)5 b(\036)1828 2802 y Fo(h)p 1730 2827 142 4 v 1751 2903 a Fl(@)g(n)1881 2729 y Fk(\023)1942 2929 y Fp(\012)1989 2902 y Fh(+)1989 2950 y(\000)2055 2846 y Fq(\()p Fl(\030)t Fq(\))19 b Fj(\000)2261 2729 y Fk(\022)2332 2790 y Fl(@)5 b(\036)2430 2802 y Fo(h)p 2332 2827 V 2353 2903 a Fl(@)g(n)2483 2729 y Fk(\023)2544 2929 y Fp(\012)2591 2902 y Fa(\000)2591 2950 y Fh(\000)2659 2846 y Fq(\()p Fl(\030)t Fq(\))28 b Fl(:)100 3143 y Fq(This)f(is)h(a)f(w)n(ell)g(kno)n (wn)g(result)g(from)h(p)r(oten)n(tial)f(theory)g([9].)37 b(F)-7 b(or)27 b Fl(\030)k Fq(a)n(w)n(a)n(y)26 b(from)h(\000,)1490 3407 y(\001)p Fl(\036)1608 3419 y Fo(h)1651 3407 y Fq(\()p Fl(\030)t Fq(\))d(=)1886 3351 y(1)p 1877 3388 60 4 v 1877 3464 a(\012)1960 3294 y Fk(Z)2007 3483 y Fp(\000)2066 3407 y Fl(h)p Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)2319 3419 y Fo(\021)2387 3407 y Fl(:)100 3680 y Fq(Th)n(us)34 b(the)i(single)f (la)n(y)n(er)e(p)r(oten)n(tial)i(is)g(harmonic)f(a)n(w)n(a)n(y)f(from)i (\000)g(if)h(and)f(only)g(if)2768 3613 y Fk(R)2807 3710 y Fp(\000)2866 3680 y Fl(h)p Fq(\()p Fl(\030)t Fq(\)d)p Fl(S)3115 3692 y Fo(\030)3188 3680 y Fq(=)g(0.)59 b(Otherwise,)100 3780 y(it)34 b(is)f(subharmonic)f(or)h(sup)r(erharmonic,)h(according)d (to)j(whether)2284 3713 y Fk(R)2323 3809 y Fp(\000)2382 3780 y Fl(h)p Fq(\()p Fl(\030)t Fq(\)d)p Fl(S)2631 3792 y Fo(\030)2702 3780 y Fq(is)f(p)r(ositiv)n(e)g(or)g(negativ)n(e.)53 b(Ev)n(ery)100 3879 y(con)n(tin)n(uous)28 b(function)i Fl(\036)h Fq(that)e(satis\014es)g(the)h(Neumann)g(b)r(oundary)f (condition,)h(and)f(is)h(harmonic)f(a)n(w)n(a)n(y)e(from)i(\000,)100 3979 y(and)e(whic)n(h)h(satis\014es)1698 4012 y Fk(Z)1744 4201 y Fp(\012)1809 4125 y Fl(\036)p Fq(\()p Fl(\030)t Fq(\)d)p Fl(\030)h Fq(=)23 b(0)1485 b(\(10)p Fl(:)p Fq(6\))100 4349 y(is)27 b(the)h(single)f(la)n(y)n(er)f(p)r(oten)n(tial)i(of)f(a)g (uniquely)h(determined)g(function)g Fl(h)f Fq(de\014ned)h(on)g(\000)f (satisfying)1653 4495 y Fk(Z)1699 4684 y Fp(\000)1758 4608 y Fl(h)p Fq(\()p Fl(\030)t Fq(\)d)p Fl(S)2007 4620 y Fo(\030)2067 4608 y Fq(=)c(0)k Fl(:)1441 b Fq(\(10)p Fl(:)p Fq(7\))100 4881 y(Indeed,)28 b(if)g Fl(\036)518 4893 y Fo(h)589 4881 y Fq(is)f(suc)n(h)h(a)f(single)g(la)n(y)n(er)f(p)r (oten)n(tial,)h(then)h(from)g(\(10.2\),)2383 4814 y Fk(R)2422 4911 y Fp(\012)2487 4881 y Fl(\036)2536 4893 y Fo(h)2580 4881 y Fq(\()p Fl(\030)t Fq(\)d)p Fl(\030)g Fq(=)22 b(0.)199 5016 y(On)32 b(the)g(other)f(hand,)i(let)g Fl(\036)f Fq(b)r(e)g(an)n(y)f(con)n(tin)n(uous)g(function)i(that)f(is)g(harmonic) e(on)i(\012)3000 4981 y Fc(\000)3000 5041 y Fp(\000)3088 5016 y Fq(and)g(\012)3314 4981 y Fp(+)3314 5041 y(\000)3369 5016 y Fq(,)h(and)f(whic)n(h)100 5116 y(satis\014es)26 b(\(10.6\).)37 b(De\014ne)28 b Fl(h)f Fq(in)h(\000)g(b)n(y)1583 5290 y Fl(h)p Fq(\()p Fl(\030)t Fq(\))c(=)1847 5173 y Fk(\024)1925 5234 y Fl(@)p 1900 5271 99 4 v 1900 5347 a(@)5 b(n)2009 5290 y(\036)2058 5173 y Fk(\025)2102 5373 y Fp(\000)2161 5290 y Fq(\()p Fl(\030)t Fq(\))29 b(;)0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)g(14:29)1377 b Fq(41)p eop %%Page: 42 42 42 41 bop 100 83 a Fq(w)n(e)27 b(refer)g(to)g(this)h(as)f(the)h (Neumann)g(data)f(for)g Fl(\036)p Fq(.)38 b(By)27 b(the)h(div)n (ergence)e(theorem,)545 239 y Fk(Z)591 428 y Fp(\000)650 352 y Fl(h)p Fq(\()p Fl(\030)t Fq(\)d)p Fl(S)899 364 y Fo(\030)959 352 y Fq(=)1047 239 y Fk(Z)1093 428 y Fp(\000)1152 352 y Fj(r)p Fl(\036)19 b Fj(\001)f Fl(n)p Fq(d)p Fl(S)1477 364 y Fo(\030)1532 352 y Fq(+)1615 239 y Fk(Z)1661 428 y Fp(\000)1720 352 y Fj(r)p Fl(\036)h Fj(\001)g Fq(\()p Fj(\000)p Fl(n)p Fq(\)d)p Fl(S)2175 364 y Fo(\030)2234 352 y Fq(=)2322 239 y Fk(Z)2368 428 y Fp(\012)2415 401 y Fa(\000)2415 449 y Fh(\000)2483 352 y Fq(\001)p Fl(\036)p Fq(d)p Fl(\030)k Fq(+)2789 239 y Fk(Z)2835 428 y Fp(\012)2882 401 y Fh(+)2882 449 y(\000)2947 352 y Fq(\001)p Fl(\036)p Fq(d)p Fl(\030)28 b Fq(=)23 b(0)k Fl(:)100 652 y Fq(Hence,)g Fl(h)h Fq(satis\014es)f(\(10.7\).)199 754 y(Notice)h(that)g Fl(\036)19 b Fj(\000)f Fl(\036)840 766 y Fo(h)911 754 y Fq(satis\014es)27 b(Neumann)h(b)r(oundary)e(conditions)h(and)1510 914 y Fk(\024)1588 975 y Fl(@)p 1563 1012 99 4 v 1563 1088 a(@)5 b(n)1672 1031 y Fq(\()p Fl(\036)19 b Fj(\000)f Fl(\036)1904 1043 y Fo(h)1948 1031 y Fq(\))1980 914 y Fk(\025)2024 1114 y Fp(\000)2083 1031 y Fq(\()p Fl(\030)t Fq(\))24 b(=)e(0)27 b Fl(:)100 1312 y Fq(This)g(means)g(that)h Fl(\036)19 b Fj(\000)f Fl(\036)923 1324 y Fo(h)994 1312 y Fq(is)28 b(a)f(constan)n(t.)36 b(Since)28 b(the)g(in)n(tegral)e(is)i (zero,)e(it)i(is)g(zero,)e(and)i(so)f Fl(\036)c Fq(=)g Fl(\036)3296 1324 y Fo(h)3339 1312 y Fq(.)199 1414 y(This)29 b(pro)n(v)n(es)e(the)j(one)e(to)h(one)g(corresp)r(ondence)e(b)r(et)n(w) n(een)i(single)g(la)n(y)n(er)e(p)r(oten)n(tials)i(of)g(functions)g Fl(h)g Fq(satisfying)100 1514 y(\(10.7\),)e(and)g(con)n(tin)n(uous)g (functions)g Fl(\036)i Fq(that)e(are)g(harmonic)g(on)g(\012)2233 1478 y Fc(\000)2233 1538 y Fp(\000)2317 1514 y Fq(and)g(\012)2538 1478 y Fp(+)2538 1538 y(\000)2593 1514 y Fq(,)h(and)f(whic)n(h)h (satisfy)f(\(10.6\).)199 1617 y(Next,)34 b(giv)n(en)d(a)g(con)n(tin)n (uous)g(function)h Fl(\036)h Fq(that)f(is)f(harmonic)g(on)h(on)f(\012) 2495 1581 y Fc(\000)2495 1641 y Fp(\000)2583 1617 y Fq(and)h(\012)2809 1581 y Fp(+)2809 1641 y(\000)2864 1617 y Fq(,)h(whether)f(or)f(not)g (\(10.6\))h(is)100 1716 y(satis\014ed,)27 b(de\014ne)h(the)g(function)g Fl(g)i Fq(on)d(\000)h(b)n(y)f Fl(g)f Fq(=)d Fl(\036)p Fj(j)1756 1728 y Fp(\000)1801 1716 y Fq(.)37 b(W)-7 b(e)28 b(naturally)f(refer)g(to)g Fl(g)k Fq(as)26 b(the)i(Diric)n(hlet)g(data) f(for)g Fl(\036)p Fq(.)199 1894 y(The)h(Neumann)g(data)f(is)1018 1777 y Fk(\024)1097 1838 y Fl(@)p 1072 1875 V 1072 1951 a(@)5 b(n)1181 1894 y(\036)1230 1777 y Fk(\025)1274 1977 y Fp(\000)1319 1894 y Fq(.)37 b(The)28 b(Diric)n(hlet{Neuman)f(op)r (erator)f Fj(T)2616 1906 y Fp(\000)2689 1894 y Fq(de\014ned)i(b)n(y) 1664 2237 y Fj(T)1709 2249 y Fp(\000)1754 2237 y Fl(g)e Fq(=)1908 2120 y Fk(\024)1986 2181 y Fl(@)p 1961 2218 V 1961 2294 a(@)5 b(n)2070 2237 y(\036)2119 2120 y Fk(\025)2163 2320 y Fp(\000)3665 2237 y Fq(\(10)p Fl(:)p Fq(8\))o Fl(:)100 2532 y Fq(where)27 b Fl(\036)h Fq(is)f(the)h(con)n(tin)n(uous) f(function)h(that)g(is)f(Harmonic)g(in)h(\012)2185 2496 y Fc(\000)2185 2556 y Fp(\000)2269 2532 y Fq(and)f(\012)2490 2496 y Fp(+)2490 2556 y(\000)2545 2532 y Fq(,)h(and)f(with)h Fl(\036)p Fj(j)3018 2544 y Fp(\000)3087 2532 y Fq(=)23 b Fl(g)s Fq(.)199 2635 y(A)29 b(simple)f(argumen)n(t)f(sho)n(ws)g(that) h Fj(T)1385 2647 y Fp(\000)1459 2635 y Fq(is)g(a)g(p)r(ositiv)n(e)f (Hermitian)h(op)r(erator.)37 b(Indeed,)29 b(let)f Fl( )j Fq(b)r(e)e(con)n(tin)n(uous)e(on)100 2734 y(\012,)g(and)h(harmonic)e (on)i(\012)912 2699 y Fc(\000)912 2759 y Fp(\000)995 2734 y Fq(and)g(\012)1217 2699 y Fp(+)1217 2759 y(\000)1272 2734 y Fq(,)g(and)f(with)h Fl( )s Fj(j)1753 2746 y Fp(\000)1822 2734 y Fq(=)22 b Fl(h)p Fq(.)37 b(Then)979 2906 y Fk(Z)1025 3095 y Fp(\000)1084 3019 y Fl(h)p Fj(T)1177 3031 y Fp(\000)1222 3019 y Fl(g)s Fq(d)p Fl(s)23 b Fq(=)1460 2906 y Fk(Z)1507 3095 y Fp(\000)1566 3019 y Fl( )1636 2902 y Fk(\024)1715 2963 y Fl(@)p 1690 3000 V 1690 3076 a(@)5 b(n)1799 3019 y(\036)1848 2902 y Fk(\025)1892 3102 y Fp(\000)1951 3019 y Fq(d)p Fl(s)1373 3273 y Fq(=)1460 3160 y Fk(Z)1507 3349 y Fp(\012)1554 3322 y Fh(+)1554 3370 y(\000)1619 3273 y Fj(r)18 b(\001)h Fq(\()p Fl( )s Fq(\()p Fj(r)p Fl(\036)p Fq(\)\)d)p Fl(\030)24 b Fq(+)2240 3160 y Fk(Z)2287 3349 y Fp(\012)2334 3322 y Fa(\000)2334 3370 y Fh(\000)2401 3273 y Fj(r)19 b(\001)f Fq(\()p Fl( )s Fq(\()p Fj(r)p Fl(\036)p Fq(\)\)d)p Fl(\030)1373 3541 y Fq(=)1460 3428 y Fk(Z)1507 3617 y Fp(\012)1572 3541 y Fj(r)p Fl( )k Fj(\001)c(r)p Fl(\036)p Fq(d)p Fl(\030)33 b(:)100 3815 y Fq(T)-7 b(aking)24 b Fl(h)f Fq(=)f(1,)k(so)e(that)h Fl( )i Fq(=)22 b(1,)j(w)n(e)g(further)g(see)g(that)g(the)h(range)e(of)h Fj(T)2365 3827 y Fp(\000)2435 3815 y Fq(is)g(orthogonal)e(to)i(the)h (constan)n(ts.)35 b(W)-7 b(e)25 b(let)100 3914 y Fj(T)145 3926 y Fp(\000)215 3914 y Fq(denote)g(the)g(F)-7 b(riedric)n(hs)24 b(extension)g(of)h Fj(T)1501 3926 y Fp(\000)1546 3914 y Fq(.)36 b(It)26 b(is)e(easy)g(to)h(see,)g(and)g(w)n(ell)f(kno)n(wn,)h (that)g(the)g(form)g(domain)f(of)h Fj(T)3755 3926 y Fp(\000)100 4014 y Fq(is)e(the)h(Sob)r(olev)f(space)g Fl(H)917 3984 y Fp(1)p Fo(=)p Fp(2)1021 4014 y Fq(\(\000\),)i(and)e(that)h(the)g(k)n (ernel)e(consists)h(exactly)g(of)g(the)h(constan)n(ts.)35 b(There)23 b(is)g(an)g(explicit)100 4114 y(form)n(ula)j(for)g(the)i(in) n(v)n(erse)e(of)h Fj(T)1087 4126 y Fp(\000)1159 4114 y Fq(restricted)g(to)g(the)g(orthogonal)e(complemen)n(t)i(of)g(the)h (constan)n(ts;)e(w)n(e)h(denote)g(this)100 4213 y(b)n(y)j Fj(S)268 4225 y Fp(\000)313 4213 y Fq(.)45 b(Indeed,)31 b(let)g Fl(v)i Fq(b)r(e)e(an)n(y)e(function)i(on)f(\000)g(with)1868 4146 y Fk(R)1907 4243 y Fp(\000)1966 4213 y Fl(v)s Fq(\()p Fl(s)p Fq(\)d)p Fl(s)e Fq(=)f(0.)44 b(Since)31 b(the)f(single)g(la)n(y) n(er)f(p)r(oten)n(tial)h Fl(\036)3631 4225 y Fo(v)3701 4213 y Fq(for)100 4313 y Fl(v)k Fq(has)c(Neumann)i(data)e Fl(v)s Fq(,)i(all)f(w)n(e)f(need)i(to)e(do)h(is)g(to)g(subtract)f(a)h (constan)n(t)f(to)h(mak)n(e)f(this)h(function)h(orthogonal)100 4412 y(to)d(the)h(constan)n(ts)e(on)i(\000,)g(instead)f(of)h(b)r(eing)f (orthogonal)f(to)h(the)h(constan)n(t)f(on)g(\012.)43 b(Therefore,)28 b(the)i(in)n(v)n(erse)e Fj(S)3669 4424 y Fp(\000)3745 4412 y Fq(is)100 4512 y(giv)n(en)e(b)n(y)603 4682 y Fj(S)653 4694 y Fp(\000)699 4682 y Fl(v)s Fq(\()p Fl(\030)t Fq(\))e(=)957 4569 y Fk(Z)1003 4757 y Fp(\000)1062 4682 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))p Fl(v)s Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)1560 4694 y Fo(\021)1621 4682 y Fj(\000)1743 4625 y Fq(1)p 1714 4663 V 1714 4739 a Fj(j)p Fq(\000)p Fj(j)1836 4569 y Fk(Z)1882 4757 y Fp(\000)1941 4569 y Fk(Z)1988 4757 y Fp(\000)2047 4682 y Fl(G)p Fq(\()p Fl(\030)t(;)g(\021)s Fq(\))p Fl(v)s Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)2545 4694 y Fo(\021)2587 4682 y Fq(d)p Fl(S)2684 4694 y Fo(\030)3053 4682 y Fl(\030)27 b Fj(2)c Fq(\000)28 b Fl(:)391 b Fq(\(10)p Fl(:)p Fq(9\))100 4915 y(It)28 b(is)g(easily)f(c)n(hec)n(k)n(ed)g(that)i(this)f(is)g (self)g(adjoin)n(t)g(on)g(the)h(orthogonal)c(complemen)n(t)j(of)h(the)f (constan)n(ts.)37 b(No)n(w)28 b(let)h Fl(h)100 5015 y Fq(b)r(e)f(an)f(arbitrary)e(smo)r(oth)j(function)g(on)f(\000)h (satisfying)1865 4948 y Fk(R)1904 5045 y Fp(\000)1963 5015 y Fl(h)p Fq(\()p Fl(s)p Fq(\)d)p Fl(s)23 b Fq(=)g(0.)37 b(Consider)26 b(the)i(single)f(la)n(y)n(er)f(p)r(oten)n(tial)1200 5298 y Fl(\036)p Fq(\()p Fl(\030)t Fq(\))e(=)1465 5185 y Fk(Z)1511 5373 y Fp(\000)1570 5298 y Fl(G)p Fq(\()p Fl(\030)t(;)14 b(\021)s Fq(\))p Fl(h)p Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)2073 5310 y Fo(\021)2447 5298 y Fl(\030)28 b Fj(2)23 b Fq(\012)28 b Fl(:)0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)h(14:29)1377 b Fq(42)p eop %%Page: 43 43 43 42 bop 100 83 a Fq(In)30 b(general,)g(the)h(Diric)n(hlet)f(data)g (for)g Fl(\036)h Fq(do)r(es)f(not)h(in)n(tegrate)e(to)i(zero)e(on)h (\000,)h(and)g(hence)f(is)g(not)h(directly)f(related)100 183 y(to)g(the)h(Neumann)g(data)f(through)g(the)h(Diric)n(hlet{Neumann) f(op)r(erator.)45 b(Ho)n(w)n(ev)n(er,)29 b(w)n(e)h(can)h(correct)e(for) h(this)h(b)n(y)100 282 y(subtracting)26 b(a)i(constan)n(t:)36 b(F)-7 b(orm)27 b(the)h(function)1409 513 y(~)1398 535 y Fl(\036)q Fq(\()p Fl(\030)t Fq(\))23 b(=)g Fl(\036)p Fq(\()p Fl(\030)t Fq(\))d Fj(\000)1957 479 y Fq(1)p 1929 516 99 4 v 1929 592 a Fj(j)p Fq(\000)p Fj(j)2050 422 y Fk(Z)2096 610 y Fp(\000)2156 535 y Fl(\036)p Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)2410 547 y Fo(\021)2479 535 y Fl(:)100 794 y Fq(Then)27 b(clearly)1750 898 y(~)1739 920 y Fl(\036)p Fj(j)1811 932 y Fp(\000)1880 920 y Fq(=)22 b Fj(S)2017 932 y Fp(\000)2063 920 y Fl(h)1739 1085 y(h)h Fq(=)f Fj(T)1942 1097 y Fp(\000)1999 1063 y Fq(~)1988 1085 y Fl(\036)2152 999 y(:)1471 b Fq(\(10)p Fl(:)p Fq(10\))100 1259 y(W)-7 b(e)28 b(can)f(no)n(w)g(express)f(the)i(v)n(ector)e (\014eld)i Fl(V)47 b Fq(driving)27 b(the)h(Mullins{Sek)n(erk)-5 b(a)26 b(\015o)n(w)h(as)1376 1519 y Fl(V)42 b Fq(=)22 b Fj(T)1598 1531 y Fp(\000)1658 1402 y Fk(\022)1719 1519 y Fl(K)i Fj(\000)1935 1462 y Fq(1)p 1907 1500 V 1907 1576 a Fj(j)p Fq(\000)p Fj(j)2028 1406 y Fk(Z)2075 1594 y Fp(\000)2134 1519 y Fl(K)6 b Fq(\()p Fl(s)p Fq(\)d)p Fl(s)2399 1402 y Fk(\023)2501 1519 y Fl(:)1122 b Fq(\(10)p 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Fq(12\))100 2397 y(where)27 b Fl(v)k Fq(is)c(a)g(function)h(on)g(\000)f(satisfying) 1655 2447 y Fk(Z)1701 2635 y Fp(\000)1760 2560 y Fl(v)s Fq(\()p Fl(\030)t Fq(\)d)p Fl(S)2004 2572 y Fo(\030)2065 2560 y Fq(=)22 b(0)28 b Fl(:)100 2785 y Fq(Notice)34 b(that)h(\()q Fj(E)631 2797 y Fp(\000)p Fo(;N)754 2785 y Fl(v)s Fq(\))15 b(\()p Fl(\030)t Fq(\))35 b(is)g(the)g(unique)g (function)g(that)g(is)g(con)n(tin)n(uous)f(on)g(\012,)j(harmonic)c(on)i (\012)p Fj(n)p Fq(\000)f(satisfying)100 2884 y(Neumann)28 b(b)r(oundary)e(conditions)h(on)h Fl(@)5 b Fq(\012,)27 b(with)h(Neuman)g(data)f Fl(v)k Fq(on)c(\003,)h(and)f(with)h(zero)f(in) n(tegral)f(o)n(v)n(er)g(\000.)199 2985 y(The)31 b Fn(Dirichlet)i (harmonic)h(extension)e(op)l(er)l(ator)f Fj(E)1835 2997 y Fp(\000)p Fo(;D)1987 2985 y Fq(is)f(de\014ned)h(b)n(y)f(setting)g Fj(E)2799 2997 y Fp(\000)p Fo(;D)2920 2985 y Fl(g)s Fq(\()p Fl(\030)t Fq(\))h(to)f(b)r(e)h(the)f(harmonic)100 3084 y(function)j Fl(\036)g Fq(on)g(\012)p Fj(n)p Fq(\000)f(with)i(Neumann)f (b)r(oundary)f(conditions)g(on)h Fl(@)5 b Fq(\000,)34 b(and)f(with)g Fl(\036)p Fj(j)2888 3096 y Fp(\000)2966 3084 y Fq(=)e Fl(g)s Fq(.)52 b(Here,)34 b(there)f(is)g(no)100 3184 y(restriction)26 b(on)i(the)g(in)n(tegral)e(of)h Fl(g)k Fq(o)n(v)n(er)25 b(\000.)199 3284 y(Naturally)-7 b(,)39 b(the)f(Diric)n(hlet)f(extension)f(can)h(b)r(e)h(expressed)d(in) j(terms)f(of)g(the)g(Neumann)h(extension)e(and)h(the)100 3384 y(Diric)n(hlet{Neumann)27 b(op)r(erator.)35 b(W)-7 b(e)28 b(ha)n(v)n(e)f(from)g(\(10.9\))g(and)g(\(10.12\))f(that)557 3649 y Fj(E)601 3661 y Fp(\000)p Fo(;D)721 3649 y Fl(g)s Fq(\()p Fl(\030)t Fq(\))e(=)e Fj(E)1023 3661 y Fp(\000)p Fo(;N)1161 3532 y Fk(\022)1222 3649 y Fj(T)1267 3661 y Fp(\000)1326 3532 y Fk(\022)1387 3649 y Fl(g)f Fj(\000)1570 3592 y Fq(1)p 1541 3629 V 1541 3706 a Fj(j)p Fq(\000)p Fj(j)1663 3536 y Fk(Z)1709 3724 y Fp(\000)1768 3649 y Fl(g)s Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)2016 3661 y Fo(\021)2057 3532 y Fk(\023\023)2193 3649 y Fq(\()p Fl(\030)t Fq(\))e(+)2438 3592 y(1)p 2409 3629 V 2409 3706 a Fj(j)p Fq(\000)p Fj(j)2531 3536 y Fk(Z)2577 3724 y Fp(\000)2636 3649 y Fl(g)s Fq(\()p Fl(\021)s Fq(\)d)p Fl(S)2884 3661 y Fo(\021)3091 3649 y Fl(\030)27 b Fj(2)d Fq(\012)j Fl(:)303 b Fq(\(10)p Fl(:)p Fq(13\))100 4027 y Fr(A.2:)41 b(The)32 b(expansion)f(in)h Fl(\025)g Fr(of)g(the)g(Laplacian)h(in)e(lo)s(cal)g(co)s(ordinates.)199 4176 y Fq(Le)d Fl(f)9 b Fq(\()p Fl(z)t(;)14 b(s)p Fq(\),)27 b Fl(z)f Fq(=)764 4143 y Fo(d)p 762 4157 40 4 v 762 4204 a(\025)811 4176 y Fq(,)i(b)r(e)g(a)f Fl(C)1109 4146 y Fp(2)1174 4176 y Fq(function)h(from)f Fl(I)-14 b(R)20 b Fj(\002)e Fq(\000)27 b(to)h Fl(I)-14 b(R)q Fq(.)37 b(Then,)28 b(in)g(dimension)f Fl(d)c Fq(=)g(2,)k(w)n(e)g(ha)n(v)n(e)g (that)656 4448 y Fl(\025)704 4414 y Fp(2)742 4448 y Fq(\001)p Fl(f)9 b Fq(\()p Fl(z)t(;)14 b(s)p Fq(\))23 b(=)1350 4392 y(1)p 1164 4429 414 4 v 1164 4505 a(1)18 b Fj(\000)g Fl(K)6 b Fq(\()p Fl(s)p Fq(\))p Fl(\025z)1602 4331 y Fk(\032)1664 4448 y Fq(\(\(1)18 b 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Fl(nK)1714 813 y Fo(n)p Fc(\000)p Fp(1)1843 848 y Fq(\()p Fl(s)p Fq(\))p Fl(z)1989 813 y Fo(n)p Fc(\000)p Fp(1)2147 848 y Fl(;)1129 1049 y(c)1165 1061 y Fo(n)p Fp(+1)1294 1049 y Fq(\()p Fl(z)t(;)14 b(s)p Fq(\))23 b(=)1597 992 y(1)p 1597 1029 42 4 v 1597 1106 a(2)1649 1049 y Fl(n)p Fq(\()p Fl(n)18 b Fj(\000)g Fq(1\))p Fl(z)1999 1014 y Fo(n)p Fc(\000)p Fp(1)2128 1049 y Fl(K)2205 1014 y Fo(n)p Fc(\000)p Fp(2)2335 1049 y Fq(\()p Fl(s)p Fq(\))2468 992 y Fl(d)p 2448 1029 83 4 v 2448 1106 a(ds)2541 1049 y(K)6 b Fq(\()p Fl(s)p Fq(\))27 b Fl(:)3646 882 y Fq(\(10)p Fl(:)p Fq(16\))1726 1276 y Fr(References)70 1447 y Fq([1])41 b(N.)f(Alik)-5 b(ak)n(os,)41 b(P)-7 b(.W.)40 b(Bates,)i(X)d(Chen,)k Fn(Conver)l(genc)l(e)e(of)h(the)e(Cahn-Hil)t(liar)l(d)k(e)l(quation)c (to)h(the)g(Hele-Shaw)199 1547 y(mo)l(del.)e Fq(Arc)n(h.)d(Rat.)h(Mec)n (h.)g(Anal.)f Fr(128)p Fq(,)27 b(165{205)e(1994)g(\(1994\).)70 1718 y([2])41 b(S.)35 b(Bastea,)f(R.)h(Esp)r(osito,)f(J.L.)g(Leb)r(o)n (witz,)i(R.)e(Marra,)g Fn(Binary)j(\015uids)f(with)h(long)f(r)l(ange)g (se)l(gr)l(e)l(gating)g(inter-)199 1818 y(action.)66 b(I:)39 b(Derivation)h(of)g(kinetic)f(and)g(hydr)l(o)l(dynamic)i(e)l (quations.)132 b Fq(J.)37 b(Stat.)66 b(Ph)n(ys.,)39 b Fr(101)p Fq(,)g(1087{1136)199 1917 y(\(2000\).)70 2089 y([3])i(L.Ca\013arelli,)32 b(N.E.)g(Muller,)h Fn(A)n(n)g Fl(L)1348 2058 y Fc(1)1452 2089 y Fn(b)l(ound)h(for)h(the)e(Cahn-Hil)t (liar)l(d)k(e)l(quation.)51 b Fq(Arc)n(h.)e(Rat.)h(Mec)n(h.)g(Anal.)199 2188 y Fr(133)59 b Fq(129{144)24 b(\(1999\).)70 2360 y([4])41 b(R.)31 b(Ca\015isc)n(h,)e Fn(The)k(\015uid)g(dynamic)l(al)g (limit)g(of)g(the)f(nonline)l(ar)g(e)l(quation.)45 b Fq(Comm.)f(Pure)30 b(and)g(Applied)g(Math.)199 2459 y Fr(33)59 b Fq(651{666)24 b(\(1980\).)70 2630 y([5])41 b(C.)22 b(Cercignani,)f(R.)g(Illner,)i(M.)e(Pulviren)n(ti,)h Fn(The)j(Mathematic)l(al)h(The)l(ory)f(of)g(Dilute)e(Gasses.)36 b Fq(Springer{V)-7 b(erlag,)199 2730 y(Berlin)27 b(\(1994\).)70 2901 y([6])41 b(X.)36 b(Chen,)i Fn(The)g(Hele-Shaw)f(Pr)l(oblem)h(and)g (A)n(r)l(e)l(a-Pr)l(eserving)f(Curve-Shortening)g(Motions.)122 b Fq(Arc)n(h.)59 b(Rat.)199 3001 y(Mec)n(h.)37 b(Anal.)g Fr(123)27 b Fq(117{151)d(\(1993\).)70 3172 y([7])41 b(X.)30 b(Chen,)g Fn(Glob)l(al)j(asymptotic)f(limit)g(of)h(solutions)e(of)h (the)g(Cahn-Hil)t(liar)l(d)i(Equation.)86 b Fq(J.)29 b(Di\013.)44 b(Eq.)d Fr(44)p Fq(,)30 b(no.)199 3272 y(2,)d(262-311)e (\(1996\))70 3443 y([8])41 b(X.)28 b(Chen,)f(J.)g(Hong,)g(F.)h(Yi,)f Fn(Existenc)l(e,)j(uniqueness,)f(and)h(r)l(e)l(gularity)g(of)g(classic) l(al)h(solutions)f(to)f(the)h(Mul)t(lins-)199 3543 y(Sekerka)h(pr)l (oblem.)77 b Fq(Comm.)36 b(P)n(artial)26 b(Di\013.)38 b(Equ.)e Fr(21)p Fq(,no)27 b(11-12,)f(1705-1726)d(\(1996\))70 3714 y([9])41 b(E.)27 b(DiBenedetto,)i Fn(Partial)i(di\013er)l(ential)g (e)l(quations.)75 b Fq(Boston)27 b(Birk)-5 b(auser,\(1995\).)29 3886 y([10])40 b(J.)28 b(Esc)n(her,)f(G)h(Simonett,)g Fn(Classic)l(al)k(solutions)f(of)f(multidimensional)i(Hele-Shaw)f(mo)l (dels.)78 b Fq(SIAM)29 b(J.)e(Math.)199 3985 y(Anal.)37 b Fr(28)p Fq(,)27 b(1028{1047)d(\(1997\).)29 4157 y([11])40 b(G.)29 b(Giacomin,)g(J.)g(Leb)r(o)n(witz,)f Fn(Phase)33 b(se)l(gr)l(e)l(gation)e(dynamics)h(in)f(p)l(article)h(systems)f(with)g (long)g(r)l(ange)g(inter)l(ac-)199 4256 y(tions)f(I:)g(macr)l(osc)l (opic)i(limits)c Fq(J.)g(Stat.)37 b(Ph)n(ys.)f Fr(87)p Fq(,)27 b(37{61)e(\(1997\).)29 4428 y([12])40 b(G.)29 b(Giacomin,)g(J.)g(Leb)r(o)n(witz,)f Fn(Phase)33 b(se)l(gr)l(e)l (gation)e(dynamics)h(in)f(p)l(article)h(systems)f(with)g(long)g(r)l (ange)g(inter)l(ac-)199 4527 y(tions)f(II:)g(interfac)l(e)h(motions)d Fq(SIAM)g(J.)g(Appl.)37 b(Math.)p Fr(58)p Fq(,)27 b(No)h(6,)f (1707{1729)c(\(1998\).)29 4698 y([13])40 b(R.L.)33 b(P)n(ego,)f Fn(F)-6 b(r)l(ont)34 b(migr)l(ation)h(in)g(the)f(nonline)l(ar)h (Cahn-Hil)t(liar)l(d)i(e)l(quation.)53 b Fq(Pro)r(c.)e(R.)33 b(So)r(c.)52 b(Lond.A)33 b Fr(422)p Fq(,)199 4798 y(261{278)25 b(\(1989\).)29 4969 y([14])40 b(P)-7 b(.)34 b(Fife)g(and)f(J.B.)h (McLeo)r(d,)h Fn(The)h(appr)l(o)l(ach)i(of)e(solutions)f(of)h(nonline)l (ar)g(di\013usion)g(e)l(quations)g(to)f(tr)l(avel)t(ling)199 5069 y(fr)l(ont)30 b(solutions.)37 b Fq(Arc)n(h.)g(Rat.)g(Mec)n(h.)f (Anal.)h Fr(65)p Fq(,)28 b(335{361,)c(\(1977\))29 5240 y([15])40 b(B.)d(Stoth,)j Fn(Conver)l(genc)l(e)g(of)f(the)g(Cahn-Hil)t (liar)l(d)i(e)l(quation)e(to)f(the)h(Mul)t(lins-Sekerka)h(pr)l(oblems)f (in)g(sp)l(eric)l(al)199 5340 y(symmetry.)f Fq(J.)27 b(Di\013.)38 b(Eqns.)e Fr(125)p Fq(,)27 b(154{183)d(\(1996\).)0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)29 b(14:29)1377 b Fq(44)p eop %%Page: 45 45 45 44 bop 29 83 a Fq([16])40 b(S.)29 b(Y)-7 b(u,)29 b Fn(Hydr)l(o)l(dynamic)j(limits)e(with)h(sho)l(ck)g(waves)h(of)f(the)f (Boltzmann)h(e)l(quation)d Fq(to)g(app)r(ear)f(in)i(Comm.)38 b(Pure)199 183 y(and)28 b(Applied)g(Math.,)g(2004)0 5539 y Fh(10)p Fg(=j)r(une=)p Fh(2004;)h(14:29)1377 b Fq(45)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF 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