Content-Type: multipart/mixed; boundary="-------------0603130738169" This is a multi-part message in MIME format. ---------------0603130738169 Content-Type: text/plain; name="06-64.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-64.keywords" Plane-like interfaces, density estimates. ---------------0603130738169 Content-Type: application/postscript; name="mattenr060313.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="mattenr060313.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: mattenr060313.dvi %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips mattenr060313 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2006.03.13:1432 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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Fp(n)2547 2856 y Fs(.)40 b(W)-8 b(e)29 b(pro)m(v)m(e)g(the)g(follo)m(wing)e(densit)m(y)168 2964 y(estimates)k(for)f(lo)s(cal)g(minimizers:)168 3092 y Fw(Theorem)37 b(1.)44 b Fi(Fix)36 b Fo(\016)e(>)d Fs(0)p Fi(.)51 b(L)-5 b(et)36 b Fo(u)f Fi(b)-5 b(e)36 b(a)g(lo)-5 b(c)g(al)37 b(minimizer)f(in)g(a)g(domain)h Fs(\012)p Fi(,)f(with)g Fr(j)p Fo(u)p Fr(j)31 b(\024)g Fs(3)p Fo(=)p Fs(2)p Fi(.)52 b(Then,)168 3200 y(ther)-5 b(e)34 b(exist)g(p)-5 b(ositive)35 b(c)-5 b(onstants)35 b Fo(c)f Fi(and)h Fo(r)1629 3214 y Fn(0)1668 3200 y Fi(,)f(dep)-5 b(ending)35 b(only)g(on)f Fo(\016)j Fi(and)e(on)f(the)g(structur)-5 b(al)35 b(c)-5 b(onstants,)168 3308 y(in)32 b(such)h(a)g(way)h(that)1600 3417 y Fo(E)1667 3436 y Fp(B)1720 3444 y Fj(r)1755 3436 y Fn(\()p Fp(\030)s Fn(\))1847 3417 y Fs(\()p Fo(u)p Fs(\))41 b Fr(\024)g Fo(c)15 b(r)2220 3380 y Fp(n)p Fl(\000)p Fn(1)2372 3417 y Fo(;)168 3550 y Fi(for)33 b(any)g Fo(r)28 b Fr(\025)d Fo(r)693 3564 y Fn(0)732 3550 y Fi(,)33 b(pr)-5 b(ovide)g(d)35 b(that)f Fo(B)1407 3565 y Fp(r)r Fn(+)p Fp(\016)1533 3550 y Fs(\()p Fo(\030)t Fs(\))26 b Fr(\022)f Fs(\012)p Fi(.)168 3678 y Fw(Theorem)39 b(2.)45 b Fi(Fix)37 b Fo(\016)g(>)c Fs(0)p Fi(.)55 b(L)-5 b(et)37 b Fo(u)g Fi(b)-5 b(e)37 b(a)h(lo)-5 b(c)g(al)38 b(minimizer)g(in)f(a)g(domain)i Fs(\012)p Fi(,)e(with)h Fr(j)p Fo(u)p Fr(j)c(\024)f Fs(1)24 b(+)f Fo(\021)3627 3645 y Fl(0)3651 3678 y Fi(,)37 b(for)168 3786 y(some)c Fo(\021)448 3753 y Fl(0)497 3786 y Fr(\025)25 b Fs(0)p Fi(.)42 b(Then,)33 b(for)g(any)g Fo(\022)1335 3800 y Fn(0)1400 3786 y Fr(2)25 b Fs(\(0)p Fo(;)15 b Fs(1\))p Fi(,)34 b(for)f(any)g Fo(\022)28 b Fr(2)d Fs([)p Fr(\000)p Fo(\022)2363 3800 y Fn(0)2402 3786 y Fo(;)15 b(\022)2485 3800 y Fn(0)2524 3786 y Fs(])33 b Fi(and)h(for)f(any)g Fo(\026)3133 3800 y Fn(0)3198 3786 y Fo(>)25 b Fs(0)p Fi(,)32 b(if)168 3961 y Fs(\(4\))1156 b Fr(L)1502 3860 y Fm(\020)1556 3961 y Fo(B)1625 3975 y Fp(K)1693 3961 y Fs(\()p Fo(\030)t Fs(\))21 b Fr(\\)f(f)p Fo(u)25 b Fr(\025)g Fo(\022)s Fr(g)2218 3860 y Fm(\021)2313 3961 y Fr(\025)40 b Fo(\026)2479 3975 y Fn(0)2533 3961 y Fo(;)168 4134 y Fi(then)25 b(ther)-5 b(e)25 b(exist)f(p)-5 b(ositive)25 b(c)-5 b(onstants)26 b Fm(b)-50 b Fo(c)p Fi(,)26 b Fo(c)1626 4101 y Fp(?)1690 4134 y Fi(and)f Fo(r)1899 4148 y Fn(0)1938 4134 y Fi(,)h(dep)-5 b(ending)26 b(on)e Fo(K)7 b Fi(,)26 b Fo(\016)s Fi(,)g Fo(\026)2818 4148 y Fn(0)2858 4134 y Fi(,)f Fo(\022)2954 4148 y Fn(0)3018 4134 y Fi(and)g(on)g(the)f (structur)-5 b(al)168 4242 y(c)g(onstants,)34 b(such)f(that)1431 4374 y Fr(L)1494 4273 y Fm(\020)1548 4374 y Fo(B)1617 4388 y Fp(r)1655 4374 y Fs(\()p Fo(\030)t Fs(\))21 b Fr(\\)f(f)p Fo(u)26 b Fr(\025)f Fo(\022)s Fr(g)2181 4273 y Fm(\021)2260 4374 y Fr(\025)g Fo(c)2395 4337 y Fp(?)2435 4374 y Fo(r)2479 4337 y Fp(n)2540 4374 y Fo(;)168 4532 y Fi(for)31 b(any)g Fo(r)d Fr(2)c Fs([)p Fo(r)703 4546 y Fn(0)743 4532 y Fo(;)15 b Fm(b)-51 b Fo(c)q(=\021)s Fs(])p Fi(,)32 b(pr)-5 b(ovide)g(d)32 b(that)g Fo(\021)i Fi(and)d Fo(\021)1842 4499 y Fl(0)1896 4532 y Fi(ar)-5 b(e)32 b(suitably)f(smal)5 b(l)31 b(\(dep)-5 b(ending)32 b(on)f Fo(n)p Fi(,)f Fo(\026)3372 4546 y Fn(0)3411 4532 y Fi(,)h Fo(\022)3513 4546 y Fn(0)3552 4532 y Fi(,)g Fo(\016)j Fi(and)168 4640 y(the)f(structur)-5 b(al)34 b(c)-5 b(onstants)35 b(of)d Fo(F)13 b Fi(\))33 b(and)h(that)g Fo(B)1809 4655 y Fp(r)r Fn(+)p Fp(\016)1935 4640 y Fs(\()p Fo(x)p Fs(\))26 b Fr(\022)f Fs(\012)p Fi(.)168 4748 y(A)n(nalo)-5 b(gously,)34 b(if)168 4918 y Fs(\(5\))1156 b Fr(L)1502 4817 y Fm(\020)1556 4918 y Fo(B)1625 4932 y Fp(K)1693 4918 y Fs(\()p Fo(\030)t Fs(\))21 b Fr(\\)f(f)p Fo(u)25 b Fr(\024)g Fo(\022)s Fr(g)2218 4817 y Fm(\021)2313 4918 y Fr(\025)40 b Fo(\026)2479 4932 y Fn(0)2533 4918 y Fo(;)168 5089 y Fi(then)1431 5203 y Fr(L)1494 5102 y Fm(\020)1548 5203 y Fo(B)1617 5217 y Fp(r)1655 5203 y Fs(\()p Fo(\030)t Fs(\))21 b Fr(\\)f(f)p Fo(u)26 b Fr(\024)f Fo(\022)s Fr(g)2181 5102 y Fm(\021)2260 5203 y Fr(\025)g Fo(c)2395 5166 y Fp(?)2435 5203 y Fo(r)2479 5166 y Fp(n)2540 5203 y Fo(;)168 5362 y Fi(for)31 b(any)g Fo(r)d Fr(2)c Fs([)p Fo(r)703 5376 y Fn(0)743 5362 y Fo(;)15 b Fm(b)-51 b Fo(c)q(=\021)s Fs(])p Fi(,)32 b(pr)-5 b(ovide)g(d)32 b(that)g Fo(\021)i Fi(and)d Fo(\021)1842 5329 y Fl(0)1896 5362 y Fi(ar)-5 b(e)32 b(suitably)f(smal)5 b(l)31 b(\(dep)-5 b(ending)32 b(on)f Fo(n)p Fi(,)f Fo(\026)3372 5376 y Fn(0)3411 5362 y Fi(,)h Fo(\022)3513 5376 y Fn(0)3552 5362 y Fi(,)g Fo(\016)j Fi(and)168 5469 y(the)f(structur)-5 b(al)34 b(c)-5 b(onstants)35 b(of)d Fo(F)13 b Fi(\))33 b(and)h(that)g Fo(B)1809 5484 y Fp(r)r Fn(+)p Fp(\016)1935 5469 y Fs(\()p Fo(\030)t Fs(\))26 b Fr(\022)f Fs(\012)p Fi(.)p eop %%Page: 3 3 3 2 bop 587 251 a Fn(THE)30 b(GEOMETR)-6 b(Y)31 b(OF)f(MESOSCOPIC)h (PHASE)f(TRANSITION)h(INTERF)-8 b(A)n(CES)552 b(3)0 450 y Fs(The)30 b(original)g(idea)g(of)h(suc)m(h)g(densit)m(y)f(estimates)h (go)s(es)h(bac)m(k)g(to)f([CC95)q(].)42 b(An)31 b(analogue)g(of)g (Theorem)g(1)0 558 y(when)23 b Fo(H)32 b Fs(=)25 b(0)f(pla)m(ys)f(also) h(an)g(imp)s(ortan)m(t)f(r^)-45 b(ole)23 b(in)g([AA)m(C01)q(].)39 b(Related)24 b(tec)m(hniques)f(ha)m(v)m(e)i(b)s(een)e(exploited)0 666 y(in)32 b([V)-8 b(al04)r(],)35 b([PV05a)q(])f(and)f([PV05b)q(].)50 b(Analogous)34 b(densit)m(y)f(estimates)h(for)g(Cacciopp)s(oli)d(sets)j (are)g(also)0 774 y(crucial)23 b(in)g(the)h(study)f(of)i(minimal)c (surface)j(functionals)e(p)s(enalized)h(b)m(y)h(a)g(v)m(olume)g(term)g (\(see)h([CdlL01]\).)0 882 y(As)32 b(a)g(consequence)g(of)g(Theorems)f (1)h(and)f(2,)i(w)m(e)f(sho)m(w)f(that,)i(once)f(the)g(minimizer)d(is)h (con)m(trolled)i(at)g(a)0 990 y(giv)m(en)k(p)s(oin)m(t,)h(the)g(lev)m (els)e(sets)i(suitably)d(far)j(from)e Fr(\006)p Fs(1)i(o)s(ccup)m(y)f (a)h(\\small)e(p)s(ortion")g(of)i(the)f(space,)j(at)0 1098 y(a)32 b(suitably)e(large)i(scale.)46 b(This)30 b(will)g(also)h(allo)m(w)h(to)h(replace)e(the)i(measure)e(theoretic)i (assumptions)d(\(4\))0 1206 y(and)g(\(5\))h(b)m(y)f(p)s(oin)m(t)m(wise) g(assumptions,)e(that)j(are)g(often)g(easier)f(to)h(deal)f(with)f(in)g (applications.)0 1345 y Fw(Theorem)42 b(3.)k Fi(Fix)40 b Fo(\016)h(>)d Fs(0)h Fi(and)i Fo(\022)1249 1359 y Fn(0)1325 1345 y Fr(2)c Fs(\(0)p Fo(;)15 b Fs(1\))p Fi(.)64 b(L)-5 b(et)40 b Fo(u)g Fi(b)-5 b(e)39 b(a)h(lo)-5 b(c)g(al)41 b(minimizer)f(in)f(a)h(domain)h Fs(\012)p Fi(,)f(with)0 1453 y Fr(j)p Fo(u)p Fr(j)33 b(\024)f Fs(1)23 b(+)g Fo(\021)448 1420 y Fl(0)471 1453 y Fi(,)38 b(for)f(some)g Fo(\021)970 1420 y Fl(0)1026 1453 y Fr(\025)31 b Fs(0)p Fi(.)54 b(Supp)-5 b(ose)38 b(that)g Fr(j)p Fo(u)p Fs(\()p Fo(x)p Fs(\))p Fr(j)33 b(\024)f Fo(\022)2198 1467 y Fn(0)2273 1453 y Fi(for)37 b(some)h Fo(x)32 b Fr(2)f Fs(\012)p Fi(.)53 b(Then,)38 b(ther)-5 b(e)37 b(exist)0 1561 y(p)-5 b(ositive)37 b(c)-5 b(onstants)37 b Fo(c)p Fi(,)f Fm(b)-50 b Fo(c)p Fi(,)37 b(and)f Fo(r)1161 1575 y Fn(0)1201 1561 y Fi(,)g(p)-5 b(ossibly)37 b(dep)-5 b(ending)37 b(on)f Fo(\022)2202 1575 y Fn(0)2241 1561 y Fi(,)h Fo(\016)i Fi(and)e(on)f(the)g(structur) -5 b(al)37 b(c)-5 b(onstants,)0 1669 y(such)33 b(that)0 1844 y Fs(\(6\))507 b(min)789 1743 y Fm(n)849 1844 y Fr(L)912 1743 y Fm(\020)966 1844 y Fo(B)1035 1858 y Fp(r)1073 1844 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)p Fo(u)26 b(>)f(\022)1559 1858 y Fn(0)1598 1844 y Fr(g)1643 1743 y Fm(\021)1713 1844 y Fo(;)40 b Fr(L)1841 1743 y Fm(\020)1895 1844 y Fo(B)1964 1858 y Fp(r)2002 1844 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)p Fo(u)26 b(<)f Fr(\000)p Fo(\022)2559 1858 y Fn(0)2598 1844 y Fr(g)2643 1743 y Fm(\021o)2798 1844 y Fr(\025)40 b Fo(cr)2992 1806 y Fp(n)0 2037 y Fi(and)0 2212 y Fs(\(7\))1076 b Fr(L)1254 2111 y Fm(\020)1308 2212 y Fo(B)1377 2226 y Fp(r)1414 2212 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(fj)p Fo(u)p Fr(j)26 b Fo(<)f(\022)1950 2226 y Fn(0)1989 2212 y Fr(g)2034 2111 y Fm(\021)2114 2212 y Fr(\025)g Fo(cr)2293 2174 y Fp(n)p Fl(\000)p Fn(1)2446 2212 y Fo(;)0 2412 y Fi(for)31 b(any)g Fo(r)d Fr(2)d Fs([)p Fo(r)536 2426 y Fn(0)575 2412 y Fo(;)15 b Fm(b)-51 b Fo(c)q(=\021)s Fs(])p Fi(,)32 b(pr)-5 b(ovide)g(d)33 b(that)e Fo(\021)j Fi(and)e Fo(\021)1675 2379 y Fl(0)1729 2412 y Fi(ar)-5 b(e)31 b(suitably)g(smal)5 b(l)31 b(\(dep)-5 b(ending)32 b(on)f Fo(n)p Fi(,)g Fo(\026)3205 2426 y Fn(0)3244 2412 y Fi(,)f Fo(\022)3345 2426 y Fn(0)3385 2412 y Fi(,)g Fo(\016)k Fi(and)0 2520 y(the)f(structur)-5 b(al)34 b(c)-5 b(onstants)35 b(of)e Fo(F)13 b Fi(\))32 b(and)i(that)g Fo(B)1641 2535 y Fp(r)r Fn(+)p Fp(\016)1767 2520 y Fs(\()p Fo(x)p Fs(\))26 b Fr(\022)f Fs(\012)p Fi(.)0 2659 y Fs(W)-8 b(e)36 b(no)m(w)g(consider)e(the)h(problem)f(of)h (\014nding)e(minimizers)f(of)k(our)e(functional)g(in)g(a)h(p)s(erio)s (dic)e(setting,)0 2767 y(whose)e(lev)m(el)g(sets)g(lie)f(in)g(a)i (strip)d(of)j(univ)m(ersal)d(width)h(and)g(assigned)g(slop)s(e.)42 b(These)31 b(kind)f(of)h(problems)0 2875 y(are)44 b(related)g(with)f(a) h(PDE)g(v)m(ersion)g(of)g(Mather)g(theory)-8 b(,)48 b(as)d(recen)m(tly) f(dev)m(elop)s(ed)f(\(among)i(others\))0 2983 y(in)28 b([Mos86)q(],)i([Ban89)r(],)f([CdlL01],)h([V)-8 b(al04)q(])29 b(and)g([RS04].)41 b(In)28 b(this)g(framew)m(ork,)h(w)m(e)h(pro)m(v)m (e)g(the)f(follo)m(wing)0 3090 y(result:)0 3230 y Fw(Theorem)34 b(4.)42 b Fi(L)-5 b(et)33 b Fo(F)45 b Fi(satisfy)34 b(the)f (assumptions)h(on)g(p)-5 b(age)33 b(1)g(and)h(supp)-5 b(ose)34 b(also)g(that)0 3400 y Fs(\(8\))1273 b Fo(F)13 b Fs(\()p Fo(x)21 b Fs(+)f Fo(k)s(;)15 b(u)p Fs(\))41 b(=)f Fo(F)13 b Fs(\()p Fo(x;)i(u)p Fs(\))0 3570 y Fi(for)33 b(any)g Fo(x)26 b Fr(2)f Fq(R)542 3537 y Fp(n)595 3570 y Fi(,)33 b Fo(u)25 b Fr(2)g Fq(R)41 b Fi(and)33 b Fo(k)c Fr(2)c Fq(Z)1320 3537 y Fp(n)1362 3570 y Fi(,)33 b(that)0 3741 y Fs(\(9\))1171 b Fo(F)13 b Fs(\()p Fo(x;)i Fr(\000)p Fs(1)21 b(+)f Fo(s)p Fs(\))41 b(=)f Fo(F)13 b Fs(\()p Fo(x;)i Fs(1)21 b(+)f Fo(s)p Fs(\))0 3911 y Fi(for)33 b(any)g Fo(s)25 b Fr(2)g Fs([)p Fr(\000)p Fo(\016)609 3925 y Fn(0)649 3911 y Fo(;)15 b(\016)729 3925 y Fn(0)769 3911 y Fs(])33 b Fi(and)h(that)0 4081 y Fs(\(10\))829 b Fo(F)1047 4095 y Fp(u)1092 4081 y Fs(\()p Fo(x;)15 b Fr(\000)p Fs(1)21 b Fr(\000)f Fo(s)p Fs(\))25 b Fr(\024)g(\000)p Fo(c)73 b Fi(and)h 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b(b)s(egin)e(with)g(a)i (tec)m(hnical)f(observ)-5 b(ation:)168 3519 y Fw(Lemma)32 b(6.)42 b Fi(Fix)33 b Fo(\027)e Fr(2)25 b Fq(N)7 b Fi(.)47 b(L)-5 b(et)33 b Fo(a)1325 3534 y Fp(k)1393 3519 y Fr(\025)25 b Fs(0)33 b Fi(b)-5 b(e)33 b(a)g(se)-5 b(quenc)g(e)32 b(such)h(that)h Fo(a)2568 3533 y Fn(1)2632 3519 y Fr(\025)25 b Fo(c)2767 3533 y Fn(0)2807 3519 y Fi(,)33 b Fo(a)2916 3534 y Fp(k)2984 3519 y Fr(\024)24 b Fo(C)3144 3533 y Fn(0)3199 3519 y Fo(L)3261 3486 y Fp(\027)3304 3519 y Fo(k)3354 3486 y Fp(n)p Fl(\000)p Fn(1)3491 3519 y Fi(,)33 b(and)168 3723 y Fs(\(19\))711 3622 y Fm(\020)823 3637 y(X)780 3834 y Fn(1)p Fl(\024)p Fp(j)t Fl(\024)p Fp(k)1011 3723 y Fo(a)1059 3737 y Fp(j)1096 3622 y Fm(\021)1150 3645 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)1461 3723 y Fr(\024)40 b Fo(C)1637 3737 y Fn(0)1692 3622 y Fm(\020)1746 3723 y Fo(a)1794 3738 y Fp(k)r Fn(+1)1947 3723 y Fs(+)2080 3637 y Fm(X)2038 3834 y Fn(1)p Fl(\024)p Fp(j)t Fl(\024)p Fp(k)2269 3723 y Fo(e)2311 3685 y Fl(\000)p Fp(L)p Fn(\()p Fp(k)r Fn(+1)p Fl(\000)p Fp(j)t Fn(\))2689 3723 y Fo(a)2737 3737 y Fp(j)2794 3723 y Fs(+)22 b(^)-47 b Fo(cL)2986 3685 y Fp(\027)3029 3723 y Fo(k)3079 3685 y Fp(n)p Fl(\000)p Fn(1)3216 3622 y Fm(\021)168 3983 y Fi(for)34 b(any)g Fo(k)c Fr(2)c Fq(N)46 b Fi(and)35 b(some)f(p)-5 b(ositive)35 b(c)-5 b(onstants)37 b Fs(^)-47 b Fo(c)p Fi(,)34 b Fo(L)p Fi(,)f Fo(c)2155 3997 y Fn(0)2195 3983 y Fi(,)g(and)i Fo(C)2499 3997 y Fn(0)2538 3983 y Fi(.)45 b(Then,)34 b(if)f Fo(L)g Fi(is)g(suitably)i(lar)-5 b(ge)34 b(\(in)168 4091 y(dep)-5 b(endenc)g(e)32 b(of)f Fo(\027)6 b Fi(,)31 b Fo(n)p Fi(,)f Fo(c)1007 4105 y Fn(0)1078 4091 y Fi(and)h Fo(C)1317 4105 y Fn(0)1357 4091 y Fi(\))g(and)i Fs(^)-47 b Fo(c)32 b Fi(is)e(suitably)i(smal)5 b(l)32 b(\(in)e(dep)-5 b(endenc)g(e)33 b(of)e Fo(\027)6 b Fi(,)30 b Fo(n)p Fi(,)h Fo(c)3322 4105 y Fn(0)3392 4091 y Fo(C)3457 4105 y Fn(0)3527 4091 y Fi(and)h Fo(L)p Fi(\),)168 4199 y(ther)-5 b(e)33 b(exists)i Fs(\026)-47 b Fo(c)26 b(>)f Fs(0)p Fi(,)33 b(dep)-5 b(ending)33 b(on)h Fo(n)p Fi(,)d Fo(c)1616 4213 y Fn(0)1689 4199 y Fi(and)i Fo(C)1930 4213 y Fn(0)1970 4199 y Fi(,)f(such)h(that)1773 4357 y Fo(a)1821 4372 y Fp(k)1889 4357 y Fr(\025)27 b Fs(\026)-47 b Fo(c)16 b(k)2090 4319 y Fp(\027)t Fl(\000)p Fn(1)168 4515 y Fi(for)33 b(any)g Fo(k)c Fr(2)24 b Fq(N)7 b Fi(.)168 4689 y(Pr)-5 b(o)g(of.)43 b Fs(The)25 b(argumen)m(t)h(w)m(e)g(presen)m(t)g(here)f (is)g(a)h(mo)s(di\014cation)e(of)i(the)f(one)h(giv)m(en)g(on)f(page)i (10)f(of)g([CC95)q(].)168 4797 y(W)-8 b(e)31 b(de\014ne)168 4975 y(\(20\))1064 b(\026)-47 b Fo(c)41 b Fs(:=)g(min)1773 4874 y Fm(n)1834 4975 y Fo(c)1873 4989 y Fn(0)1928 4975 y Fo(;)2227 4914 y Fs(1)p 2003 4954 493 4 v 2003 5042 a(2)2048 5016 y Fp(n)2091 4997 y Fe(2)2130 5042 y Fo(C)2202 5011 y Fp(n)2195 5068 y Fn(0)2249 5042 y Fo(n)2304 5016 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))2505 4874 y Fm(o)2581 4975 y Fo(:)168 5190 y Fs(W)-8 b(e)31 b(also)g(supp)s(ose)e(that)h Fo(L)h Fs(is)e(so)i(large)f(that)168 5410 y(\(21\))1217 b Fo(L)1607 5373 y Fp(\027)1650 5410 y Fo(e)1692 5373 y Fl(\000)p Fp(L)1840 5410 y Fr(\024)2049 5349 y Fs(\026)-47 b Fo(c)2086 5316 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)p 1961 5390 482 4 v 1961 5477 a Fs(8)p Fo(C)2078 5445 y Fn(2)2071 5503 y(0)2117 5477 y Fo(n)2172 5450 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)p eop %%Page: 7 7 7 6 bop 587 251 a Fn(THE)30 b(GEOMETR)-6 b(Y)31 b(OF)f(MESOSCOPIC)h (PHASE)f(TRANSITION)h(INTERF)-8 b(A)n(CES)552 b(7)0 450 y Fs(and)0 623 y(\(22\))1505 b Fo(e)1707 586 y Fp(L)1800 623 y Fr(\025)40 b Fs(2)15 b Fo(:)0 796 y Fs(F)-8 b(urther,)30 b(w)m(e)h(assume)f(that)j(^)-47 b Fo(c)31 b Fs(is)e(so)i(small)e(that)0 1032 y(\(23\))1242 b(^)-47 b Fo(c)41 b Fr(\024)1753 970 y Fs(\026)-47 b Fo(c)1790 937 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)p 1601 1011 610 4 v 1601 1098 a Fs(4)p Fo(C)1711 1112 y Fn(0)1766 1098 y Fo(L)1828 1072 y Fp(\027)1886 1098 y Fo(n)1941 1072 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)2236 1032 y Fo(:)0 1253 y Fs(The)21 b(pro)s(of)g(is)f(b)m(y)h (induction.)36 b(If)21 b Fo(k)28 b Fs(=)d(1,)f(the)e(claim)e(is)h (true,)i(thanks)e(to)h(\(20\))r(.)37 b(Th)m(us,)23 b(w)m(e)f(no)m(w)f (tak)m(e)i Fo(k)29 b Fr(\025)c Fs(1,)0 1361 y(w)m(e)j(supp)s(ose)f(the) h(claim)f(to)i(hold)d(for)i(an)m(y)g Fo(j)j Fs(=)25 b(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(k)32 b Fs(and)27 b(w)m(e)i(pro)m(v)m(e)f(it)g (for)g Fo(a)2791 1376 y Fp(k)r Fn(+1)2924 1361 y Fs(.)39 b(T)-8 b(o)29 b(this)e(e\013ect,)j(w)m(e)0 1469 y(observ)m(e)h(that,)g (from)f(the)h(inductiv)m(e)e(h)m(yp)s(othesis,)799 1588 y Fm(\020)911 1602 y(X)868 1800 y Fn(1)p Fl(\024)p Fp(j)t Fl(\024)p Fp(k)1099 1689 y Fo(a)1147 1703 y Fp(j)1184 1588 y Fm(\021)1238 1610 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)1591 1689 y Fr(\025)85 b Fs(\026)-47 b Fo(c)1784 1651 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)2055 1588 y Fm(\020)2166 1602 y(X)2124 1800 y Fn(1)p Fl(\024)p Fp(j)t Fl(\024)p Fp(k)2355 1689 y Fo(j)2397 1651 y Fp(n)p Fl(\000)p Fn(1)2535 1588 y Fm(\021)2589 1610 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)1591 2001 y Fr(\025)85 b Fs(\026)-47 b Fo(c)1784 1963 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)2055 1900 y Fm(\020)2124 1877 y(Z)2215 1903 y Fp(k)2175 2083 y Fn(0)2273 2001 y Fo(t)2306 1963 y Fp(n)p Fl(\000)p Fn(1)2458 2001 y Fo(dt)2538 1900 y Fm(\021)2592 1923 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)1591 2263 y Fs(=)1765 2201 y(\026)g Fo(c)1802 2168 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)p 1755 2242 325 4 v 1755 2329 a Fo(n)1810 2303 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)2110 2263 y Fr(\001)20 b Fo(k)2205 2225 y Fp(n)p Fl(\000)p Fn(1)2358 2263 y Fo(:)0 2466 y Fs(On)30 b(the)g(other)h(hand,)859 2566 y Fm(X)817 2764 y Fn(1)p Fl(\024)p Fp(j)t Fl(\024)p Fp(k)1048 2652 y Fo(e)1090 2615 y Fl(\000)p Fp(L)p Fn(\()p Fp(k)r Fn(+1)p Fl(\000)p Fp(j)t Fn(\))1468 2652 y Fo(a)1516 2666 y Fp(j)1635 2652 y Fr(\024)83 b Fo(C)1854 2666 y Fn(0)1894 2652 y Fo(L)1956 2615 y Fp(\027)2056 2566 y Fm(X)2014 2764 y Fn(1)p Fl(\024)p Fp(j)t Fl(\024)p Fp(k)2245 2652 y Fo(e)2287 2615 y Fl(\000)p Fp(L)p Fn(\()p Fp(k)r Fn(+1)p Fl(\000)p Fp(j)t Fn(\))2665 2652 y Fo(j)2707 2615 y Fp(n)p Fl(\000)p Fn(1)1635 2916 y Fr(\024)g Fo(C)1854 2930 y Fn(0)1894 2916 y Fo(L)1956 2879 y Fp(\027)1999 2916 y Fo(k)2049 2879 y Fp(n)p Fl(\000)p Fn(1)2186 2916 y Fo(e)2228 2879 y Fl(\000)p Fp(L)2350 2830 y Fm(X)2359 3026 y Fp(i)p Fl(\025)p Fn(0)2497 2916 y Fo(e)2539 2879 y Fl(\000)p Fp(Li)1635 3164 y Fr(\024)g Fs(2)p Fo(C)1899 3178 y Fn(0)1939 3164 y Fo(L)2001 3127 y Fp(\027)2044 3164 y Fo(k)2094 3127 y Fp(n)p Fl(\000)p Fn(1)2231 3164 y Fo(e)2273 3127 y Fl(\000)p Fp(L)2396 3164 y Fo(;)0 3338 y Fs(due)30 b(to)h(\(22\))q(.)0 3445 y(By)g(collecting)f(the)g(ab)s(o)m(v)m(e)i(estimates,)f(w)m(e)g(th)m (us)f(deduce)g(from)g(\(19\))i(that)849 3686 y Fo(a)897 3701 y Fp(k)r Fn(+1)1113 3686 y Fr(\025)1266 3585 y Fm(\020)1400 3625 y Fs(\026)-47 b Fo(c)1437 3592 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)p 1331 3665 445 4 v 1331 3752 a Fo(C)1396 3766 y Fn(0)1450 3752 y Fo(n)1505 3726 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)1805 3686 y Fr(\000)20 b Fs(2)p Fo(C)2006 3700 y Fn(0)2046 3686 y Fo(L)2108 3649 y Fp(\027)2151 3686 y Fo(e)2193 3649 y Fl(\000)p Fp(L)2321 3686 y Fr(\000)h Fs(^)-47 b Fo(c)q(L)2513 3649 y Fp(\027)2556 3585 y Fm(\021)2625 3686 y Fo(k)2675 3649 y Fp(n)p Fl(\000)p Fn(1)1113 3945 y Fr(\025)1376 3884 y Fs(\026)g Fo(c)1413 3851 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)p 1276 3925 505 4 v 1276 4012 a Fs(2)15 b Fo(C)1401 4026 y Fn(0)1457 4012 y Fo(n)1512 3985 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)1811 3945 y Fr(\001)21 b Fo(k)1907 3908 y Fp(n)p Fl(\000)p Fn(1)0 4167 y Fs(due)30 b(to)h(\(21\))h(and)e(\(23\))q(.)0 4275 y(W)-8 b(e)32 b(also)e(notice)h(that)g(\(20\))g(and)f(the)h(fact)g(that)g Fo(k)d Fr(\025)d Fs(1)31 b(imply)d(that)1127 4496 y(\026)-47 b Fo(c)1164 4458 y Fn(1)p Fp(=n)1322 4496 y Fr(\024)1673 4434 y Fs(1)p 1443 4475 V 1443 4562 a(2)15 b Fo(C)1568 4576 y Fn(0)1623 4562 y Fo(n)1678 4536 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)1978 4496 y Fr(\001)2024 4395 y Fm(\020)2166 4434 y Fo(k)p 2088 4475 207 4 v 2088 4558 a(k)23 b Fs(+)d(1)2305 4395 y Fm(\021)2359 4418 y Fp(n)p Fl(\000)p Fn(1)2511 4496 y Fo(:)0 4725 y Fs(Then,)30 b(the)g(ab)s(o)m(v)m(e)i(inequalities)c(giv)m(e)i(that)h Fo(a)1586 4740 y Fp(k)r Fn(+1)1745 4725 y Fr(\025)26 b Fs(\026)-47 b Fo(c)q Fs(\()p Fo(k)24 b Fs(+)19 b(1\))2156 4692 y Fp(n)p Fl(\000)p Fn(1)2294 4725 y Fs(,)31 b(as)g(desired.)824 b Ff(\003)0 4919 y Fs(W)-8 b(e)25 b(no)m(w)f(deal)f(with)g(the)g(pro)s (of)g(of)h(the)g(\014rst)f(claim)g(in)g(Theorem)g(2,)j(the)e(second)g (claim)e(b)s(eing)h(analogous.)0 5027 y(F)-8 b(or)35 b(this,)f(w)m(e)g(b)s(orro)m(w)f(sev)m(eral)i(ideas)e(from)h([CC95])g (and)f([V)-8 b(al04)r(].)52 b(First,)34 b(w)m(e)h(observ)m(e)f(that,)i (with)c(no)0 5135 y(loss)37 b(of)g(generalit)m(y)-8 b(,)40 b(w)m(e)d(ma)m(y)h(assume)f Fo(\022)j Fs(to)e(b)s(e)e(as)i(close)f(to)h Fr(\000)p Fs(1)g(as)f(w)m(e)h(wish.)60 b(Indeed:)53 b(assume)37 b(the)0 5243 y(result)29 b(to)i(b)s(e)f(true)g(for)g Fo(\022)865 5210 y Fp(?)934 5243 y Fs(\(sa)m(y)-8 b(,)32 b(close)f(to)g Fr(\000)p Fs(1\),)g(and)f(let)g Fo(\022)e Fr(2)d Fs([)p Fr(\000)p Fo(\022)2284 5257 y Fn(0)2323 5243 y Fo(;)15 b(\022)2406 5257 y Fn(0)2445 5243 y Fs(],)31 b(with)e Fo(\022)2779 5210 y Fp(?)2843 5243 y Fr(\024)c(\000)p Fo(\022)3053 5257 y Fn(0)3092 5243 y Fs(.)41 b(Then,)903 5444 y Fo(\026)958 5458 y Fn(0)1023 5444 y Fr(\024)25 b(L)1182 5343 y Fm(\020)1236 5444 y Fr(f)p Fo(u)h Fr(\025)f Fo(\022)s Fr(g)19 b(\\)h Fo(B)1715 5458 y Fp(K)1784 5343 y Fm(\021)1863 5444 y Fr(\024)25 b(L)2022 5343 y Fm(\020)2076 5444 y Fr(f)p Fo(u)h Fr(\025)f Fo(\022)2341 5406 y Fp(?)2379 5444 y Fr(g)c(\\)f Fo(B)2595 5458 y Fp(K)2663 5343 y Fm(\021)2733 5444 y Fo(;)p eop %%Page: 8 8 8 7 bop 168 251 a Fn(8)1011 b(MA)-6 b(TTEO)23 b(NO)n(V)-8 b(A)n(GA)23 b(AND)g(ENRICO)h(V)-8 b(ALDINOCI)168 450 y Fs(therefore,)31 b(using)e(the)h(result)g(for)g Fo(\022)1408 417 y Fp(?)1477 450 y Fs(and)f(Theorem)h(1,)h(w)m(e)g(conclude)f(that) 820 660 y(const)15 b Fo(r)1086 622 y Fp(n)1216 660 y Fr(\024)83 b(L)1433 559 y Fm(\020)1487 660 y Fr(f)p Fo(u)25 b Fr(\025)g Fo(\022)1751 622 y Fp(?)1790 660 y Fr(g)c(\\)f Fo(B)2006 674 y Fp(r)2043 559 y Fm(\021)1216 859 y Fr(\024)83 b(L)1433 758 y Fm(\020)1487 859 y Fr(f)p Fo(u)25 b Fr(\025)g Fo(\022)s Fr(g)20 b(\\)g Fo(B)1966 873 y Fp(r)2004 758 y Fm(\021)2079 859 y Fs(+)f Fr(L)2232 758 y Fm(\020)2286 859 y Fr(f)p Fo(\022)2377 821 y Fp(?)2442 859 y Fr(\024)25 b Fo(u)g(<)g(\022)s Fr(g)20 b(\\)g Fo(B)2972 873 y Fp(r)3010 758 y Fm(\021)1216 1077 y Fr(\024)83 b(L)1433 976 y Fm(\020)1487 1077 y Fr(f)p Fo(u)25 b Fr(\025)g Fo(\022)s Fr(g)20 b(\\)g Fo(B)1966 1091 y Fp(r)2004 976 y Fm(\021)2079 1077 y Fs(+)2342 1015 y(1)p 2179 1056 372 4 v 2267 1139 a(inf)2179 1204 y Fp(u)p Fl(2)p Fn([)p Fp(\022)2322 1185 y Fj(?)2358 1204 y Fp(;\022)2411 1213 y Fe(0)2445 1204 y Fn(])2480 1139 y Fo(F)2576 953 y Fm(Z)2626 1159 y Fp(B)2679 1167 y Fj(r)2734 1077 y Fo(F)13 b Fs(\()p Fo(\030)t(;)i(u)p Fs(\))g Fo(d\030)1216 1370 y Fr(\024)83 b(L)1433 1269 y Fm(\020)1487 1370 y Fr(f)p Fo(u)25 b Fr(\025)g Fo(\022)s Fr(g)20 b(\\)g Fo(B)1966 1384 y Fp(r)2004 1269 y Fm(\021)2079 1370 y Fs(+)35 b(const)15 b Fo(E)2474 1384 y Fp(B)2527 1392 y Fj(r)2566 1370 y Fs(\()p Fo(u)p Fs(\))21 b(+)35 b(const)16 b Fo(\021)j(r)3146 1332 y Fp(n)1216 1569 y Fr(\024)83 b(L)1433 1468 y Fm(\020)1487 1569 y Fr(f)p Fo(u)25 b Fr(\025)g Fo(\022)s Fr(g)20 b(\\)g Fo(B)1966 1583 y Fp(r)2004 1468 y Fm(\021)2079 1569 y Fs(+)35 b(const)15 b(\()p Fo(r)2486 1531 y Fp(n)p Fl(\000)p Fn(1)2644 1569 y Fs(+)k Fo(\021)g(r)2842 1531 y Fp(n)2889 1569 y Fs(\))c Fo(;)168 1779 y Fs(whic)m(h)29 b(giv)m(es)i(that)1429 1951 y Fr(L)1492 1850 y Fm(\020)1546 1951 y Fr(f)p Fo(u)26 b Fr(\025)f Fo(\022)s Fr(g)20 b(\\)g Fo(B)2026 1965 y Fp(r)2064 1850 y Fm(\021)2143 1951 y Fr(\025)40 b Fs(const)16 b Fo(r)2521 1913 y Fp(n)168 2142 y Fs(for)33 b(large)i Fo(r)h Fs(and)d(small)g Fo(\021)s Fs(.)52 b(Th)m(us,)34 b(in)f(the)h(rest)g(of)g(the)g(pro)s(of,)h(w)m(e)f(ma)m(y)h(and)e(do)h (assume)g(that)h Fo(\022)h Fs(is)d(as)168 2249 y(close)d(to)i Fr(\000)p Fs(1)e(as)h(w)m(e)f(wish.)168 2437 y(In)25 b(what)i(follo)m(ws,)e Fo(A)i Fs(is)e(a)i(suitably)d(large)j(p)s (ositiv)m(e)e(parameter;)j(w)m(e)f(will)d(also)i(mak)m(e)h(use)f(of)h (t)m(w)m(o)g(further)168 2545 y(parameters)33 b(\002)f(and)g Fo(T)13 b Fs(:)45 b(w)m(e)33 b(will)d(\014x)i(\002)h(small)e(enough)h (and)h(then)f(c)m(ho)s(ose)i Fo(T)45 b Fs(so)33 b(that)g(\002)p Fo(T)45 b Fs(is)32 b(suitably)168 2653 y(large)e(\(p)s(ossibly)e(dep)s (ending)g(on)i Fo(\022)1381 2667 y Fn(0)1420 2653 y Fs(\).)41 b(W)-8 b(e)32 b(also)e(set)168 2846 y(\(24\))1653 2822 y(\026)1646 2846 y Fo(\022)43 b Fs(=)d Fo(\022)22 b Fr(\000)e Fo(C)2064 2860 y Fl(\003)2103 2846 y Fo(e)2145 2808 y Fl(\000)p Fn(\002)p Fp(T)2326 2846 y Fo(;)168 3033 y Fs(where)30 b Fo(C)496 3047 y Fl(\003)565 3033 y Fs(denotes)h(a)g (suitably)d(large)j(constan)m(t.)168 3144 y(Let)d Fo(k)g Fr(2)d Fq(N)7 b Fs(.)46 b(On)27 b(page)h(183)h(of)f([V)-8 b(al04)r(],)29 b(a)f(function)1997 3120 y(~)1996 3144 y Fo(h)e Fr(2)f Fo(C)2232 3111 y Fn(1)p Fp(;)p Fn(1)2325 3144 y Fs(\([0)p Fo(;)15 b Fs(\()p Fo(k)21 b Fs(+)15 b(1\))p Fo(T)e Fs(]\))28 b(w)m(as)g(constructed)g(so)g(that)168 3259 y Fr(\000)p Fs(1)d Fr(\024)406 3235 y Fs(~)405 3259 y Fo(h)h Fr(\024)f Fs(1,)681 3235 y(~)680 3259 y Fo(h)p Fs(\(\()p Fo(k)f Fs(+)c(1\))p Fo(T)13 b Fs(\))26 b(=)f(1,)1369 3235 y(~)1368 3259 y Fo(h)1420 3226 y Fl(0)1444 3259 y Fs(\(0\))h(=)f(0,)168 3457 y(\(25\))1408 3433 y(~)1407 3457 y Fo(h)q Fs(\()p Fo(\034)10 b Fs(\))21 b(+)f(1)41 b Fr(\024)f Fs(const)15 b Fo(e)2153 3420 y Fl(\000)p Fn(\002)p Fp(T)10 b Fn(\()p Fp(k)r Fn(+1)p Fl(\000)p Fp(j)t Fn(\))168 3644 y Fs(if)29 b Fo(\034)35 b Fr(2)25 b Fs([\()p Fo(j)i Fr(\000)20 b Fs(1\))p Fo(T)8 b(;)31 b(j)5 b(T)13 b Fs(],)31 b(for)f Fo(j)h Fs(=)25 b(1)p Fo(;)15 b(:)g(:)g(:)i(;)e(k)24 b Fs(+)c(1,)1435 3839 y Fr(j)1461 3815 y Fs(~)1460 3839 y Fo(h)1512 3802 y Fl(0)1536 3839 y Fs(\()p Fo(\034)10 b Fs(\))p Fr(j)26 b(\024)40 b Fs(const)16 b(\002)p Fo(\034)10 b Fs(\()2198 3815 y(~)2197 3839 y Fo(h)p Fs(\()p Fo(\034)g Fs(\))21 b(+)f(1\))168 4027 y(if)29 b Fo(\034)35 b Fr(2)25 b Fs([0)p Fo(;)15 b Fs(1],)1460 4184 y Fr(j)1486 4160 y Fs(~)1485 4184 y Fo(h)1537 4146 y Fl(0)1561 4184 y Fs(\()p Fo(\034)10 b Fs(\))p Fr(j)26 b(\024)40 b Fs(const)16 b(\002\()2173 4160 y(~)2172 4184 y Fo(h)p Fs(\()p Fo(\034)10 b Fs(\))21 b(+)f(1\))168 4352 y(if)29 b Fo(\034)35 b Fr(2)25 b Fs([1)p Fo(;)15 b Fs(\()p Fo(k)25 b Fs(+)20 b(1\))p Fo(T)13 b Fs(],)31 b(and)168 4547 y(\(26\))1122 b Fr(j)1476 4523 y Fs(~)1475 4547 y Fo(h)1527 4510 y Fl(00)1570 4547 y Fs(\()p Fo(\034)10 b Fs(\))p Fr(j)27 b(\024)40 b Fs(const)15 b(\002\()2182 4523 y(~)2181 4547 y Fo(h)q Fs(\()p Fo(\034)10 b Fs(\))21 b(+)f(1\))168 4734 y(if)29 b Fo(\034)35 b Fr(2)25 b Fs([0)p Fo(;)15 b Fs(\()p Fo(k)25 b Fs(+)20 b(1\))p Fo(T)13 b Fs(].)41 b(W)-8 b(e)32 b(then)e(de\014ne)748 4929 y Fo(h)p Fs(\()p Fo(x)p Fs(\))42 b(:=)e(\(1)21 b(+)f Fo(\021)1340 4892 y Fl(0)1364 4929 y Fs(\)\()1435 4905 y(~)1434 4929 y Fo(h)q Fs(\()p Fr(j)p Fo(x)p Fr(j)p Fs(\))h(+)f(1\))h Fr(\000)f Fs(1)15 b Fo(;)198 b(\033)s Fs(\()p Fo(x)p Fs(\))41 b(:=)g(min)n Fr(f)p Fo(u)p Fs(\()p Fo(x)p Fs(\))p Fo(;)32 b(h)p Fs(\()p Fo(x)p Fs(\))p Fr(g)748 5077 y Fs(and)182 b Fo(\014)5 b Fs(\()p Fo(x)p Fs(\))41 b(:=)f(min)o Fr(f)p Fo(u)p Fs(\()p Fo(x)p Fs(\))21 b Fr(\000)f Fo(\033)s Fs(\()p Fo(x)p Fs(\))p Fo(;)32 b Fs(1)20 b(+)2313 5053 y(\026)2305 5077 y Fo(\022)r Fr(g)15 b Fo(:)168 5271 y Fs(Since)29 b Fo(h)d Fr(\025)f Fs(1)20 b(+)g Fo(\021)783 5238 y Fl(0)832 5271 y Fr(\025)25 b Fo(u)30 b Fs(on)h Fo(@)5 b(B)1259 5289 y Fn(\()p Fp(k)r Fn(+1\))p Fp(T)1498 5271 y Fs(,)30 b(it)g(follo)m(ws)g(that)h Fo(\033)d Fs(=)d Fo(u)30 b Fs(on)h Fo(@)5 b(B)2647 5289 y Fn(\()p Fp(k)r Fn(+1\))p Fp(T)2916 5271 y Fs(and)30 b(so)168 5462 y(\(27\))1138 b Fo(E)1533 5476 y Fp(B)1586 5490 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1798 5462 y Fs(\()p Fo(u)p Fs(\))41 b Fr(\024)g Fo(E)2140 5476 y Fp(B)2193 5490 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)2405 5462 y Fs(\()p Fo(\033)s Fs(\))p eop %%Page: 9 9 9 8 bop 587 251 a Fn(THE)30 b(GEOMETR)-6 b(Y)31 b(OF)f(MESOSCOPIC)h (PHASE)f(TRANSITION)h(INTERF)-8 b(A)n(CES)552 b(9)0 450 y Fs(as)34 b(long)f(as)h Fo(B)499 469 y Fn(\()p Fp(k)r Fn(+1\))p Fp(T)768 450 y Fr(\032)c Fs(\012,)35 b(due)e(to)h(\(3\))q(.) 50 b(W)-8 b(e)35 b(use)f(the)f(Cauc)m(h)m(y)h(and)f(Sob)s(olev)g (Inequalities)f(and)h(\(27\))q(,)0 558 y(to)e(gather)g(that)102 696 y Fm(\020)171 674 y(Z)221 880 y Fp(B)274 894 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)502 797 y Fo(\014)592 727 y Fe(2)p Fj(n)p 568 739 117 3 v 568 781 a(n)p Fd(\000)p Fe(1)699 696 y Fm(\021)763 691 y Fj(n)p Fd(\000)p Fe(1)p 763 704 V 802 745 a Fj(n)977 797 y Fr(\024)98 b Fs(const)1384 674 y Fm(Z)1434 880 y Fp(B)1487 894 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1695 880 y Fl(\\f)p Fp(u)p Fl(\000)p Fp(\033)r Fl(\024)p Fn(1+)2066 862 y(\026)2060 880 y Fp(\022)s Fl(g)2150 797 y Fr(j)p Fo(\014)5 b Fr(j)15 b(jr)p Fo(\014)5 b Fr(j)977 1066 y(\024)98 b Fs(const)15 b Fo(A)1436 965 y Fm(\020)1506 942 y(Z)1557 1148 y Fp(B)1610 1162 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1817 1148 y Fl(\\f)p Fp(u>\033)r Fl(g)2078 1066 y Fs(\()p Fr(jr)p Fo(u)p Fr(j)2291 1028 y Fn(2)2351 1066 y Fr(\000)20 b(jr)p Fo(\033)s Fr(j)2623 1028 y Fn(2)2683 1066 y Fr(\000)g Fs(2)p Fr(r)p Fs(\()p Fo(u)h Fr(\000)f Fo(\033)s Fs(\))g Fr(\001)h(r)p Fo(\033)s Fs(\))3416 965 y Fm(\021)1131 1334 y Fs(+)1226 1273 y(const)p 1212 1313 238 4 v 1296 1397 a Fo(A)1474 1210 y Fm(Z)1525 1417 y Fp(B)1578 1431 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1786 1417 y Fl(\\f)p Fp(u)p Fl(\000)p Fp(\033)r Fl(\024)p Fn(1+)2157 1399 y(\026)2151 1417 y Fp(\022)r Fl(g)2226 1334 y Fs(\()p Fo(u)f Fr(\000)g Fo(\033)s Fs(\))2514 1297 y Fn(2)977 1603 y Fs(=)98 b(const)15 b Fo(A)1436 1502 y Fm(\020)1506 1479 y(Z)1557 1685 y Fp(B)1610 1699 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1821 1603 y Fs(\()p Fr(jr)p Fo(u)p Fr(j)2034 1565 y Fn(2)2095 1603 y Fr(\000)20 b(jr)p Fo(\033)s Fr(j)2367 1565 y Fn(2)2407 1603 y Fs(\))-2442 b(\(28\))1131 1871 y(+2)1262 1747 y Fm(Z)1312 1953 y Fp(B)1365 1967 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1573 1953 y Fl(\\f)p Fp(u>\033)r Fl(g)1833 1871 y Fs(\()p Fo(u)21 b Fr(\000)f Fo(\033)s Fs(\))15 b(\001)p Fo(\033)2268 1770 y Fm(\021)1131 2139 y Fs(+)1226 2078 y(const)p 1212 2118 V 1296 2202 a Fo(A)1474 2016 y Fm(Z)1525 2222 y Fp(B)1578 2236 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1786 2222 y Fl(\\f)p Fp(u)p Fl(\000)p Fp(\033)r Fl(\024)p Fn(1+)2157 2204 y(\026)2151 2222 y Fp(\022)r Fl(g)2226 2139 y Fs(\()p Fo(u)20 b Fr(\000)g Fo(\033)s Fs(\))2514 2102 y Fn(2)977 2408 y Fr(\024)98 b Fs(const)15 b Fo(A)1436 2307 y Fm(h)1495 2284 y(Z)1545 2490 y Fp(B)1598 2504 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1806 2490 y Fl(\\f)p Fp(u>\033)r Fl(g)2081 2307 y Fm(\020)2136 2408 y Fo(F)e Fs(\()p Fo(x;)i(\033)s Fs(\))21 b Fr(\000)f Fo(F)13 b Fs(\()p Fo(x;)i(u)p Fs(\))22 b(+)d Fo(H)7 b Fs(\()p Fo(x)p Fs(\)\()p Fo(\033)25 b Fr(\000)20 b Fo(u)p Fs(\))3428 2307 y Fm(\021)1131 2676 y Fs(+2)1262 2552 y Fm(Z)1312 2759 y Fp(B)1365 2773 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1577 2676 y Fs(\()p Fo(u)h Fr(\000)f Fo(\033)s Fs(\))15 b(\001)p Fo(\033)2012 2575 y Fm(i)2076 2676 y Fs(+)2192 2615 y(const)p 2176 2655 V 2261 2739 a Fo(A)2439 2552 y Fm(Z)2490 2759 y Fp(B)2543 2773 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)2751 2759 y Fl(\\f)p Fp(u)p Fl(\000)p Fp(\033)r Fl(\024)p Fn(1+)3122 2741 y(\026)3116 2759 y Fp(\022)s Fl(g)3191 2676 y Fs(\()p Fo(u)21 b Fr(\000)e Fo(\033)s Fs(\))3479 2639 y Fn(2)3535 2676 y Fo(:)0 2939 y Fs(W)-8 b(e)29 b(no)m(w)f(estimate)g(the)g(left)g (hand)f(side)f(of)35 b(\(28\))r(.)40 b(If)27 b(\002)p Fo(T)40 b Fs(is)27 b(large)h(enough)g(and)f Fo(\021)2846 2906 y Fl(0)2897 2939 y Fs(is)g(small)g(enough,)h(w)m(e)0 3047 y(see)j(from)f(\(25\))i(that)f Fo(\022)22 b Fr(\000)e Fo(h)26 b Fr(\025)f Fs(\(1)c Fr(\000)f Fo(\022)1316 3061 y Fn(0)1355 3047 y Fs(\))p Fo(=)p Fs(2)31 b(in)e Fo(B)1686 3062 y Fp(k)r(T)1780 3047 y Fs(.)40 b(Consequen)m(tly)-8 b(,)0 3277 y(\(29\))1085 b Fo(\014)31 b Fr(\025)1433 3216 y Fs(1)20 b Fr(\000)g Fo(\022)1632 3230 y Fn(0)p 1433 3256 239 4 v 1529 3340 a Fs(2)1712 3277 y(in)29 b Fo(B)1887 3292 y Fp(k)r(T)2001 3277 y Fr(\\)19 b(f)p Fo(u)26 b(>)f(\022)s Fr(g)p Fs(.)0 3486 y(Th)m(us,)30 b(giv)m(en)g Fo(\032)25 b Fr(\025)g Fs(0,)31 b(if)e(w)m(e)i(set)1281 3654 y Fo(V)20 b Fs(\()p Fo(\032)p Fs(\))42 b(:=)e Fr(L)1712 3553 y Fm(\020)1766 3654 y Fo(B)1835 3668 y Fp(\032)1895 3654 y Fr(\\)20 b(f)p Fo(u)26 b(>)f(\022)s Fr(g)2286 3553 y Fm(\021)2355 3654 y Fo(;)0 3846 y Fs(w)m(e)31 b(deduce)f(from)g(\(29\))i(that)f(the)f(left)g(hand)g(side)f(of)38 b(\(28\))31 b(is)f(bigger)g(than)1506 4049 y(const)16 b Fo(V)k Fs(\()p Fo(k)s(T)13 b Fs(\))1998 3984 y Fj(n)p Fd(\000)p Fe(1)p 1999 3997 117 3 v 2038 4038 a Fj(n)2145 4049 y Fo(:)0 4232 y Fs(Let)36 b(us)e(no)m(w)h(estimate)h(the)f(righ)m (t)g(hand)f(side)g(of)42 b(\(28\))q(.)55 b(T)-8 b(o)36 b(this)e(exten)m(t,)j(w)m(e)f(denote)g(the)f(righ)m(t)f(hand)0 4340 y(side)29 b(of)38 b(\(28\))32 b(b)m(y)1676 4485 y Fo(I)1716 4499 y Fn(1)1775 4485 y Fs(+)20 b Fo(I)1906 4499 y Fn(2)1961 4485 y Fo(;)0 4649 y Fs(with)445 4856 y Fo(I)485 4870 y Fn(1)608 4856 y Fs(:=)98 b(const)16 b Fo(A)1123 4733 y Fm(Z)1174 4939 y Fp(B)1227 4953 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1434 4939 y Fl(\\f)p Fp(u>\033)r Fl(g)1710 4756 y Fm(\020)1764 4856 y Fo(H)7 b Fs(\()p Fo(x)p Fs(\)\()p Fo(\033)24 b Fr(\000)c Fo(u)p Fs(\))2258 4756 y Fm(\021)2525 4856 y Fs(and)445 5125 y Fo(I)485 5139 y Fn(2)608 5125 y Fs(:=)98 b(const)16 b Fo(A)1093 5024 y Fm(h)1151 5001 y(Z)1201 5207 y Fp(B)1254 5221 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1462 5207 y Fl(\\f)p Fp(u>\033)r Fl(g)1738 5024 y Fm(\020)1792 5125 y Fo(F)d Fs(\()p Fo(x;)i(\033)s Fs(\))21 b Fr(\000)f Fo(F)13 b Fs(\()p Fo(x;)i(u)p Fs(\))2477 5024 y Fm(\021)787 5393 y Fs(+2)918 5270 y Fm(Z)969 5476 y Fp(B)1022 5490 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1234 5393 y Fs(\()p Fo(u)20 b Fr(\000)g Fo(\033)s Fs(\))15 b(\001)p Fo(\033)1668 5292 y Fm(i)1732 5393 y Fs(+)1848 5332 y(const)p 1833 5372 238 4 v 1918 5456 a Fo(A)2096 5270 y Fm(Z)2146 5476 y Fp(B)2199 5490 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)2407 5476 y Fl(\\f)p Fp(u)p Fl(\000)p Fp(\033)r Fl(\024)p Fn(1+)2778 5458 y(\026)2772 5476 y Fp(\022)s Fl(g)2847 5393 y Fs(\()p Fo(u)21 b Fr(\000)f Fo(\033)s Fs(\))3136 5356 y Fn(2)3191 5393 y Fo(:)p eop %%Page: 10 10 10 9 bop 168 251 a Fn(10)976 b(MA)-6 b(TTEO)23 b(NO)n(V)-8 b(A)n(GA)23 b(AND)g(ENRICO)h(V)-8 b(ALDINOCI)168 450 y Fs(First)29 b(of)i(all,)e(w)m(e)i(estimate)g Fo(I)1182 464 y Fn(1)1222 450 y Fs(.)41 b(T)-8 b(o)30 b(this)g(e\013ect,)i(w)m(e) f(recall)e(that)i Fo(r)d Fs(:=)d(\()p Fo(k)f Fs(+)c(1\))p Fo(T)39 b Fr(2)25 b Fs([)p Fo(r)3148 464 y Fn(0)3187 450 y Fo(;)15 b Fm(b)-51 b Fo(c)q(=\021)s Fs(])31 b(and)f(so)938 600 y Fo(I)978 614 y Fn(1)1100 600 y Fr(\024)98 b Fs(const)16 b Fo(\021)j Fr(L)p Fs(\()p Fo(B)1723 618 y Fn(\()p Fp(k)r Fn(+1\))p Fp(T)1961 600 y Fs(\))51 b Fr(\024)65 b Fs(const)16 b Fo(\021)j Fs(\()p Fo(k)24 b Fs(+)19 b(1\))2746 562 y Fp(n)2794 600 y Fo(T)2860 562 y Fp(n)1100 751 y Fr(\024)98 b Fs(const)16 b Fm(b)-51 b Fo(c)16 b Fs(\()p Fo(k)23 b Fs(+)d(1\))1823 713 y Fp(n)p Fl(\000)p Fn(1)1961 751 y Fo(T)2027 713 y Fp(n)p Fl(\000)p Fn(1)2215 751 y Fr(\024)65 b Fs(const)15 b Fm(b)-50 b Fo(c)15 b(k)2678 713 y Fp(n)p Fl(\000)p Fn(1)2815 751 y Fo(T)2881 713 y Fp(n)p Fl(\000)p Fn(1)3034 751 y Fo(:)168 900 y Fs(W)-8 b(e)27 b(no)m(w)g(estimate)g Fo(I)906 914 y Fn(2)946 900 y Fs(.)39 b(F)-8 b(or)27 b(this)e(scop)s(e,)j(w)m(e)f(\014rst)f(consider)f(the)i(con)m (tribution)e(of)h Fo(I)3080 914 y Fn(2)3146 900 y Fs(in)f Fr(f)p Fo(u)h Fr(\024)f Fo(\022)s Fr(g)p Fs(.)39 b(Since)168 1008 y Fo(h)25 b Fr(\025)g(\000)p Fs(1,)31 b(w)m(e)g(ha)m(v)m(e)g(that) g Fr(\000)p Fs(1)26 b Fr(\024)f Fo(h)g Fs(=)g Fo(\033)k Fr(\024)c Fo(u)30 b Fs(at)h(an)m(y)g(p)s(oin)m(t)e(of)i Fr(f)p Fo(u)26 b(>)f(\033)s Fr(g)p Fs(,)31 b(and)f(so)1488 1201 y(\()p Fo(u)21 b Fs(+)f(1\))1767 1164 y Fn(2)1827 1201 y Fr(\000)g Fs(\()p Fo(\033)k Fs(+)c(1\))2200 1164 y Fn(2)2260 1201 y Fr(\000)2361 1140 y Fs(1)p 2361 1180 46 4 v 2361 1263 a(2)2417 1201 y(\()p Fo(u)g Fr(\000)g Fo(\033)s Fs(\))2705 1164 y Fn(2)1335 1431 y Fs(=)82 b(\()p Fo(u)21 b Fr(\000)f Fo(\033)s Fs(\))1808 1302 y Fm(\022)1885 1369 y Fs(1)p 1885 1410 V 1885 1493 a(2)1940 1431 y Fo(u)g Fs(+)2113 1369 y(3)p 2113 1410 V 2113 1493 a(2)2169 1431 y Fo(\033)j Fs(+)d(2)2380 1302 y Fm(\023)2488 1431 y Fr(\025)40 b Fs(0)168 1635 y(in)29 b Fr(f)p Fo(u)d(>)e(\033)s Fr(g)p Fs(.)42 b(Accordingly)-8 b(,)30 b(in)f Fr(f)p Fo(\033)g(<)c(u)g Fr(\024)g Fo(\022)s Fr(g)p Fs(,)999 1834 y Fo(F)13 b Fs(\()p Fo(x;)i(u)p Fs(\))21 b Fr(\000)f Fo(F)13 b Fs(\()p Fo(x;)i(\033)s Fs(\))85 b(=)1922 1710 y Fm(Z)2013 1737 y Fp(u)1973 1916 y(\033)2073 1834 y Fo(F)2131 1848 y Fp(u)2177 1834 y Fs(\()p Fo(x;)15 b(\020)7 b Fs(\))15 b Fo(d\020)1769 2078 y Fr(\025)98 b Fs(const)2175 1954 y Fm(Z)2266 1980 y Fp(u)2226 2160 y(\033)2311 2078 y Fs(\()p Fo(\020)27 b Fs(+)20 b(1\))15 b Fo(d\020)1769 2300 y Fs(=)98 b(const)2160 2199 y Fm(h)2203 2300 y Fs(\()p Fo(u)21 b Fs(+)f(1\))2482 2262 y Fn(2)2542 2300 y Fr(\000)g Fs(\()p Fo(\033)k Fs(+)c(1\))2915 2262 y Fn(2)2955 2199 y Fm(i)1769 2475 y Fr(\025)98 b Fs(const)15 b(\()p Fo(u)21 b Fr(\000)f Fo(\033)s Fs(\))2449 2437 y Fn(2)2504 2475 y Fo(:)168 2624 y Fs(The)30 b(latter)g(estimate)h(and)f(\(26\))i(imply) c(that)j(the)g(con)m(tribution)e(of)h Fo(I)2605 2638 y Fn(2)2675 2624 y Fs(in)f Fr(f)p Fo(u)d Fr(\024)f Fo(\022)s Fr(g)30 b Fs(is)f(con)m(trolled)h(b)m(y)168 2821 y(\(30\))709 2697 y Fm(Z)760 2903 y Fp(B)813 2917 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)1021 2903 y Fl(\\f)p Fp(\033)r()g(\022)s Fr(g)p Fs(.)51 b(The)33 b(con)m(tribution)g(in)f Fo(B)3230 3844 y Fn(\()p Fp(k)r Fn(+1\))p Fp(T)3492 3826 y Fr(n)23 b Fo(B)3629 3841 y Fp(k)r(T)3756 3826 y Fs(of)168 3934 y(suc)m(h)30 b(term)g(is)g(b)s(ounded)e(b)m(y)592 4002 y Fm(Z)642 4208 y Fn(\()p Fp(B)722 4222 y Fe(\()p Fj(k)q Fe(+1\))p Fj(T)931 4208 y Fl(n)p Fp(B)1019 4220 y Fj(k)q(T)1102 4208 y Fn(\))p Fl(\\f)p Fp(u>\022)r Fl(g)1397 4024 y Fm(\020)1451 4125 y Fr(j)p Fo(F)13 b Fs(\()p Fo(x;)i(\033)s Fs(\))22 b Fr(\000)e Fo(F)13 b Fs(\()p Fo(x;)i(u)p Fs(\))p Fr(j)21 b Fs(+)f(\()p Fo(\033)k Fs(+)c(1\)\()p Fo(u)h Fr(\000)f Fo(\033)s Fs(\))h(+)f(\()p Fo(u)g Fr(\000)g Fo(\033)s Fs(\))3270 4088 y Fn(2)3310 4024 y Fm(\021)3380 4125 y Fo(;)168 4353 y Fs(thanks)30 b(to)h(\(26\))q(.)41 b(The)30 b(ab)s(o)m(v)m(e)i(quan)m(tit)m(y)e(is)g(then)g(b)s(ounded)e (b)m(y)1513 4530 y Fr(L)1576 4429 y Fm(\020)1630 4530 y Fr(f)p Fo(u)d(>)g(\022)s Fr(g)20 b(\\)g Fs(\()p Fo(B)2144 4549 y Fn(\()p Fp(k)r Fn(+1\))p Fp(T)2403 4530 y Fr(n)h Fo(B)2538 4545 y Fp(k)r(T)2631 4530 y Fs(\))2666 4429 y Fm(\021)1359 4693 y Fs(=)83 b Fo(V)20 b Fs(\(\()p Fo(k)k Fs(+)c(1\))p Fo(T)13 b Fs(\))21 b Fr(\000)f Fo(V)g Fs(\()p Fo(k)s(T)13 b Fs(\))i Fo(:)168 4843 y Fs(Let)31 b(us)e(no)m(w)i(lo)s (ok)f(at)h(the)f(con)m(tribution)f(of)i Fo(I)1763 4857 y Fn(2)1832 4843 y Fs(in)f Fr(f)p Fo(u)25 b(>)g(\022)s Fr(g)20 b(\\)g Fo(B)2418 4858 y Fp(k)r(T)2511 4843 y Fs(.)41 b(W)-8 b(e)32 b(observ)m(e)f(that)1007 4992 y Fo(B)1076 5007 y Fp(k)r(T)1190 4992 y Fr(\\)19 b(f)p Fo(\033)29 b(<)c(u)h Fr(\024)f Fo(\033)e Fs(+)d(1)g(+)1996 4968 y(\026)1988 4992 y Fo(\022)s Fr(g)25 b(\022)g Fo(B)2269 5007 y Fp(k)r(T)2383 4992 y Fr(\\)20 b(f)p Fo(\033)29 b(<)c(u)g Fr(\024)g Fo(\022)s Fr(g)15 b Fo(;)168 5142 y Fs(due)29 b(to)j(\(25\))q(,)f(pro)m(vided)e(that)i Fo(C)1304 5156 y Fl(\003)1373 5142 y Fs(in)e(\(24\))j(is)e(large)g (enough.)168 5250 y(Consequen)m(tly)-8 b(,)1298 5280 y Fm(Z)1349 5486 y Fp(B)1402 5498 y Fj(k)q(T)1485 5486 y Fl(\\f)p Fp(u)p Fl(\000)p Fp(\033)r Fl(\024)p Fn(1+)1856 5469 y(\026)1850 5486 y Fp(\022)r Fl(g\\f)p Fp(u>\022)r Fl(g)2173 5404 y Fs(\()p Fo(u)21 b Fr(\000)f Fo(\033)s Fs(\))2462 5366 y Fn(2)2542 5404 y Fs(=)40 b(0)p eop %%Page: 11 11 11 10 bop 587 251 a Fn(THE)30 b(GEOMETR)-6 b(Y)31 b(OF)f(MESOSCOPIC)h (PHASE)f(TRANSITION)h(INTERF)-8 b(A)n(CES)517 b(11)0 450 y Fs(and)30 b(so)g(the)h(con)m(tribution)e(of)h Fo(I)1106 464 y Fn(2)1176 450 y Fs(in)f Fr(f)p Fo(u)d(>)f(\022)s Fr(g)20 b(\\)g Fo(B)1762 465 y Fp(k)r(T)1885 450 y Fs(is)30 b(con)m(trolled)g(b)m(y)1119 569 y Fm(Z)1170 775 y Fp(B)1223 787 y Fj(k)q(T)1305 775 y Fl(\\f)p Fp(u>\022)r Fl(g)1573 591 y Fm(\020)1627 692 y Fo(F)13 b Fs(\()p Fo(x;)i(\033)s Fs(\))22 b Fr(\000)e Fo(F)13 b Fs(\()p Fo(x;)i(u)p Fs(\))21 b(+)f Fr(j)p Fs(\001)p Fo(h)p Fr(j)2603 591 y Fm(\021)2683 692 y Fr(\024)965 996 y(\024)1165 882 y Fp(k)1119 909 y Fm(X)1123 1105 y Fp(j)t Fn(=1)1266 872 y Fm(Z)1316 1078 y Fp(B)1369 1089 y Fj(j)s(T)1446 1078 y Fl(n)p Fp(B)1534 1092 y Fe(\()p Fj(j)s Fd(\000)p Fe(1\))p Fj(T)1738 1078 y Fl(\\f)p Fp(u>\022)r Fl(g)2006 895 y Fm(\020)2060 996 y Fo(F)13 b Fs(\()p Fo(x;)i(h)p Fs(\))22 b(+)e Fr(j)p Fs(\001)p Fo(h)p Fr(j)2636 895 y Fm(\021)2706 996 y Fo(:)-2731 b Fs(\(31\))0 1286 y(By)31 b(our)f(assumption)e(on)j Fo(F)13 b Fs(,)30 b(w)m(e)h(ha)m(v)m(e)h(that)1358 1481 y Fo(F)13 b Fs(\()p Fo(x;)i Fr(\000)p Fs(1)21 b(+)f Fo(s)p Fs(\))25 b Fr(\024)40 b Fs(const)16 b Fo(s)f(;)0 1676 y Fs(pro)m(vided)29 b(that)i Fo(s)25 b(>)g Fs(0)31 b(is)e(small)g (enough.)40 b(Th)m(us,)30 b(w)m(e)h(b)s(ound)d(the)j(ab)s(o)m(v)m(e)g (term)g(in)e(\(31\))j(b)m(y)1080 1850 y Fp(k)1034 1877 y Fm(X)1038 2073 y Fp(j)t Fn(=1)1180 1964 y Fo(e)1222 1926 y Fl(\000)p Fn(\002)p Fp(T)10 b Fn(\()p Fp(k)r Fn(+1)p Fl(\000)p Fp(j)t Fn(\))1658 1863 y Fm(h)1701 1964 y Fo(V)20 b Fs(\()p Fo(j)5 b(T)13 b Fs(\))22 b Fr(\000)e Fo(V)g Fs(\(\()p Fo(j)26 b Fr(\000)20 b Fs(1\))p Fo(T)13 b Fs(\))2543 1863 y Fm(i)2602 1964 y Fo(;)0 2259 y Fs(thanks)40 b(to)h(\(25\))r(.)71 b(Th)m(us,)43 b(the)d(quan)m(tit)m(y)h(ab)s(o)m(v)m(e)h(pro)m(vides)d (a)i(b)s(ound)e(for)h(the)h(con)m(tribution)e(of)i Fo(I)3506 2273 y Fn(2)3586 2259 y Fs(in)0 2367 y Fr(f)p Fo(u)26 b(>)f(\022)s Fr(g)20 b(\\)f Fo(B)479 2382 y Fp(k)r(T)573 2367 y Fs(.)0 2475 y(By)31 b(collecting)f(all)f(theses)i(estimates,)g (w)m(e)g(get)g(that)688 2686 y(const)16 b(\()p Fo(V)k Fs(\()p Fo(k)s(T)13 b Fs(\)\))1250 2620 y Fj(n)p Fd(\000)p Fe(1)p 1251 2633 117 3 v 1290 2674 a Fj(n)519 2913 y Fr(\024)83 b Fo(V)20 b Fs(\(\()p Fo(k)k Fs(+)c(1\))p Fo(T)13 b Fs(\))21 b Fr(\000)f Fo(V)g Fs(\()p Fo(k)s(T)13 b Fs(\))36 b(+)1719 2799 y Fp(k)1672 2826 y Fm(X)1677 3022 y Fp(j)t Fn(=1)1819 2913 y Fo(e)1861 2875 y Fl(\000)p Fn(\002)p Fp(T)10 b Fn(\()p Fp(k)r Fn(+1)p Fl(\000)p Fp(j)t Fn(\))2297 2812 y Fm(h)2340 2913 y Fo(V)20 b Fs(\()p Fo(j)5 b(T)13 b Fs(\))21 b Fr(\000)f Fo(V)g Fs(\(\()p Fo(j)27 b Fr(\000)20 b Fs(1\))p Fo(T)13 b Fs(\))3182 2812 y Fm(i)519 3159 y Fs(+)82 b Fm(b)-50 b Fo(c)15 b(k)777 3122 y Fp(n)p Fl(\000)p Fn(1)914 3159 y Fo(T)980 3122 y Fp(n)p Fl(\000)p Fn(1)1133 3159 y Fo(:)0 3355 y Fs(Then,)26 b(the)g(desired)f(result)g(follo)m(ws)g(from)h(Lemma)g(6,)h(applied)d (here)i(with)f Fo(a)2645 3369 y Fp(j)2707 3355 y Fs(:=)g Fo(V)20 b Fs(\()p Fo(j)5 b(T)13 b Fs(\))f Fr(\000)g Fo(V)20 b Fs(\(\()p Fo(j)e Fr(\000)12 b Fs(1\))p Fo(T)h Fs(\).)1354 3680 y Ft(Pr)n(oof)34 b(of)g(Theorem)f(3)0 3842 y Fs(This)c(is)g(a)i (mo)s(di\014cation)d(of)j(some)g(argumen)m(ts)f(on)h(pages)g(167{169)i (of)e([V)-8 b(al04)q(].)0 3960 y(W)g(e)39 b(\014rst)d(pro)m(v)m(e)j (\(6\))q(.)62 b(T)-8 b(o)38 b(this)e(e\013ect,)41 b(w)m(e)d(de\014ne) 1831 3936 y(^)1824 3960 y Fo(\022)h Fs(:=)e(\(1)26 b(+)f Fo(\022)2284 3974 y Fn(0)2323 3960 y Fs(\))p Fo(=)p Fs(2.)63 b(Exploiting)37 b(\(14\))i(and)e(in)m(terior)0 4068 y(elliptic)g (regularit)m(y)h(theory)h(\(see,)j(e.g.)68 b(Theorem)38 b(3.13)j(in)c([HL97)r(]\),)k(w)m(e)f(ha)m(v)m(e)g(that)f Fo(u)g Fs(is)f(uniformly)0 4176 y(Lipsc)m(hitz)29 b(con)m(tin)m(uous)h (in)f Fo(B)1018 4190 y Fn(1)1058 4176 y Fs(\()p Fo(x)p Fs(\),)i(with)e(Lipsc)m(hitz)g(constan)m(t,)j(sa)m(y)-8 b(,)32 b(\003)25 b Fr(\025)g Fs(1.)41 b(Th)m(us,)1226 4382 y Fr(j)p Fo(u)p Fs(\()p Fo(y)s Fs(\))p Fr(j)26 b(\024)f(j)p Fo(u)p Fs(\()p Fo(x)p Fs(\))p Fr(j)c Fs(+)f(\003)p Fr(j)p Fo(x)h Fr(\000)f Fo(y)s Fr(j)25 b Fo(<)2358 4358 y Fs(^)2350 4382 y Fo(\022)17 b(;)0 4577 y Fs(as)31 b(long)f(as)g Fr(j)p Fo(x)21 b Fr(\000)f Fo(y)s Fr(j)25 b Fo(<)g Fs(\(1)c Fr(\000)f Fo(\022)1038 4591 y Fn(0)1077 4577 y Fs(\))p Fo(=)p Fs(\(2\003\))27 b(=:)e Fo(K)7 b Fs(.)41 b(Then,)477 4800 y(min)643 4699 y Fm(n)704 4800 y Fr(L)767 4699 y Fm(\020)821 4800 y Fo(B)890 4814 y Fp(K)958 4800 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)p Fo(u)26 b Fr(\025)f(\000)1479 4776 y Fs(^)1472 4800 y Fo(\022)r Fr(g)1562 4699 y Fm(\021)1631 4800 y Fo(;)41 b Fr(L)1760 4699 y Fm(\020)1814 4800 y Fo(B)1883 4814 y Fp(K)1951 4800 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)p Fo(u)26 b Fr(\024)2401 4776 y Fs(^)2394 4800 y Fo(\022)r Fr(g)2484 4699 y Fm(\021)q(o)2640 4800 y Fs(=)40 b Fr(L)p Fs(\()p Fo(B)2918 4814 y Fp(K)2986 4800 y Fs(\()p Fo(x)p Fs(\)\))15 b Fo(;)0 5023 y Fs(whic)m(h)29 b(giv)m(es)i(the)f(analogous)h(of)g(assumptions)d(\(4\))k(and)e(\(5\))q (.)40 b(Accordingly)-8 b(,)30 b(b)m(y)h(Theorem)f(2,)547 5246 y(min)714 5145 y Fm(n)774 5246 y Fr(L)837 5145 y Fm(\020)891 5246 y Fo(B)960 5260 y Fp(r)998 5246 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)p Fo(u)26 b Fr(\025)f(\000)1519 5222 y Fs(^)1512 5246 y Fo(\022)r Fr(g)1602 5145 y Fm(\021)1672 5246 y Fo(;)40 b Fr(L)1800 5145 y Fm(\020)1854 5246 y Fo(B)1923 5260 y Fp(r)1961 5246 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)p Fo(u)26 b Fr(\024)2411 5222 y Fs(^)2404 5246 y Fo(\022)r Fr(g)2494 5145 y Fm(\021o)2649 5246 y Fr(\025)41 b Fs(const)15 b Fo(r)3027 5209 y Fp(n)3089 5246 y Fo(;)0 5469 y Fs(for)30 b Fo(r)e Fr(2)d Fs([)p Fo(r)360 5483 y Fn(0)399 5469 y Fo(;)15 b Fm(b)-51 b Fo(c)q(=\021)s Fs(].)p eop %%Page: 12 12 12 11 bop 168 251 a Fn(12)976 b(MA)-6 b(TTEO)23 b(NO)n(V)-8 b(A)n(GA)23 b(AND)g(ENRICO)h(V)-8 b(ALDINOCI)168 450 y Fs(Consequen)m(tly)g(,)30 b(exploiting)f(Theorems)h(1)g(and)g(2,)1078 682 y Fr(L)1141 581 y Fm(\020)1195 682 y Fo(B)1264 696 y Fp(r)1302 682 y Fs(\()p Fo(x)p Fs(\))20 b Fr(\\)g(f)p Fo(u)26 b(>)f(\022)1787 696 y Fn(0)1826 682 y Fr(g)1871 581 y Fm(\021)924 880 y Fr(\025)83 b(L)1141 780 y Fm(\020)1195 880 y Fo(B)1264 894 y Fp(r)1302 880 y Fs(\()p Fo(x)p Fs(\))20 b Fr(\\)g(f)p Fo(u)26 b Fr(\025)f(\000)1822 856 y Fs(^)1815 880 y Fo(\022)r Fr(g)1905 780 y Fm(\021)1980 880 y Fr(\000)20 b(L)2134 780 y Fm(\020)2188 880 y Fo(B)2257 894 y Fp(r)2295 880 y Fs(\()p Fo(x)p Fs(\))g Fr(\\)g(f)p Fo(\022)2606 894 y Fn(0)2671 880 y Fr(\025)25 b Fo(u)g Fr(\025)g(\000)3018 856 y Fs(^)3011 880 y Fo(\022)r Fr(g)3101 780 y Fm(\021)924 1099 y Fr(\025)98 b Fs(const)16 b Fo(r)1360 1061 y Fp(n)1426 1099 y Fr(\000)1757 1037 y Fs(1)p 1527 1078 505 4 v 1527 1161 a(inf)1638 1193 y Fp(u)p Fl(2)p Fn([)p Fl(\000)1807 1176 y Fn(^)1801 1193 y Fp(\022)q(;\022)1888 1202 y Fe(0)1922 1193 y Fn(])1961 1161 y Fo(F)2057 975 y Fm(Z)2108 1181 y Fp(B)2161 1189 y Fj(r)2196 1181 y Fn(\()p Fp(x)p Fn(\))p Fl(\\f)p Fp(\022)2405 1190 y Fe(0)2440 1181 y Fl(\025)p Fp(u)p Fl(\025\000)2652 1163 y Fn(^)2646 1181 y Fp(\022)q Fl(g)2735 1099 y Fo(F)d Fs(\()p Fo(x;)i(u)p Fs(\))g Fo(dx)924 1313 y Fr(\025)98 b Fs(const)16 b Fo(r)1360 1276 y Fp(n)1426 1313 y Fr(\000)35 b Fs(const)16 b Fo(E)1822 1332 y Fp(B)1875 1340 y Fj(r)1910 1332 y Fn(\()p Fp(x)p Fn(\))2009 1313 y Fs(\()p Fo(u)p Fs(\))21 b Fr(\000)35 b Fs(const)16 b Fo(\021)i(r)2588 1276 y Fp(n)924 1464 y Fr(\025)98 b Fs(const)16 b Fo(r)1360 1427 y Fp(n)1426 1464 y Fr(\000)35 b Fs(const)16 b Fo(r)1799 1427 y Fp(n)p Fl(\000)p Fn(1)924 1599 y Fr(\025)98 b Fs(const)16 b Fo(r)1360 1561 y Fp(n)1421 1599 y Fo(;)168 1808 y Fs(for)30 b(large)g Fo(r)j Fs(and)d(small)f Fo(\021)s Fs(.)168 1916 y(Analogously)-8 b(,)1286 2109 y Fr(L)1349 2009 y Fm(\020)1403 2109 y Fo(B)1472 2123 y Fp(r)1510 2109 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)p Fo(u)26 b(<)f Fr(\000)p Fo(\022)2067 2123 y Fn(0)2106 2109 y Fr(g)2151 2009 y Fm(\021)2246 2109 y Fr(\025)40 b Fs(const)16 b Fo(r)2624 2072 y Fp(n)2685 2109 y Fo(;)168 2327 y Fs(as)30 b(desired.)40 b(The)29 b(latter)i(t)m(w)m(o)h(estimates)f(complete)g (the)f(pro)s(of)g(of)37 b(\(6\))r(.)168 2435 y(W)-8 b(e)29 b(no)m(w)f(pro)m(v)m(e)g(\(7\))r(.)39 b(F)-8 b(or)29 b(this)e(scop)s(e,)h(w)m(e)h(denote)f(b)m(y)43 b(P)m(er)2248 2449 y Fp(U)2307 2435 y Fs(\()p Fo(E)5 b Fs(\))29 b(the)f(p)s(erimeter) e(of)i(the)g(\(Cacciopp)s(oli\))168 2543 y(set)j Fo(E)k Fs(in)29 b(the)i(\(op)s(en\))f(set)h Fo(U)41 b Fs(\(see,)31 b(e.g.,)h([Giu84]\).)42 b(W)-8 b(e)31 b(also)f(de\014ne)1329 2861 y(\026)-51 b Fo(u)p Fs(\()p Fo(x)p Fs(\))41 b(:=)1675 2675 y Fm(8)1675 2757 y(<)1675 2920 y(:)1755 2752 y Fo(u)p Fs(\()p Fo(x)p Fs(\))198 b(if)29 b Fr(j)p Fo(u)p Fs(\()p Fo(x)p Fs(\))p Fr(j)d(\024)f Fo(\022)2599 2766 y Fn(0)2638 2752 y Fs(,)1802 2860 y Fo(\022)1845 2874 y Fn(0)2152 2860 y Fs(if)k Fo(u)p Fs(\()p Fo(x)p Fs(\))d Fo(>)f(\022)2574 2874 y Fn(0)2613 2860 y Fs(,)1766 2968 y Fr(\000)p Fo(\022)1880 2982 y Fn(0)2129 2968 y Fs(if)k Fo(u)p Fs(\()p Fo(x)p Fs(\))d Fo(<)f Fr(\000)p Fo(\022)2622 2982 y Fn(0)168 3174 y Fs(and)801 3350 y Fo(\026)p Fs(\()p Fo(t;)15 b(r)s Fs(\))25 b(:=)g(min)1356 3250 y Fm(n)1417 3350 y Fr(L)1480 3250 y Fm(\020)1534 3350 y Fo(B)1603 3364 y Fp(r)1641 3350 y Fs(\()p Fo(x)p Fs(\))20 b Fr(\\)g(f)6 b Fs(\026)-51 b Fo(u)26 b Fr(\025)f Fo(t)p Fr(g)2161 3250 y Fm(\021)2231 3350 y Fo(;)40 b Fr(L)2359 3250 y Fm(\020)2413 3350 y Fo(B)2482 3364 y Fp(r)2520 3350 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)6 b Fs(\026)-51 b Fo(u)26 b(<)f(t)p Fr(g)3041 3250 y Fm(\021o)3171 3350 y Fo(:)168 3568 y Fs(Exploiting)k(\(6\))q(,)i (w)m(e)g(ha)m(v)m(e)g(that,)h(if)d Fo(t)c Fr(2)g Fs(\()p Fr(\000)p Fo(\022)1726 3582 y Fn(0)1765 3568 y Fo(;)15 b(\022)1848 3582 y Fn(0)1887 3568 y Fs(\),)691 3805 y Fo(\026)p Fs(\()p Fo(t;)g(r)s Fs(\))83 b Fr(\025)g Fs(min)1337 3704 y Fm(n)1397 3805 y Fr(L)1460 3704 y Fm(\020)1514 3805 y Fo(B)1583 3819 y Fp(r)1621 3805 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)6 b Fs(\026)-51 b Fo(u)26 b Fr(\025)f Fo(\022)2107 3819 y Fn(0)2146 3805 y Fr(g)2191 3704 y Fm(\021)2261 3805 y Fo(;)40 b Fr(L)2389 3704 y Fm(\020)2443 3805 y Fo(B)2512 3819 y Fp(r)2550 3805 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)6 b Fs(\026)-51 b Fo(u)26 b Fr(\024)f(\000)p Fo(\022)3107 3819 y Fn(0)3145 3805 y Fr(g)3190 3704 y Fm(\021)q(o)1016 4004 y Fr(\025)83 b Fs(min)1337 3903 y Fm(n)1397 4004 y Fr(L)1460 3903 y Fm(\020)1514 4004 y Fo(B)1583 4018 y Fp(r)1621 4004 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)p Fo(u)26 b(>)f(\022)2107 4018 y Fn(0)2146 4004 y Fr(g)2191 3903 y Fm(\021)2261 4004 y Fo(;)40 b Fr(L)2389 3903 y Fm(\020)2443 4004 y Fo(B)2512 4018 y Fp(r)2550 4004 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(f)p Fo(u)26 b(<)f Fr(\000)p Fo(\022)3107 4018 y Fn(0)3145 4004 y Fr(g)3190 3903 y Fm(\021)q(o)1016 4164 y Fr(\025)98 b Fs(const)16 b Fo(r)1452 4126 y Fp(n)1514 4164 y Fo(:)168 4373 y Fs(W)-8 b(e)24 b(no)m(w)e(use)h(the)f(ab)s(o)m(v)m(e)i(estimate) g(and)e(the)h(Coarea)g(and)f(Isop)s(erimetric)f(F)-8 b(orm)m(ulas)23 b(\(see,)i(e.g.,)f([Giu84)q(]\))168 4481 y(to)31 b(deduce)f(that)924 4590 y Fm(Z)974 4796 y Fp(B)1027 4804 y Fj(r)1062 4796 y Fn(\()p Fp(x)p Fn(\))p Fl(\\fj)p Fp(u)p Fl(j)p Fp(<\022)1407 4805 y Fe(0)1442 4796 y Fl(g)1496 4714 y Fr(jr)p Fo(u)p Fr(j)84 b Fs(=)1911 4590 y Fm(Z)1962 4796 y Fp(B)2015 4804 y Fj(r)2050 4796 y Fn(\()p Fp(x)p Fn(\))2164 4714 y Fr(jr)6 b Fs(\026)-51 b Fo(u)p Fr(j)1758 4992 y(\025)1911 4868 y Fm(Z)2002 4894 y Fp(\022)2035 4903 y Fe(0)1962 5074 y Fl(\000)p Fp(\022)2050 5083 y Fe(0)2119 4992 y Fs(P)m(er)2269 5010 y Fp(B)2322 5018 y Fj(r)2357 5010 y Fn(\()p Fp(x)p Fn(\))2456 4891 y Fm(\020)2510 4992 y Fr(f)6 b Fs(\026)-51 b Fo(u)26 b(<)f(t)p Fr(g)2807 4891 y Fm(\021)2877 4992 y Fo(dt)1758 5262 y Fr(\025)98 b Fs(const)2164 5138 y Fm(Z)2255 5165 y Fp(\022)2288 5174 y Fe(0)2215 5345 y Fl(\000)p Fp(\022)2303 5354 y Fe(0)2357 5161 y Fm(\020)2411 5262 y Fo(\026)p Fs(\()p Fo(t;)15 b(r)s Fs(\))2653 5161 y Fm(\021)2707 5184 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))p Fp(=n)2993 5262 y Fo(dt)1758 5469 y Fr(\025)98 b Fs(const)15 b Fo(r)2193 5432 y Fp(n)p Fl(\000)p Fn(1)2345 5469 y Fo(:)p eop %%Page: 13 13 13 12 bop 587 251 a Fn(THE)30 b(GEOMETR)-6 b(Y)31 b(OF)f(MESOSCOPIC)h (PHASE)f(TRANSITION)h(INTERF)-8 b(A)n(CES)517 b(13)0 450 y Fs(Consequen)m(tly)-8 b(,)29 b(taking)g(a)g(suitably)e(large)i (additional)e(parameter)i Fo(A)p Fs(,)h(b)m(y)f(the)g(Cauc)m(h)m(y)g (Inequalit)m(y)f(and)0 558 y(Theorem)i(1,)h(w)m(e)g(ha)m(v)m(e)h(that) 480 765 y(const)16 b Fo(r)747 727 y Fp(n)p Fl(\000)p Fn(1)967 765 y Fr(\024)1142 703 y Fs(1)p 1130 744 69 4 v 1130 827 a Fo(A)1224 641 y Fm(Z)1274 847 y Fp(B)1327 855 y Fj(r)1362 847 y Fn(\()p Fp(x)p Fn(\))1476 765 y Fr(jr)p Fo(u)p Fr(j)1654 727 y Fn(2)1714 765 y Fs(+)k Fo(A)15 b Fr(L)1951 664 y Fm(\020)2005 765 y Fo(B)2074 779 y Fp(r)2112 765 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(fj)p Fo(u)p Fr(j)26 b Fo(<)f(\022)2648 779 y Fn(0)2687 765 y Fr(g)2732 664 y Fm(\021)967 1019 y Fr(\024)1142 958 y Fs(1)p 1130 998 V 1130 1082 a Fo(A)1224 1019 y(E)1291 1038 y Fp(B)1344 1046 y Fj(r)1379 1038 y Fn(\()p Fp(x)p Fn(\))1477 1019 y Fs(\()p Fo(u)p Fs(\))c(+)35 b(const)16 b Fo(\021)j(r)2057 982 y Fp(n)2123 1019 y Fs(+)h Fo(A)15 b Fr(L)2360 918 y Fm(\020)2415 1019 y Fo(B)2484 1033 y Fp(r)2522 1019 y Fs(\()p Fo(x)p Fs(\))20 b Fr(\\)g(fj)p Fo(u)p Fr(j)26 b Fo(<)f(\022)3057 1033 y Fn(0)3097 1019 y Fr(g)3142 918 y Fm(\021)967 1254 y Fr(\024)1146 1193 y Fs(const)15 b Fo(r)1412 1160 y Fp(n)p Fl(\000)p Fn(1)p 1130 1233 419 4 v 1306 1317 a Fo(A)1579 1254 y Fs(+)35 b(const)16 b Fo(\021)i(r)2015 1217 y Fp(n)2082 1254 y Fs(+)i Fo(A)15 b Fr(L)2319 1153 y Fm(\020)2373 1254 y Fo(B)2442 1268 y Fp(r)2480 1254 y Fs(\()p Fo(x)p Fs(\))21 b Fr(\\)f(fj)p Fo(u)p Fr(j)26 b Fo(<)f(\022)3016 1268 y Fn(0)3055 1254 y Fr(g)3100 1153 y Fm(\021)3170 1254 y Fo(:)0 1450 y Fs(Using)32 b(that)i Fo(\021)s(r)i Fs(is)c(assumed)g (to)i(b)s(e)e(small)f(and)i(c)m(ho)s(osing)g Fo(A)g Fs(appropriately)e (large,)i(\(7\))h(follo)m(ws.)48 b(This)0 1558 y(ends)30 b(the)g(pro)s(of)g(of)g(Theorem)g(3.)1354 1797 y Ft(Pr)n(oof)k(of)g (Theorem)f(4)0 1959 y Fs(Let)e Fo(Q)25 b Fs(:=)g([0)p Fo(;)15 b Fs(1])561 1926 y Fp(n)610 1959 y Fs(.)41 b(W)-8 b(e)31 b(de\014ne)f(the)g Fo(Q)p Fs(-p)s(erio)s(dic)e(functions)h(in)g Fo(W)2299 1914 y Fn(1)p Fp(;)p Fn(2)2286 1988 y(lo)r(c)2393 1959 y Fs(\()p Fq(R)2488 1926 y Fp(n)2541 1959 y Fs(\))i(b)m(y)837 2158 y Fo(W)936 2120 y Fn(1)p Fp(;)p Fn(2)923 2180 y(p)r(er)1030 2158 y Fs(\()p Fo(Q)p Fs(\))41 b(:=)1350 2057 y Fm(n)1425 2158 y Fo(u)26 b Fr(2)e Fo(W)1687 2114 y Fn(1)p Fp(;)p Fn(2)1674 2187 y(lo)r(c)1781 2158 y Fs(\()p Fq(R)1876 2120 y Fp(n)1929 2158 y Fs(\))41 b(suc)m(h)30 b(that)1425 2357 y Fo(u)p Fs(\()p Fo(x)21 b Fs(+)f Fo(e)1718 2371 y Fp(j)1755 2357 y Fs(\))25 b(=)g Fo(u)p Fs(\()p Fo(x)p Fs(\))41 b(for)30 b(an)m(y)41 b Fo(x)25 b Fr(2)g Fq(R)2670 2319 y Fp(n)2723 2256 y Fm(o)2799 2357 y Fo(;)0 2258 y Fs(\(32\))0 2549 y(where)30 b Fr(f)p Fo(e)350 2563 y Fn(1)390 2549 y Fo(;)15 b(:)g(:)g(:)i(;)e(e)634 2563 y Fp(n)681 2549 y Fr(g)31 b Fs(is)f(the)g(standard)g(Euclidean)e(base)j (of)f Fq(R)2171 2516 y Fp(n)2224 2549 y Fs(.)40 b(With)30 b(this)g(setting,)g(w)m(e)h(ha)m(v)m(e:)0 2696 y Fw(Lemma)38 b(7.)45 b Fi(The)37 b(functional)h Fo(E)1191 2710 y Fp(Q)1287 2696 y Fi(attains)h(its)e(minimum)h(in)e Fo(W)2356 2652 y Fn(1)p Fp(;)p Fn(2)2343 2708 y(p)r(er)2450 2696 y Fs(\()p Fo(Q)p Fs(\))p Fi(.)56 b(A)n(lso,)38 b(if)e Fo(u)h Fi(is)g(any)h(of)f (such)0 2804 y(minimizers,)c(then)g(it)g(is)g(c)-5 b(ontinuous,)33 b(its)g(mo)-5 b(dulus)34 b(of)f(c)-5 b(ontinuity)33 b(is)g(uniformly)h (b)-5 b(ounde)g(d,)34 b(and)0 2992 y Fs(\(33\))1489 2887 y Fm(\014)1489 2942 y(\014)1489 2996 y(\014)1519 2992 y Fr(j)p Fo(u)p Fs(\()p Fo(x)p Fs(\))p Fr(j)22 b(\000)d Fs(1)1900 2887 y Fm(\014)1900 2942 y(\014)1900 2996 y(\014)1956 2992 y Fr(\024)25 b Fo(\016)2092 3006 y Fn(0)2147 2992 y Fo(;)0 3179 y Fi(for)33 b(any)g Fo(x)26 b Fr(2)f Fo(Q)p Fi(,)32 b(as)h(long)g(as)g Fo(\021)j Fi(is)d(smal)5 b(l)33 b(enough.)0 3362 y(Pr)-5 b(o)g(of.)43 b Fs(Let)31 b Fo(u)496 3377 y Fp(k)569 3362 y Fs(b)s(e)f(a)g(minimizing)d(sequence.)41 b(W)-8 b(e)32 b(ma)m(y)f(supp)s(ose)e(that)0 3527 y(\(34\))1209 b Fo(E)1436 3541 y Fp(Q)1495 3527 y Fs(\()p Fo(u)1582 3542 y Fp(k)1626 3527 y Fs(\))40 b Fr(\024)g Fo(E)1879 3541 y Fp(Q)1939 3527 y Fs(\(1\))i(=)e(0)15 b Fo(;)0 3692 y Fs(due)30 b(to)h(\(2\))q(.)0 3800 y(Also,)f(it)g(follo)m(ws)g (from)g(\(10\))i(that)142 3993 y(min)309 3892 y Fm(n)369 3993 y Fo(F)13 b Fs(\()p Fo(x;)i Fs(1)22 b(+)e Fo(s)p Fs(\))g Fr(\000)g Fo(F)13 b Fs(\()p Fo(x;)i Fs(1)21 b(+)f Fo(\016)1309 4007 y Fn(0)1349 3993 y Fs(\))p Fo(;)31 b(F)13 b Fs(\()p Fo(x;)i Fr(\000)p Fs(1)21 b Fr(\000)f Fo(s)p Fs(\))g Fr(\000)g Fo(F)13 b Fs(\()p Fo(x;)i Fr(\000)p Fs(1)21 b Fr(\000)f Fo(\016)2521 4007 y Fn(0)2561 3993 y Fs(\))2596 3892 y Fm(o)2740 3993 y Fr(\025)82 b Fo(c)15 b Fs(\()p Fo(s)21 b Fr(\000)f Fo(\016)3177 4007 y Fn(0)3217 3993 y Fs(\))2740 4192 y Fr(\025)2893 4087 y Fm(\014)2893 4142 y(\014)2893 4196 y(\014)2924 4192 y Fo(H)7 b Fs(\()p Fo(x)p Fs(\))15 b(\()p Fo(\016)3219 4206 y Fn(0)3280 4192 y Fr(\000)20 b Fo(s)p Fs(\))3449 4087 y Fm(\014)3449 4142 y(\014)3449 4196 y(\014)3494 4192 y Fo(;)-3519 b Fs(\(35\))0 4380 y(for)30 b(an)m(y)h Fo(s)25 b Fr(\025)g Fo(\016)515 4394 y Fn(0)555 4380 y Fs(,)30 b(and)1393 4507 y Fo(F)13 b Fs(\()p Fo(x;)i(r)s Fs(\))21 b(+)f Fo(H)7 b Fs(\()p Fo(x)p Fs(\))p Fo(r)43 b Fr(\025)e Fs(0)15 b Fo(;)0 4653 y Fs(as)29 b(long)f(as)h Fr(j)p Fo(r)s Fr(j)d(\025)f Fo(C)696 4667 y Fn(0)735 4653 y Fs(,)k(with)f Fo(C)1060 4667 y Fn(0)1128 4653 y Fs(appropriately)f(large,)i(if)f Fo(\021)k Fs(is)c(small)f(enough.)40 b(Consequen)m(tly)-8 b(,)29 b(b)m(y)h(\(34\))q(,)0 4865 y(\(36\))900 4741 y Fm(Z)950 4947 y Fp(Q)1025 4865 y Fr(jr)p Fo(u)1178 4880 y Fp(k)1221 4865 y Fr(j)1246 4827 y Fn(2)1326 4865 y Fr(\024)1437 4741 y Fm(Z)1488 4947 y Fp(Q)p Fl(\\fj)p Fp(u)1687 4959 y Fj(k)1724 4947 y Fl(j\024)p Fp(C)1849 4956 y Fe(0)1883 4947 y Fl(g)1938 4865 y Fr(j)p Fo(H)22 b(u)2113 4880 y Fp(k)2156 4865 y Fr(j)41 b(\024)f Fo(C)2398 4879 y Fn(0)2452 4865 y Fr(L)p Fs(\()p Fo(Q)p Fs(\))15 b Fo(\021)k(:)0 5092 y Fs(F)-8 b(urthermore,)31 b(if)e(w)m(e)i (de\014ne)959 5362 y Fo(u)1011 5324 y Fp(?)1011 5385 y(k)1054 5362 y Fs(\()p Fo(x)p Fs(\))41 b(:=)1354 5175 y Fm(8)1354 5257 y(<)1354 5421 y(:)1479 5253 y Fo(u)1531 5268 y Fp(k)1574 5253 y Fs(\()p Fo(x)p Fs(\))252 b(if)29 b Fr(j)p Fo(u)2108 5268 y Fp(k)2151 5253 y Fs(\()p Fo(x)p Fs(\))p Fr(j)d Fo(<)f Fs(1)c(+)f Fo(\016)2617 5267 y Fn(0)2657 5253 y Fs(,)1470 5361 y(1)g(+)g Fo(\016)1666 5375 y Fn(0)1973 5361 y Fs(if)29 b Fo(u)2108 5376 y Fp(k)2151 5361 y Fs(\()p Fo(x)p Fs(\))d Fr(\025)f Fs(1)c(+)f Fo(\016)2592 5375 y Fn(0)2631 5361 y Fs(,)1434 5469 y Fr(\000)p Fs(1)h Fr(\000)f Fo(\016)1702 5483 y Fn(0)1938 5469 y Fs(if)29 b Fo(u)2073 5484 y Fp(k)2116 5469 y Fs(\()p Fo(x)p Fs(\))d Fr(\024)f(\000)p Fs(1)20 b Fr(\000)g Fo(\016)2627 5483 y Fn(0)2667 5469 y Fs(,)p eop %%Page: 14 14 14 13 bop 168 251 a Fn(14)976 b(MA)-6 b(TTEO)23 b(NO)n(V)-8 b(A)n(GA)23 b(AND)g(ENRICO)h(V)-8 b(ALDINOCI)168 450 y Fs(then)34 b Fo(E)446 464 y Fp(Q)506 450 y Fs(\()p Fo(u)593 417 y Fp(?)593 478 y(k)636 450 y Fs(\))g Fr(\024)f Fo(E)876 464 y Fp(Q)935 450 y Fs(\()p Fo(u)1022 465 y Fp(k)1065 450 y Fs(\),)k(thanks)e(to)h(\(35\))q(.)55 b(Accordingly)-8 b(,)36 b(b)m(y)f(p)s(ossibly)d(replacing)i Fo(u)3280 465 y Fp(k)3358 450 y Fs(with)f Fo(u)3621 417 y Fp(?)3621 478 y(k)3664 450 y Fs(,)k(w)m(e)168 558 y(ma)m(y)31 b(assume)f(that)168 704 y(\(37\))1399 b Fr(j)p Fo(u)1804 719 y Fp(k)1847 704 y Fr(j)25 b(\024)g Fs(1)c(+)f Fo(\016)2190 718 y Fn(0)2245 704 y Fo(:)168 850 y Fs(As)38 b(a)g(consequence)h(of)45 b(\(36\))40 b(and)d(\(37\))r(,)j(the)e(compact)h(em)m(b)s(edding)e(of)h Fo(W)2856 817 y Fn(1)p Fp(;)p Fn(2)2950 850 y Fs(\()p Fo(Q)p Fs(\))g(in)m(to)g Fo(L)3384 817 y Fn(2)3423 850 y Fs(\()p Fo(Q)p Fs(\))h(yields)168 958 y(that)i Fo(u)427 973 y Fp(k)509 958 y Fs(con)m(v)m(erges)i(to)e(some)f Fo(u)g Fs(in)f Fo(L)1555 925 y Fn(2)1595 958 y Fs(\()p Fo(Q)p Fs(\),)k(w)m(eakly)d(in)f Fo(W)2330 925 y Fn(1)p Fp(;)p Fn(2)2424 958 y Fs(\()p Fo(Q)p Fs(\))i(and)f(almost)g(ev)m (erywhere,)j(up)c(to)168 1076 y(subsequences.)h(Accordingly)-8 b(,)30 b Fo(u)25 b Fr(2)g Fo(W)1538 1031 y Fn(1)p Fp(;)p Fn(2)1525 1088 y(p)r(er)1632 1076 y Fs(\()p Fo(Q)p Fs(\))30 b(and)1404 1270 y(lim)15 b(inf)1413 1332 y Fp(k)r Fl(!)p Fn(+)p Fl(1)1672 1147 y Fm(Z)1722 1353 y Fp(Q)1797 1270 y Fr(jr)p Fo(u)1950 1285 y Fp(k)1993 1270 y Fr(j)2018 1233 y Fn(2)2098 1270 y Fr(\025)2209 1147 y Fm(Z)2259 1353 y Fp(Q)2334 1270 y Fr(jr)p Fo(u)p Fr(j)2512 1233 y Fn(2)2567 1270 y Fo(:)168 1475 y Fs(Then,)29 b(b)m(y)i(F)-8 b(atou)31 b(Lemma,)1296 1621 y(inf)1216 1701 y Fp(W)1293 1668 y Fe(1)p Fj(;)p Fe(2)1283 1711 y(p)r(er)1376 1701 y Fn(\()p Fp(Q)p Fn(\))1502 1621 y Fo(E)1569 1635 y Fp(Q)1669 1621 y Fs(=)40 b(lim)15 b(inf)1789 1682 y Fp(k)r Fl(!)p Fn(+)p Fl(1)2047 1621 y Fo(E)2114 1635 y Fp(Q)2174 1621 y Fs(\()p Fo(u)2261 1636 y Fp(k)2304 1621 y Fs(\))41 b Fr(\025)f Fo(E)2558 1635 y Fp(Q)2618 1621 y Fs(\()p Fo(u)p Fs(\))15 b Fo(;)168 1839 y Fs(th)m(us)30 b Fo(u)g Fs(is)f(the)i(desired)e(minimizer.)168 1947 y(The)i(fact)h(that)g(the)g (minimizers)c(are)k(con)m(tin)m(uous)g(follo)m(ws)e(from)h(standard)g (elliptic)e(regularit)m(y)i(theory)168 2055 y(\(see,)g(e.g.)42 b(Theorem)30 b(3.13)i(in)d([HL97)q(]\).)168 2173 y(W)-8 b(e)38 b(no)m(w)e(pro)m(v)m(e)i(\(33\))q(.)60 b(F)-8 b(or)37 b(this,)h(w)m(e)f(assume)f(that)h Fo(u)f Fr(2)f Fo(W)2350 2129 y Fn(1)p Fp(;)p Fn(2)2337 2185 y(p)r(er)2444 2173 y Fs(\()p Fo(Q)p Fs(\))i(is)f(a)h(minimizer)d(for)i Fo(E)3446 2187 y Fp(Q)3543 2173 y Fs(and)g(w)m(e)168 2281 y(de\014ne)1150 2476 y Fo(u)1202 2438 y Fp(?)1242 2476 y Fs(\()p Fo(x)p Fs(\))41 b(:=)1541 2290 y Fm(8)1541 2371 y(<)1541 2535 y(:)1688 2367 y Fo(u)p Fs(\()p Fo(x)p Fs(\))273 b(if)30 b Fr(j)p Fo(u)p Fs(\()p Fo(x)p Fs(\))p Fr(j)c Fo(<)f Fs(1)c(+)f Fo(\016)2762 2381 y Fn(0)2801 2367 y Fs(,)1657 2475 y(1)h(+)f Fo(\016)1854 2489 y Fn(0)2161 2475 y Fs(if)29 b Fo(u)p Fs(\()p Fo(x)p Fs(\))d Fr(\025)f Fs(1)20 b(+)g Fo(\016)2736 2489 y Fn(0)2776 2475 y Fs(,)1622 2583 y Fr(\000)p Fs(1)g Fr(\000)g Fo(\016)1889 2597 y Fn(0)2125 2583 y Fs(if)30 b Fo(u)p Fs(\()p Fo(x)p Fs(\))c Fr(\024)e(\000)p Fs(1)d Fr(\000)f Fo(\016)2772 2597 y Fn(0)2812 2583 y Fs(.)168 2712 y(Then,)29 b(b)m(y)j(\(35\))g(and)e(the) g(minimalit)m(y)e(of)i Fo(u)p Fs(,)h(w)m(e)g(ha)m(v)m(e)342 2905 y(0)26 b Fr(\024)f Fo(E)576 2919 y Fp(Q)635 2905 y Fs(\()p Fo(u)722 2867 y Fp(?)762 2905 y Fs(\))c Fr(\000)f Fo(E)976 2919 y Fp(Q)1036 2905 y Fs(\()p Fo(u)p Fs(\))26 b Fr(\024)e(\000)1363 2844 y Fo(c)p 1360 2884 46 4 v 1360 2967 a Fs(2)1416 2804 y Fm(h)1474 2781 y(Z)1524 2987 y Fl(f)p Fp(u>)p Fn(1+)p Fp(\016)1776 2996 y Fe(0)1811 2987 y Fl(g)1851 2905 y Fs(\()p Fo(u)c Fr(\000)g Fs(1)h Fr(\000)f Fo(\016)2246 2919 y Fn(0)2286 2905 y Fs(\))g(+)2432 2781 y Fm(Z)2483 2987 y Fl(f)p Fp(u<)p Fl(\000)p Fn(1)p Fl(\000)p Fp(\016)2790 2996 y Fe(0)2824 2987 y Fl(g)2864 2905 y Fs(\()p Fr(\000)p Fo(u)g Fr(\000)g Fs(1)h Fr(\000)f Fo(\016)3330 2919 y Fn(0)3369 2905 y Fs(\))3404 2804 y Fm(i)3473 2905 y Fr(\024)25 b Fs(0)15 b Fo(;)198 3118 y Fs(whic)m(h)29 b(sa)m(ys)i(that)g Fr(j)p Fo(u)p Fr(j)26 b(\024)f Fs(1)20 b(+)g Fo(\016)1268 3132 y Fn(0)1308 3118 y Fs(.)41 b(Moreo)m(v)m(er,)33 b(if,)c(b)m(y)h(con)m(tradiction,) 1498 3264 y Fr(\000)p Fs(1)21 b(+)f Fo(\016)1766 3278 y Fn(0)1831 3264 y Fr(\024)25 b Fo(u)p Fs(\()p Fo(x)2066 3278 y Fn(0)2105 3264 y Fs(\))h Fr(\024)f Fs(1)c Fr(\000)f Fo(\016)2459 3278 y Fn(0)168 3410 y Fs(for)30 b(some)h Fo(x)587 3424 y Fn(0)651 3410 y Fr(2)25 b Fo(Q)p Fs(,)31 b(then)f(the)g(uniform)e(con)m(tin)m(uit)m(y)j(of)f Fo(u)h Fs(yields)d(that)1498 3599 y Fr(\000)p Fs(1)21 b(+)1735 3537 y Fo(\016)1775 3551 y Fn(0)p 1735 3578 80 4 v 1753 3661 a Fs(2)1850 3599 y Fr(\024)k Fo(u)p Fs(\()p Fo(x)p Fs(\))h Fr(\024)f Fs(1)c Fr(\000)2409 3537 y Fo(\016)2449 3551 y Fn(0)p 2409 3578 V 2426 3661 a Fs(2)168 3775 y(for)38 b(an)m(y)h Fo(x)g Fr(2)g Fo(B)755 3789 y Fp(\032)795 3775 y Fs(\()p Fo(x)882 3789 y Fn(0)922 3775 y Fs(\),)i(for)d(a)h (suitable,)h(univ)m(ersal)c Fo(\032)j(>)g Fs(0.)66 b(Accordingly)-8 b(,)40 b Fo(F)13 b Fs(\()p Fo(x;)i(u)p Fs(\()p Fo(x)p Fs(\)\))41 b Fr(\025)54 b Fs(const)g(for)168 3883 y Fo(x)25 b Fr(2)g Fo(B)400 3897 y Fp(\032)440 3883 y Fs(\()p Fo(x)527 3897 y Fn(0)567 3883 y Fs(\),)31 b(whic)m(h)e(implies)e(that)859 4029 y Fo(E)926 4043 y Fp(Q)985 4029 y Fs(\()p Fo(u)p Fs(\))f Fr(\025)40 b Fs(const)16 b Fr(L)p Fs(\()p Fo(B)1634 4043 y Fp(\032)1674 4029 y Fs(\()p Fo(x)1761 4043 y Fn(0)1801 4029 y Fs(\)\))21 b Fr(\000)f Fo(\021)s Fr(L)p Fs(\()p Fo(Q)p Fs(\))26 b Fo(>)f Fs(0)h(=)f Fo(E)2592 4043 y Fp(Q)2651 4029 y Fs(\(1\))i Fr(\025)e Fo(E)2956 4043 y Fp(Q)3016 4029 y Fs(\()p Fo(u)p Fs(\))168 4175 y(and)k(this)h(con)m (tradiction)g(ends)g(the)g(pro)s(of)g(of)37 b(\(33\))r(.)1784 b Ff(\003)168 4333 y Fs(In)32 b(the)i(ligh)m(t)e(of)41 b(\(9\))34 b(and)f(Lemma)g(7,)i(w)m(e)f(deduce)f(that)h(the)f (functional)f Fo(E)2830 4347 y Fp(Q)2923 4333 y Fs(admits)g(t)m(w)m(o)j (minimizers)168 4455 y(in)29 b Fo(W)373 4411 y Fn(1)p Fp(;)p Fn(2)360 4467 y(p)r(er)467 4455 y Fs(\()p Fo(Q)p Fs(\),)i(sa)m(y)g Fo(u)874 4469 y Fl(\006)933 4455 y Fs(,)f(so)h(that)g Fo(u)1349 4469 y Fn(+)1433 4455 y Fs(=)25 b Fo(u)1581 4469 y Fl(\000)1661 4455 y Fs(+)19 b(2,)32 b(satisfying)168 4631 y(\(38\))1668 4526 y Fm(\014)1668 4581 y(\014)1668 4635 y(\014)1714 4631 y Fo(u)1766 4645 y Fl(\006)1860 4631 y Fr(\007)j Fs(1)2026 4526 y Fm(\014)2026 4581 y(\014)2026 4635 y(\014)2097 4631 y Fr(\024)40 b Fo(\016)2248 4645 y Fn(0)2303 4631 y Fo(:)168 4812 y Fs(By)34 b(elliptic)d(regularit)m(y)i(theory)h(\(see,)h(e.g.,)h([GT83)q (])e(or)f([HL97)q(]\),)j(w)m(e)e(also)f(ha)m(v)m(e)i(that)f Fo(u)3304 4826 y Fl(\006)3394 4812 y Fr(2)c Fo(C)3557 4779 y Fn(1)p Fp(;\013)3661 4812 y Fs(\()p Fo(Q)p Fs(\),)168 4920 y(for)43 b(all)g Fo(\013)48 b(<)f Fs(1.)81 b(Let)44 b(us)f(notice)h(that,)k(if)42 b Fo(F)13 b Fs(\()p Fo(x;)i Fr(\001)p Fs(\))45 b(is)e(strictly)g(con)m(v)m(ex)i(in)d([1)30 b Fr(\000)f Fo(\016)3160 4934 y Fn(0)3200 4920 y Fo(;)15 b Fs(1)30 b(+)f Fo(\016)3455 4934 y Fn(0)3494 4920 y Fs(])44 b(and)f(in)168 5038 y([)p Fr(\000)p Fs(1)14 b Fr(\000)g Fo(\016)448 5052 y Fn(0)487 5038 y Fo(;)h Fr(\000)p Fs(1)f(+)g Fo(\016)782 5052 y Fn(0)821 5038 y Fs(],)28 b(suc)m(h)f(minimizers)d(are)k(the)f(only)f(global)h(minimizers)d(of)j Fo(E)2945 5052 y Fp(Q)3032 5038 y Fs(in)f Fo(W)3234 4994 y Fn(1)p Fp(;)p Fn(2)3221 5050 y(p)r(er)3327 5038 y Fs(\()p Fo(Q)p Fs(\).)40 b(W)-8 b(e)29 b(will)168 5146 y(use)h(these)h (minimizers)c(to)k(construct)g(a)g(reduced)e(energy)i(functional)e (\(see)i(\(49\))h(b)s(elo)m(w\).)168 5254 y(W)-8 b(e)37 b(no)m(w)e(con)m(tin)m(ue)h(with)f(the)g(pro)s(of)g(of)h(Theorem)g(4.) 56 b(F)-8 b(or)37 b(this)d(scop)s(e,)k(w)m(e)e(tak)m(e)h Fo(!)g Fr(2)d Fq(Q)3358 5221 y Fp(n)3435 5254 y Fr(n)24 b(f)p Fs(0)p Fr(g)p Fs(,)39 b(the)168 5362 y(irrational)d(case)i(b)s (eing)f(then)g(easily)g(obtained)g(b)m(y)h(a)g(limit)e(argumen)m(t.)63 b(W)-8 b(e)39 b(consider)e(the)h(follo)m(wing)168 5469 y(equiv)-5 b(alence)41 b(relation)f Fr(\030)i Fs(induced)d(b)m(y)j Fo(!)s Fs(:)63 b(w)m(e)42 b(sa)m(y)g(that)g Fo(x)i Fr(\030)f Fo(y)i Fs(if)40 b(and)h(only)g(if)f Fo(x)28 b Fr(\000)f Fo(y)47 b Fr(2)d Fq(Z)3599 5436 y Fp(n)3683 5469 y Fs(and)p eop %%Page: 15 15 15 14 bop 587 251 a Fn(THE)30 b(GEOMETR)-6 b(Y)31 b(OF)f(MESOSCOPIC)h (PHASE)f(TRANSITION)h(INTERF)-8 b(A)n(CES)517 b(15)0 450 y Fo(!)16 b Fr(\001)d Fs(\()p Fo(x)g Fr(\000)g Fo(y)s Fs(\))27 b(=)e(0.)40 b(W)-8 b(e)28 b(will)d(denote)i(b)m(y)g Fq(R)1405 417 y Fp(n)1458 450 y Fo(=)10 b Fr(\030)27 b Fs(the)g(quotien)m(t)g(space,)i(whic)m(h,)d(of)h(course,)h(is)e(top)s (ologically)0 558 y(equiv)-5 b(alen)m(t)30 b(to)h(the)f(pro)s(duct)g (of)g(the)h(\()p Fo(n)20 b Fr(\000)g Fs(1\)-dimensional)28 b(torus)i(and)g(the)h(real)f(line.)0 666 y(The)k(equiv)-5 b(alence)33 b(relation)h Fr(\030)f Fs(ma)m(y)i(b)s(e)f(made)g(explicit) e(b)m(y)i(taking)g(an)g(in)m(teger)h(base)f(of)h Fq(R)3237 633 y Fp(n)3324 666 y Fs(giv)m(en)f(b)m(y)0 775 y(suitable)j(m)m (utually)f(orthogonal)j(v)m(ectors)g Fo(K)1601 742 y Fn(\(1\))1695 775 y Fo(;)15 b(:)g(:)g(:)i(;)e(K)1981 742 y Fn(\()p Fp(n)p Fn(\))2121 775 y Fr(2)37 b Fq(Z)2284 742 y Fp(n)2365 775 y Fs(in)g(suc)m(h)g(a)i(w)m(a)m(y)g(that)f Fo(!)j Fs(is)c(parallel)0 884 y(to)31 b Fo(K)195 851 y Fn(\()p Fp(n)p Fn(\))327 884 y Fs(and)f Fo(K)588 851 y Fn(\(1\))682 884 y Fo(;)15 b(:)g(:)g(:)h(;)f(K)967 851 y Fn(\()p Fp(n)p Fl(\000)p Fn(1\))1190 884 y Fs(span)29 b(the)i(set)g(of)f(the)h(in)m(teger)g(v)m(ectors)h(orthogonal)e(to)i Fo(!)s Fs(.)0 991 y(In)e(this)f(setting,)i(giv)m(en)f Fo(\027)h Fr(2)25 b Fq(N)6 b Fs(,)37 b(w)m(e)31 b(consider)e(the)i (rectangle)481 1213 y Fr(R)558 1175 y Fp(!)558 1235 y(\027)649 1213 y Fs(:=)785 1112 y Fm(n)905 1099 y Fp(n)861 1126 y Fm(X)865 1322 y Fp(j)t Fn(=1)1007 1213 y Fo(t)1040 1227 y Fp(j)1077 1213 y Fo(K)1161 1175 y Fn(\()p Fp(j)t Fn(\))1267 1213 y Fo(;)56 b Fs(0)26 b Fr(\024)f Fo(t)1548 1227 y Fn(1)1612 1213 y Fo(<)g Fs(1)15 b Fo(;)g(:)g(:)g(:)32 b(;)15 b Fs(0)27 b Fr(\024)d Fo(t)2185 1227 y Fp(n)p Fl(\000)p Fn(1)2348 1213 y Fo(<)h Fs(1)15 b Fo(;)56 b Fr(\000)p Fo(\027)31 b Fr(\024)25 b Fo(t)2861 1227 y Fp(n)2933 1213 y Fo(<)g(\027)3080 1112 y Fm(o)3155 1213 y Fo(:)0 1459 y Fs(W)-8 b(e)38 b(will)c(no)m(w)i(sho)m(w)g(that)h(the)g (minimizers)d(constructed)i(in)g(Lemma)g(7)h(are)g(also)f(minimizers)e (under)0 1567 y(the)d(p)s(erio)s(dicit)m(y)c(induced)i(b)m(y)h Fr(R)1153 1534 y Fp(!)1153 1589 y(\027)1203 1567 y Fs(.)41 b(That)30 b(is,)g(in)f(analogy)i(with)e(\(32\))r(,)h(w)m(e)h(de\014ne) 295 1746 y Fo(W)394 1709 y Fn(1)p Fp(;)p Fn(2)381 1769 y(p)r(er)487 1746 y Fs(\()p Fr(R)599 1709 y Fp(!)599 1769 y(\027)650 1746 y Fs(\))84 b(:=)948 1645 y Fm(n)1008 1746 y Fo(u)26 b Fr(2)e Fo(W)1270 1709 y Fn(1)p Fp(;)p Fn(2)1364 1746 y Fs(\()p Fr(R)1476 1709 y Fp(!)1476 1769 y(\027)1527 1746 y Fs(\))56 b(suc)m(h)30 b(that)948 1945 y Fo(u)p Fs(\()p Fo(x)p Fs(\))c(=)f Fo(u)p Fs(\()p Fo(x)20 b Fs(+)g Fo(K)1578 1908 y Fn(\(1\))1672 1945 y Fs(\))26 b(=)f Fr(\001)15 b(\001)g(\001)26 b Fs(=)f Fo(u)p Fs(\()p Fo(x)20 b Fs(+)g Fo(K)2390 1908 y Fn(\()p Fp(n)p 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Fp(k)1473 4189 y Fr(g)p Fs(\))26 b(=)f Fo(E)1742 4203 y Fl(R)1802 4184 y Fj(!)1802 4220 y(\027)1851 4189 y Fs(\(max)q Fr(f)p Fo(v)s(;)15 b(v)2232 4204 y Fp(k)2275 4189 y Fr(g)p Fs(\))27 b(=)d Fo(E)2544 4203 y Fl(R)2604 4184 y Fj(!)2604 4220 y(\027)2654 4189 y Fs(\()p Fo(v)s Fs(\))0 4340 y(and)30 b(so)h(b)s(oth)f(min)n Fr(f)p Fo(v)s(;)15 b(v)831 4355 y Fp(k)875 4340 y Fr(g)31 b Fs(and)f(max)p Fr(f)p Fo(v)s(;)15 b(v)1473 4355 y Fp(k)1517 4340 y Fr(g)31 b Fs(minimize)d Fo(E)2049 4354 y Fl(R)2109 4335 y Fj(!)2109 4371 y(\027)2189 4340 y Fs(in)h Fo(W)2394 4296 y Fn(1)p Fp(;)p Fn(2)2381 4352 y(p)r(er)2488 4340 y Fs(\()p Fr(R)2600 4307 y Fp(!)2600 4363 y(\027)2651 4340 y Fs(\).)41 b(By)31 b(rep)s(eating)f(the)h(argu-) 0 4448 y(men)m(t,)g(w)m(e)g(see)g(that)g(if)e Fr(Z)38 b Fs(is)29 b(an)m(y)i(\014nite)e(subset)h(of)g Fq(Z)1862 4415 y Fp(n)1905 4448 y Fs(,)g(w)m(e)h(ha)m(v)m(e)h(that)f(the)f (function)1154 4622 y Fo(v)1198 4636 y Fl(Z)1258 4622 y Fs(\()p Fo(x)p Fs(\))41 b(:=)g(min)1724 4521 y Fm(n)1785 4622 y Fo(v)s Fs(\()p Fo(x)21 b Fs(+)f Fo(k)s Fs(\))15 b Fo(;)56 b(k)29 b Fr(2)c(Z)2447 4521 y Fm(o)0 4821 y Fs(also)30 b(minimizes)e Fo(E)674 4835 y Fl(R)734 4816 y Fj(!)734 4851 y(\027)814 4821 y Fs(in)h Fo(W)1019 4776 y Fn(1)p Fp(;)p Fn(2)1006 4832 y(p)r(er)1113 4821 y Fs(\()p Fr(R)1225 4788 y Fp(!)1225 4843 y(\027)1275 4821 y Fs(\).)0 4928 y(W)-8 b(e)32 b(no)m(w)e(c)m(ho)s(ose)h Fr(Z)38 b Fs(to)31 b(b)s(e)f(the)g(set)h(of)g(all)e(v)m(ectors)j(in)d Fq(Z)1981 4896 y Fp(n)2044 4928 y Fr(\\)19 b(R)2201 4896 y Fp(!)2201 4951 y(\027)2252 4928 y Fs(.)40 b(Since)394 5090 y Fr(R)471 5053 y Fp(!)471 5113 y(\027)542 5090 y Fs(+)20 b Fo(m)713 5104 y Fn(1)752 5090 y Fo(K)836 5053 y Fn(\(1\))950 5090 y Fs(+)g Fr(\001)15 b(\001)g(\001)21 b Fs(+)f Fo(m)1338 5104 y Fp(n)p Fl(\000)p Fn(1)1475 5090 y Fo(K)1559 5053 y Fp(n)p Fl(\000)p Fn(1)1716 5090 y Fs(+)g(2)p Fo(\027)6 b(m)1983 5104 y Fp(n)2030 5090 y Fo(K)2114 5053 y Fn(\()p Fp(n)p Fn(\))2231 5090 y Fo(;)252 b Fs(for)31 b Fo(m)2728 5104 y Fn(1)2767 5090 y Fo(;)15 b(:)g(:)g(:)i(;)e(m)3049 5104 y Fp(n)3121 5090 y Fr(2)25 b Fq(Z)0 5242 y Fs(is)32 b(a)i(tiling)d(of)j Fq(R)579 5209 y Fp(n)632 5242 y Fs(,)h(w)m(e)e(ha)m(v)m(e)i(that)f(for)f(an)m(y) h Fo(k)f Fr(2)d Fo(e)1771 5256 y Fn(1)1833 5242 y Fs(+)22 b Fr(Z)40 b Fs(there)33 b(exists)g(a)h(unique)d Fo(\024)p Fs(\()p Fo(k)s Fs(\))h Fr(2)e(Z)40 b Fs(in)32 b(suc)m(h)h(a)0 5350 y(w)m(a)m(y)811 5469 y Fo(k)24 b Fr(\000)c Fo(\024)p Fs(\()p Fo(k)s Fs(\))26 b(=)f Fo(m)1347 5483 y Fn(1)1386 5469 y Fo(K)1470 5432 y Fn(\(1\))1585 5469 y Fs(+)20 b Fr(\001)15 b(\001)g(\001)21 b Fs(+)f Fo(m)1973 5483 y Fp(n)p Fl(\000)p Fn(1)2110 5469 y Fo(K)2194 5432 y Fp(n)p Fl(\000)p Fn(1)2351 5469 y Fs(+)g(2)p Fo(\027)6 b(m)2618 5483 y Fp(n)2665 5469 y Fo(K)2749 5432 y Fn(\()p Fp(n)p Fn(\))p eop %%Page: 16 16 16 15 bop 168 251 a Fn(16)976 b(MA)-6 b(TTEO)23 b(NO)n(V)-8 b(A)n(GA)23 b(AND)g(ENRICO)h(V)-8 b(ALDINOCI)168 450 y Fs(for)30 b(suitable)f Fo(m)726 464 y Fn(1)765 450 y Fo(;)15 b(:)g(:)g(:)i(;)e(m)1047 464 y Fp(n)1119 450 y Fr(2)25 b Fq(Z)h Fs(and,)k(vicev)m(ersa,)h(the)g(set)g Fr(f)p Fo(\024)p Fs(\()p Fo(k)s Fs(\))15 b Fo(;)42 b(k)29 b Fr(2)c Fo(e)2712 464 y Fn(1)2772 450 y Fs(+)19 b Fr(Z)7 b(g)31 b Fs(agrees)h(with)d Fr(Z)7 b Fs(.)168 558 y(Consequen)m(tly)-8 b(,)1037 739 y Fo(v)1081 753 y Fl(Z)1142 739 y Fs(\()p Fo(x)20 b Fs(+)g Fo(e)1382 753 y Fn(1)1422 739 y Fs(\))83 b(=)g(min)1861 638 y Fm(n)1921 739 y Fo(v)s Fs(\()p Fo(x)21 b Fs(+)f Fo(k)s Fs(\))15 b Fo(;)56 b(k)29 b Fr(2)c Fo(e)2552 753 y Fn(1)2612 739 y Fs(+)20 b Fr(Z)2776 638 y Fm(o)1540 938 y Fs(=)83 b(min)1861 837 y Fm(n)1921 938 y Fo(v)s Fs(\()p Fo(x)21 b Fs(+)f Fo(\024)p Fs(\()p Fo(k)s Fs(\)\))15 b Fo(;)58 b(k)28 b Fr(2)d Fo(e)2675 952 y Fn(1)2735 938 y Fs(+)20 b Fr(Z)2899 837 y Fm(o)1540 1137 y Fs(=)83 b(min)1861 1036 y Fm(n)1921 1137 y Fo(v)s Fs(\()p Fo(x)21 b Fs(+)f Fo(h)p Fs(\))15 b Fo(;)57 b(h)25 b Fr(2)g(Z)2587 1036 y Fm(o)1540 1300 y Fs(=)83 b Fo(v)1738 1314 y Fl(Z)1798 1300 y Fs(\()p 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b(next)h(target)i(is)e(to)h (sho)m(w)f(that)168 2275 y(\(41\))1195 b Fo(E)1590 2289 y Fl(R)1650 2271 y Fj(!)1650 2306 y(\027)1699 2275 y Fs(\()p Fo(w)r Fs(\))27 b(=)1964 2189 y Fm(X)1959 2387 y Fp(k)r Fl(2Z)2115 2275 y Fo(E)2182 2290 y Fp(k)r Fn(+)p Fp(Q)2336 2275 y Fs(\()p Fo(w)r Fs(\))168 2546 y(for)f(an)m(y)h Fo(w)h Fr(2)c Fo(W)748 2502 y Fn(1)p Fp(;)p Fn(2)735 2558 y(p)r(er)842 2546 y Fs(\()p Fr(R)954 2513 y Fp(!)954 2568 y(\027)1005 2546 y Fs(\).)40 b(Though)25 b(form)m(ula)h(\(40\))i (is)e(v)m(ery)g(close)h(to)g(common)g(in)m(tuition)d(\(one)k(ma)m(y)f (just)168 2654 y(lo)s(ok)k(at)h(some)g(pa)m(v)m(emen)m(t)h(decorations) f(to)g(get)h(con)m(vinced\),)f(w)m(e)g(pro)m(vide)f(a)g(rigorous)g(pro) s(of)g(of)g(it)h(\(the)168 2762 y(exp)s(ert)27 b(reader)h(go)s(es)g (straigh)m(t)g(to)g(\(46\))r(\).)40 b(T)-8 b(o)28 b(c)m(hec)m(k)h (\(41\))r(,)f(w)m(e)g(\014rst)f(demonstrate)h(that)g(for)g(an)m(y)g Fo(\030)h Fr(2)c(R)3779 2729 y Fp(!)3779 2784 y(\027)168 2870 y Fs(there)30 b(exist)g Fo(k)f Fr(2)c(Z)37 b Fs(and)30 b Fo(`)1095 2884 y Fn(1)1134 2870 y Fo(;)15 b(:)g(:)g(:)i(;)e(`)1374 2884 y Fp(n)1446 2870 y Fr(2)25 b Fq(Z)h Fs(in)j(suc)m(h)h(a)h(w)m(a)m (y)g(that)168 3112 y(\(42\))948 b Fo(\030)24 b Fr(\000)c Fo(k)j Fs(+)1592 2998 y Fp(n)p Fl(\000)p Fn(1)1593 3026 y Fm(X)1597 3221 y Fp(j)t Fn(=1)1740 3112 y Fo(`)1778 3126 y Fp(j)1815 3112 y Fo(K)1899 3074 y Fn(\()p Fp(j)t Fn(\))2010 3112 y Fs(+)d(2)p Fo(\027)6 b(`)2235 3126 y Fp(n)2282 3112 y Fo(K)2366 3074 y Fn(\()p Fp(n)p Fn(\))2508 3112 y Fr(2)40 b Fo(Q)15 b(:)168 3370 y Fs(T)-8 b(o)31 b(con\014rm)e(this,)h(let)g([)p Fr(\001)p Fs(])h(denote)g(the)f(in)m (teger)h(part)f(of)h(a)g(real)f(n)m(um)m(b)s(er)f(and)1554 3557 y([)p Fo(\030)t Fs(])41 b(:=)1825 3456 y Fm(\020)1879 3557 y Fs([)p Fo(\030)1944 3571 y Fn(1)1984 3557 y Fs(])p Fo(;)15 b(:)g(:)g(:)i(;)e Fs([)p Fo(\030)2276 3571 y Fp(n)2323 3557 y Fs(])2348 3456 y Fm(\021)2418 3557 y Fo(:)168 3738 y Fs(Let)31 b Fo(`)369 3752 y Fp(j)435 3738 y Fs(b)s(e)f(the)h(unique)d(in)m(teger)j(for)f(whic)m(h)1150 3968 y([)p Fo(\030)t Fs(])21 b Fr(\001)1366 3906 y Fo(K)1450 3873 y Fn(\()p Fp(j)t Fn(\))p 1321 3947 266 4 v 1321 4034 a Fr(j)p Fo(K)1430 4008 y Fn(\()p Fp(j)t Fn(\))1521 4034 y Fr(j)1546 4008 y Fn(2)1616 3968 y Fs(+)f Fo(`)1745 3982 y Fp(j)1806 3968 y Fr(2)25 b Fs([0)p Fo(;)15 b Fs(1\))73 b(for)30 b(1)c Fr(\024)f Fo(j)31 b Fr(\024)25 b Fo(n)19 b Fr(\000)h Fs(1)31 b(and)1150 4236 y([)p Fo(\030)t Fs(])21 b Fr(\001)1421 4175 y Fo(K)1505 4142 y Fn(\()p Fp(n)p Fn(\))p 1321 4215 387 4 v 1321 4302 a Fs(2)p Fo(\027)g Fr(j)p Fo(K)1541 4276 y Fn(\()p Fp(n)p Fn(\))1643 4302 y Fr(j)1668 4276 y Fn(2)1738 4236 y Fs(+)e Fo(`)1866 4250 y Fp(n)1939 4236 y Fr(2)2024 4135 y Fm(h)2088 4236 y Fr(\000)2188 4175 y Fs(1)p 2188 4215 46 4 v 2188 4299 a(2)2244 4236 y Fo(;)2294 4175 y Fs(1)p 2294 4215 V 2294 4299 a(2)2350 4135 y Fm(\021)2419 4236 y Fo(:)168 4446 y Fs(Let)168 4671 y(\(43\))1011 b Fo(k)29 b Fs(:=)c([)p Fo(\030)t Fs(])c(+)1742 4557 y Fp(n)p Fl(\000)p Fn(1)1743 4585 y Fm(X)1747 4780 y Fp(j)t Fn(=1)1890 4671 y Fo(`)1928 4685 y Fp(j)1964 4671 y Fo(K)2048 4633 y Fn(\()p Fp(j)t Fn(\))2160 4671 y Fs(+)e(2)p Fo(\027)6 b(`)2384 4685 y Fp(n)2431 4671 y Fo(K)2515 4633 y Fn(\()p Fp(n)p Fn(\))2632 4671 y Fo(:)168 4924 y Fs(Then,)29 b Fo(k)g Fr(2)c Fq(Z)657 4891 y Fp(n)730 4924 y Fs(and,)30 b(moreo)m(v)m(er,)1169 5145 y Fo(k)24 b Fr(\001)1320 5084 y Fo(K)1404 5051 y Fn(\()p Fp(j)t Fn(\))p 1295 5124 226 4 v 1295 5211 a Fr(j)p Fo(K)1404 5185 y Fn(\()p Fp(j)t Fn(\))1495 5211 y Fr(j)1556 5145 y(2)1642 5044 y Fm(h)1685 5145 y Fs(0)p Fo(;)31 b Fr(j)p Fo(K)1895 5108 y Fn(\()p Fp(j)t Fn(\))1986 5145 y Fr(j)2011 5044 y Fm(\021)2136 5145 y Fs(for)f(1)c Fr(\024)f Fo(j)31 b Fr(\024)25 b Fo(n)20 b Fr(\000)g Fs(1)30 b(and)1169 5414 y Fo(k)24 b Fr(\001)1320 5352 y Fo(K)1404 5319 y Fn(\()p Fp(n)p Fn(\))p 1295 5393 237 4 v 1295 5480 a Fr(j)p Fo(K)1404 5454 y Fn(\()p Fp(n)p Fn(\))1506 5480 y Fr(j)1566 5414 y(2)1652 5313 y Fm(h)1715 5414 y Fr(\000)c Fo(\027)h Fr(j)p Fo(K)1981 5376 y Fn(\()p Fp(n)p Fn(\))2083 5414 y Fr(j)p Fo(;)31 b(\027)20 b Fr(j)p Fo(K)2338 5376 y Fn(\()p Fp(n)p Fn(\))2440 5414 y Fr(j)2465 5313 y Fm(\021)2535 5414 y Fo(;)p eop %%Page: 17 17 17 16 bop 587 251 a Fn(THE)30 b(GEOMETR)-6 b(Y)31 b(OF)f(MESOSCOPIC)h (PHASE)f(TRANSITION)h(INTERF)-8 b(A)n(CES)517 b(17)0 450 y Fs(hence)32 b Fo(k)g Fr(2)27 b(R)498 417 y Fp(!)498 473 y(\027)581 450 y Fs(and)k(so)i Fo(k)e Fr(2)d(Z)7 b Fs(.)46 b(Moreo)m(v)m(er,)34 b(the)f(v)m(ector)g(on)f(the)h(left)e (hand)g(side)g(of)40 b(\(42\))33 b(agrees)h(with)0 558 y Fo(\030)14 b Fr(\000)c Fs([)p Fo(\030)t Fs(],)27 b(due)e(to)h(\(43\)) q(,)h(and)e(so)g(it)g(has)g(co)s(ordinates)g(lying)f(in)g([0)p Fo(;)15 b Fs(1\),)28 b(th)m(us)d(completing)g(the)g(pro)s(of)g(of)33 b(\(42\))q(.)0 666 y(W)-8 b(e)32 b(no)m(w)e(denote)h Fr(\030)710 680 y Fp(\027)783 666 y Fs(the)g(equiv)-5 b(alence)29 b(relation)h(stating)h(that)g Fo(x)25 b Fr(\030)2402 680 y Fp(\027)2470 666 y Fo(y)33 b Fs(if)c(and)h(only)g(if)1227 924 y Fo(x)20 b Fr(\000)g Fo(y)28 b Fs(=)1559 810 y Fp(n)p Fl(\000)p Fn(1)1560 838 y Fm(X)1564 1033 y Fp(j)t Fn(=1)1707 924 y Fo(`)1745 938 y Fp(j)1782 924 y Fo(K)1866 886 y Fn(\()p Fp(j)t Fn(\))1977 924 y Fs(+)20 b(2)p Fo(\027)6 b(`)2202 938 y Fp(n)2249 924 y Fo(K)2333 886 y Fn(\()p Fp(n)p Fn(\))0 1193 y Fs(for)30 b(some)h Fo(`)405 1207 y Fn(1)444 1193 y Fo(;)15 b(:)g(:)g(:)i(;)e(`)684 1207 y Fp(n)756 1193 y Fr(2)25 b Fq(Z)p Fs(.)36 b(Let)31 b Fo(\031)1183 1207 y Fp(\027)1257 1193 y Fs(b)s(e)e(the)i(natural)e(pro) 5 b(jection)30 b(induced)f(b)m(y)h Fr(\030)2823 1207 y Fp(\027)2866 1193 y Fs(.)41 b(Let)1461 1381 y Fr(R)f Fs(:=)1735 1295 y Fm([)1715 1492 y Fp(k)r Fl(2Z)1856 1381 y Fs(\()p Fo(k)24 b Fs(+)c Fo(Q)p Fs(\))15 b Fo(:)0 1649 y Fs(Then,)29 b(\(42\))j(states)f(that)f Fo(\031)960 1663 y Fp(\027)1003 1649 y Fs(\()p Fr(R)p Fs(\))c(=)f Fr(R)1349 1616 y Fp(!)1349 1671 y(\027)1399 1649 y Fo(=)10 b Fr(\030)1525 1663 y Fp(\027)1599 1649 y Fs(\(and)29 b(w)m(e)i(ma)m(y)f(iden)m(tify)e(the)i(latter)h(with)d Fr(R)3157 1616 y Fp(!)3157 1671 y(\027)3237 1649 y Fs(itself)7 b(\).)40 b(W)-8 b(e)0 1757 y(no)m(w)30 b(sho)m(w)h(that)0 1931 y(\(44\))1115 b Fo(\031)1327 1945 y Fp(\027)1400 1931 y Fs(is,)30 b(in)f(fact,)i(injectiv)m(e)f(on)h Fr(R)p Fs(.)0 2105 y(Indeed,)c(assume)g(that)g Fo(\031)874 2119 y Fp(\027)917 2105 y Fs(\()p Fo(x)p Fs(\))f(=)f Fo(\031)1213 2119 y Fp(\027)1256 2105 y Fs(\()p Fo(x)1343 2072 y Fl(0)1367 2105 y Fs(\))i(with)e Fo(x;)31 b(x)1792 2072 y Fl(0)1840 2105 y Fr(2)25 b(R)p Fs(.)40 b(Then,)26 b Fo(x)f Fs(=)g Fo(q)16 b Fs(+)d Fo(k)s Fs(,)28 b Fo(x)2796 2072 y Fl(0)2845 2105 y Fs(=)d Fo(q)2985 2072 y Fl(0)3021 2105 y Fs(+)13 b Fo(k)3155 2072 y Fl(0)3205 2105 y Fs(with)26 b Fo(q)s(;)k(q)3552 2072 y Fl(0)3601 2105 y Fr(2)0 2213 y Fo(Q)p Fs(,)g Fo(k)s(;)h(k)283 2180 y Fl(0)332 2213 y Fr(2)25 b(Z)37 b Fs(and)1213 2434 y Fo(x)20 b Fr(\000)g Fo(x)1428 2396 y Fl(0)1477 2434 y Fs(=)1573 2320 y Fp(n)p Fl(\000)p Fn(1)1574 2347 y Fm(X)1578 2543 y Fp(j)t Fn(=1)1721 2434 y Fo(`)1759 2448 y Fp(j)1795 2434 y Fo(K)1879 2396 y Fn(\()p Fp(j)t Fn(\))1991 2434 y Fs(+)g(2)p Fo(\027)6 b(`)2216 2448 y Fp(n)2263 2434 y Fo(K)2347 2396 y Fn(\()p Fp(n)p Fn(\))0 2691 y Fs(for)27 b(some)g Fo(`)398 2705 y Fn(1)438 2691 y Fo(;)15 b(:)g(:)g(:)h(;)f(`)677 2705 y Fp(n)750 2691 y Fr(2)25 b Fq(Z)p Fs(.)35 b(In)26 b(particular,)g Fo(q)17 b Fr(\000)d Fo(q)1700 2658 y Fl(0)1747 2691 y Fr(2)25 b Fq(Z)1898 2658 y Fp(n)1968 2691 y Fs(and)h Fo(q)17 b Fr(\001)d Fo(e)2280 2706 y Fp(k)2323 2691 y Fo(;)30 b(q)2422 2658 y Fl(0)2459 2691 y Fr(\001)14 b Fo(e)2540 2706 y Fp(k)2608 2691 y Fr(2)25 b Fs([0)p Fo(;)15 b Fs(1\),)30 b(for)d(an)m(y)h(1)d Fr(\024)g Fo(k)k Fr(\024)c Fo(n)p Fs(.)0 2798 y(Th)m(us,)30 b(\()p Fo(q)23 b Fr(\000)d Fo(q)490 2765 y Fl(0)513 2798 y Fs(\))h Fr(\001)f Fo(e)656 2813 y Fp(k)724 2798 y Fr(2)25 b Fq(Z)16 b Fr(\\)k Fs(\()p Fr(\000)p Fs(1)p Fo(;)15 b Fs(1\))27 b(=)e Fr(f)p Fs(0)p Fr(g)p Fs(,)32 b(and)d(so)i Fo(q)d Fs(=)d Fo(q)2055 2765 y Fl(0)2078 2798 y Fs(.)41 b(Accordingly)-8 b(,)0 3062 y(\(45\))1035 b Fo(k)23 b Fr(\000)d Fo(k)1406 3024 y Fl(0)1455 3062 y Fs(=)1551 2948 y Fp(n)p Fl(\000)p Fn(1)1552 2975 y Fm(X)1556 3171 y Fp(j)t Fn(=1)1699 3062 y Fo(`)1737 3076 y Fp(j)1773 3062 y Fo(K)1857 3024 y Fn(\()p Fp(j)t Fn(\))1969 3062 y Fs(+)g(2)p Fo(\027)6 b(`)2194 3076 y Fp(n)2241 3062 y Fo(K)2325 3024 y Fn(\()p Fp(n)p Fn(\))2441 3062 y Fo(:)0 3331 y Fs(Since)29 b Fo(k)g Fr(2)c(Z)7 b Fs(,)30 b(w)m(e)h(ha)m(v)m(e)h (that)1001 3567 y Fo(k)24 b Fr(\001)1152 3506 y Fo(K)1236 3473 y Fn(\()p Fp(j)t Fn(\))p 1127 3546 226 4 v 1127 3634 a Fr(j)p Fo(K)1236 3607 y Fn(\()p Fp(j)t Fn(\))1328 3634 y Fr(j)1388 3567 y(2)1474 3466 y Fm(h)1517 3567 y Fs(0)p Fo(;)31 b Fr(j)p Fo(K)1727 3530 y Fn(\()p Fp(j)t Fn(\))1818 3567 y Fr(j)1843 3466 y Fm(\021)1968 3567 y Fs(for)g(1)25 b Fr(\024)g Fo(j)31 b Fr(\024)25 b Fo(n)20 b Fr(\000)g Fs(1)31 b(and)1001 3836 y Fo(k)24 b Fr(\001)1152 3774 y Fo(K)1236 3741 y Fn(\()p Fp(n)p Fn(\))p 1127 3815 237 4 v 1127 3902 a Fr(j)p Fo(K)1236 3876 y Fn(\()p Fp(n)p Fn(\))1338 3902 y Fr(j)1399 3836 y(2)1484 3735 y Fm(h)1548 3836 y Fr(\000)19 b Fo(\027)i Fr(j)p Fo(K)1813 3798 y Fn(\()p Fp(n)p Fn(\))1915 3836 y Fr(j)p Fo(;)31 b(\027)21 b Fr(j)p Fo(K)2171 3798 y Fn(\()p Fp(n)p Fn(\))2273 3836 y Fr(j)2298 3735 y Fm(\021)2367 3836 y Fo(;)0 4070 y Fs(for)43 b(an)m(y)h(1)j Fr(\024)f Fo(j)53 b Fr(\024)46 b Fo(n)d Fs(\(and)g(the)h(same)f(holds)f(for)h Fo(k)1939 4037 y Fl(0)1963 4070 y Fs(\).)80 b(This)41 b(and)i(\(45\))i(yield)c (that)j Fo(`)3198 4084 y Fp(j)3281 4070 y Fr(2)j Fs(\()p Fr(\000)p Fs(1)p Fo(;)15 b Fs(1\))0 4178 y(for)30 b(1)c Fr(\024)f Fo(j)31 b Fr(\024)25 b Fo(n)p Fs(,)30 b(so)g Fo(`)729 4192 y Fp(j)791 4178 y Fs(=)25 b(0.)41 b(Consequen)m(tly)-8 b(,)30 b Fo(x)c Fs(=)f Fo(x)1805 4145 y Fl(0)1828 4178 y Fs(,)30 b(pro)m(ving)h(\(44\))r(.)0 4299 y(As)f(a)h(consequence)g(of) 38 b(\(44\))q(,)31 b(w)m(e)g(ha)m(v)m(e)g(that,)h(if)d Fo(w)f Fr(2)d Fo(W)1978 4255 y Fn(1)p Fp(;)p Fn(2)1965 4311 y(p)r(er)2071 4299 y Fs(\()p Fr(R)2183 4266 y Fp(!)2183 4322 y(\027)2234 4299 y Fs(\),)31 b(then)609 4494 y Fo(E)676 4508 y Fl(R)736 4490 y Fj(!)736 4525 y(\027)786 4494 y Fs(\()p Fo(w)r Fs(\))26 b(=)f Fo(E)1112 4513 y Fl(R)1172 4494 y Fj(!)1172 4530 y(\027)1217 4513 y Fp(=)-12 b Fl(\030)1295 4521 y Fj(\027)1338 4494 y Fs(\()p Fo(w)r Fs(\))27 b(=)e Fo(E)1665 4513 y Fl(R)p Fp(=)-12 b Fl(\030)1803 4521 y Fj(\027)1846 4494 y Fs(\()p Fo(w)r Fs(\))27 b(=)e Fo(E)2173 4508 y Fl(R)2237 4494 y Fs(\()p Fo(w)r Fs(\))i(=)2502 4408 y Fm(X)2497 4606 y Fp(k)r Fl(2Z)2653 4494 y Fo(E)2720 4509 y Fp(k)r Fn(+)p Fp(Q)2873 4494 y Fs(\()p Fo(w)r Fs(\))15 b Fo(;)0 4762 y Fs(that)31 b(is,)f(\(40\))q(.)0 4880 y(Then,)g(using)g(\(40\))i(and)d(the)i(p)s(erio)s(dicit)m(y)d (relations)h(in)g Fo(W)2056 4836 y Fn(1)p Fp(;)p Fn(2)2043 4892 y(p)r(er)2150 4880 y Fs(\()p Fo(Q)p Fs(\),)i(w)m(e)g(gather)g (that)0 5075 y(\(46\))145 b Fo(E)372 5089 y Fl(R)432 5070 y Fj(!)432 5106 y(\027)481 5075 y Fs(\()p Fo(u)p Fs(\))26 b(=)730 4989 y Fm(X)725 5186 y Fp(k)r Fl(2Z)882 5075 y Fo(E)949 5090 y Fp(k)r Fn(+)p Fp(Q)1102 5075 y Fs(\()p Fo(u)p Fs(\))g(=)1351 4989 y Fm(X)1346 5186 y Fp(k)r Fl(2Z)1503 5075 y Fo(E)1570 5089 y Fp(Q)1630 5075 y Fs(\()p Fo(u)p Fs(\))g Fr(\024)1879 4989 y Fm(X)1874 5186 y Fp(k)r Fl(2Z)2030 5075 y Fo(E)2097 5089 y Fp(Q)2157 5075 y Fs(\()p Fo(v)2236 5089 y Fl(Z)2297 5075 y Fs(\))f(=)2458 4989 y Fm(X)2453 5186 y Fp(k)r Fl(2Z)2610 5075 y Fo(E)2677 5090 y Fp(k)r Fn(+)p Fp(Q)2830 5075 y Fs(\()p Fo(v)2909 5089 y Fl(Z)2970 5075 y Fs(\))g(=)g Fo(E)3193 5089 y Fl(R)3253 5070 y Fj(!)3253 5106 y(\027)3303 5075 y Fs(\()p Fo(v)3382 5089 y Fl(Z)3442 5075 y Fs(\))15 b Fo(:)0 5362 y Fs(W)-8 b(e)23 b(infer)d(from)h(this)f(and)h(\(40\))j(that)e Fo(E)1330 5376 y Fl(R)1390 5357 y Fj(!)1390 5392 y(\027)1439 5362 y Fs(\()p Fo(u)p Fs(\))k Fr(\024)f Fo(E)1750 5376 y Fl(R)1810 5357 y Fj(!)1810 5392 y(\027)1859 5362 y Fs(\()p Fo(v)s Fs(\).)39 b(Since)21 b Fo(W)2368 5317 y Fn(1)p Fp(;)p Fn(2)2355 5373 y(p)r(er)2461 5362 y Fs(\()p Fo(Q)p Fs(\))26 b Fr(\022)f Fo(W)2824 5317 y Fn(1)p Fp(;)p Fn(2)2811 5373 y(p)r(er)2918 5362 y Fs(\()p Fr(R)3030 5329 y Fp(!)3030 5384 y(\027)3081 5362 y Fs(\),)f(w)m(e)e(ob)m(viously) 0 5469 y(ha)m(v)m(e)32 b(also)e(the)g(rev)m(erse)i(inequalit)m(y)-8 b(.)39 b(This)29 b(yields)f(the)j(pro)s(of)e(of)38 b(\(39\))r(,)30 b(as)h(desired.)687 b Ff(\003)p eop %%Page: 18 18 18 17 bop 168 251 a Fn(18)976 b(MA)-6 b(TTEO)23 b(NO)n(V)-8 b(A)n(GA)23 b(AND)g(ENRICO)h(V)-8 b(ALDINOCI)168 454 y Fs(W)g(e)36 b(no)m(w)e(address)g(the)h(problem)e(of)i(comparing)f (the)h(energy)g(of)g(the)g(minimizers)c(in)j Fo(W)3326 410 y Fn(1)p Fp(;)p Fn(2)3313 466 y(p)r(er)3420 454 y Fs(\()p Fr(R)3532 421 y Fp(!)3532 477 y(\027)3582 454 y Fs(\))h(with)168 563 y(the)h(ones)g(in)f Fo(W)749 530 y Fn(1)p Fp(;)p Fn(2)843 563 y Fs(\()p Fr(R)955 530 y Fp(!)955 585 y(\027)1006 563 y Fo(=)20 b Fr(\030)p Fs(\),)37 b(where)f Fr(\030)g Fs(is)f(the)h(equiv)-5 b(alence)35 b(relation)h(in)m(tro)s(duced)e(on)i(page)h(14,)h(that)168 671 y(is,)30 b(w)m(e)h(estimate)h(ho)m(w)f(m)m(uc)m(h)g(the)g(p)s(erio) s(dicit)m(y)e(conditions)g(in)h(the)h(direction)e(of)j Fo(!)h Fs(a\013ect)g(the)e(minimal)168 779 y(energy)-8 b(.)48 b(F)-8 b(or)33 b(this,)f(w)m(e)h(will)d(pro)m(v)m(e)k(an)e (existence)h(result)f(for)g Fo(W)2434 746 y Fn(1)p Fp(;)p Fn(2)2528 779 y Fs(\()p Fr(R)2640 746 y Fp(!)2640 801 y(\027)2691 779 y Fo(=)14 b Fr(\030)p Fs(\)-minimizers)30 b(in)h(Lemma)i(9)168 887 y(b)s(elo)m(w)c(and)h(then)g(p)s(erform)f(the) i(necessary)f(energy)h(estimates)g(in)e(Lemma)i(10.)168 1015 y Fw(Lemma)k(9.)43 b Fi(The)34 b(functional)h Fo(E)1348 1029 y Fl(R)1408 1010 y Fj(!)1408 1046 y(\027)1492 1015 y Fi(attains)h(the)f(minimum)g(in)f Fo(W)2574 982 y Fn(1)p Fp(;)p Fn(2)2668 1015 y Fs(\()p Fr(R)2780 982 y Fp(!)2780 1038 y(\027)2831 1015 y Fo(=)13 b Fr(\030)p Fs(\))35 b Fi(at)g(a)g(suitable)f Fo(u)3605 1029 y Fp(\027)3683 1015 y Fi(sat-)168 1123 y(isfying)168 1272 y Fs(\(47\))1283 b Fr(k)p Fo(u)1708 1286 y Fp(\027)1751 1272 y Fr(k)1796 1292 y Fp(C)1851 1273 y Fe(1)1886 1292 y Fn(\()p Fl(R)1973 1269 y Fj(!)1973 1313 y(\027)s Fd(\000)p Fe(1)2091 1292 y Fn(\))2163 1272 y Fr(\024)40 b Fo(C)22 b(;)168 1433 y Fi(for)33 b(a)g(suitable)g(universal)g Fo(C)f(>)25 b Fs(0)p Fi(.)168 1595 y(Pr)-5 b(o)g(of.)43 b Fs(By)c(p)s(erforming)d (a)j(standard)f(minimization)e(argumen)m(t)j(as)g(in)e(form)m(ulas)h (\(34\))r({\(37\))r(,)j(w)m(e)e(get)168 1703 y(the)45 b(existence)h(of)g(a)g(minimizer)c Fo(u)1442 1717 y Fp(\027)1536 1703 y Fr(2)50 b Fo(W)1746 1670 y Fn(1)p Fp(;)p Fn(2)1840 1703 y Fs(\()p Fr(R)1952 1670 y Fp(!)1952 1726 y(\027)2003 1703 y Fo(=)35 b Fr(\030)p Fs(\))46 b(whic)m(h)e(is)g(p)s(oin)m(t)m (wise)g(uniformly)f(b)s(ounded.)168 1811 y(Then,)30 b(\(47\))j(is)e(a)h (consequence)g(of)f(the)h(in)m(terior)e(elliptic)f(regularit)m(y)i (theory)h(\(see,)g(e.g.)45 b(Theorem)32 b(3.13)168 1919 y(in)d([HL97)q(]\).)3158 b Ff(\003)168 2082 y Fw(Lemma)42 b(10.)k Fi(L)-5 b(et)40 b Fo(\027)45 b Fr(\025)38 b Fs(4)p Fi(.)64 b(L)-5 b(et)41 b Fo(u)1444 2096 y Fp(\027)1527 2082 y Fi(a)f(minimizer)h(for)f Fo(E)2273 2096 y Fl(R)2333 2077 y Fj(!)2333 2112 y(\027)2423 2082 y Fi(in)g Fo(W)2641 2049 y Fn(1)p Fp(;)p Fn(2)2734 2082 y Fs(\()p Fr(R)2846 2049 y Fp(!)2846 2104 y(\027)2897 2082 y Fo(=)24 b Fr(\030)p Fs(\))p Fi(,)42 b(as)e(c)-5 b(onstructe)g(d)42 b(in)168 2189 y(L)-5 b(emma)34 b(9.)42 b(Then,)1424 2301 y Fo(E)1491 2315 y Fl(R)1551 2296 y Fj(!)1551 2332 y(\027)1600 2301 y Fs(\()p Fo(u)1687 2315 y Fn(+)1747 2301 y Fs(\))e Fr(\024)h Fo(E)2001 2315 y Fl(R)2061 2296 y Fj(!)2061 2332 y(\027)2110 2301 y Fs(\()p Fo(u)2197 2315 y Fp(\027)2240 2301 y Fs(\))36 b(+)f Fo(C)2482 2315 y Fp(!)2548 2301 y Fo(;)168 2431 y Fi(for)44 b(a)g(suitable)g Fo(C)820 2445 y Fp(!)915 2431 y Fo(>)h Fs(0)f Fi(p)-5 b(ossibly)45 b(dep)-5 b(ending)45 b(on)f Fo(!)s Fi(,)i Fo(n)d Fi(and)h(on)g(the)g(structur)-5 b(al)45 b(c)-5 b(onstants)46 b(of)e(the)168 2539 y(pr)-5 b(oblem)34 b(\(but)f(indep)-5 b(endent)34 b(of)f Fo(\027)6 b Fi(\).)168 2702 y(Pr)-5 b(o)g(of.)43 b Fs(Let)d Fo(\034)50 b Fs(b)s(e)39 b(a)h(smo)s(oth)f(cut-o\013)i(functions,)g(so)f(that)g(0) i Fr(\024)e Fo(\034)51 b Fr(\024)41 b Fs(1,)h Fr(jr)p Fo(\034)10 b Fr(j)41 b(\024)g Fs(10,)i Fo(\034)10 b Fs(\()p Fo(x)p Fs(\))42 b(=)e(1)g(for)168 2809 y(an)m(y)30 b Fo(x)25 b Fr(2)g(R)579 2776 y Fp(!)579 2834 y(\027)t Fl(\000)p Fn(2)743 2809 y Fs(and)k Fo(\034)10 b Fs(\()p Fo(x)p Fs(\))26 b(=)f(0)30 b(for)g(an)m(y)h Fo(x)25 b Fr(2)g(R)1839 2776 y Fp(!)1839 2832 y(\027)1909 2809 y Fr(n)20 b(R)2051 2776 y Fp(!)2051 2834 y(\027)t Fl(\000)p Fn(1)2184 2809 y Fs(.)41 b(Let)30 b Fo(v)2456 2823 y Fp(\027)2525 2809 y Fs(:=)25 b Fo(\034)10 b(u)2748 2823 y Fp(\027)2791 2809 y Fs(.)41 b(By)30 b(construction,)g Fo(v)3589 2823 y Fp(\027)3662 2809 y Fs(ma)m(y)168 2936 y(b)s(e)i(extended)h(p)s(erio)s(dically)d(in)i(the)h Fo(!)s Fs(-direction)f(outside)g Fr(R)2312 2903 y Fp(!)2312 2959 y(\027)2363 2936 y Fs(,)i(that)f(is,)h(there)f(exists)j(~)-48 b Fo(v)3275 2950 y Fp(\027)3348 2936 y Fr(2)29 b Fo(W)3537 2892 y Fn(1)p Fp(;)p Fn(2)3524 2948 y(p)r(er)3631 2936 y Fs(\()p Fr(R)3743 2903 y Fp(!)3743 2959 y(\027)3794 2936 y Fs(\))168 3044 y(so)h(that)35 b(~)-49 b Fo(v)520 3058 y Fp(\027)589 3044 y Fs(=)25 b Fo(v)729 3058 y Fp(\027)802 3044 y Fs(in)k Fr(R)985 3011 y Fp(!)985 3066 y(\027)1036 3044 y Fs(.)40 b(As)31 b(a)g(consequence,)168 3193 y(\(48\))985 b Fo(E)1380 3207 y Fl(R)1440 3188 y Fj(!)1440 3224 y(\027)1490 3193 y Fs(\()p Fo(v)1569 3207 y Fp(\027)1612 3193 y Fs(\))41 b(=)f Fo(E)1866 3207 y Fl(R)1926 3188 y Fj(!)1926 3224 y(\027)1975 3193 y Fs(\()s(~)-48 b Fo(v)2054 3207 y Fp(\027)2098 3193 y Fs(\))41 b Fr(\025)f Fo(E)2352 3207 y Fl(R)2412 3188 y Fj(!)2412 3224 y(\027)2461 3193 y Fs(\()p Fo(u)2548 3207 y Fn(+)2608 3193 y Fs(\))15 b Fo(;)168 3342 y Fs(thanks)30 b(to)h(Lemma)f(8.)168 3450 y(On)f(the)i(other)f(hand,)g(recalling)g (\(47\))r(,)1589 3600 y Fo(E)1656 3618 y Fl(R)1716 3599 y 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Fn(+)1492 5444 y Fs(\()p Fo(x)p Fs(\))p Fr(j)1639 5406 y Fn(2)1699 5444 y Fr(\000)g Fo(F)13 b Fs(\()p Fo(x;)i(u)2040 5458 y Fn(+)2100 5444 y Fs(\()p Fo(x)p Fs(\)\))21 b Fr(\000)f Fo(H)7 b Fs(\()p Fo(x)p Fs(\))15 b Fo(u)2641 5458 y Fn(+)2701 5444 y Fs(\()p Fo(x)p Fs(\))2823 5343 y Fm(\021)2893 5444 y Fo(dx)g(:)168 5315 y Fs(\(49\))p eop %%Page: 19 19 19 18 bop 587 251 a Fn(THE)30 b(GEOMETR)-6 b(Y)31 b(OF)f(MESOSCOPIC)h (PHASE)f(TRANSITION)h(INTERF)-8 b(A)n(CES)517 b(19)0 450 y Fs(W)-8 b(e)34 b(consider)d(the)i(space)h(of)e(p)s(erio)s(dic)e (\(with)i(resp)s(ect)h(to)g(the)g(iden)m(ti\014cation)e Fr(\030)p Fs(\))i(functions)e(for)i(whic)m(h)0 558 y(the)e(ab)s(o)m(v)m (e)g(functional)e(is)g(w)m(ell-de\014ned,)g(that)i(is,)f(w)m(e)h (de\014ne)518 727 y Fr(D)588 741 y Fp(!)679 727 y Fs(:=)816 626 y Fm(n)876 727 y Fo(u)26 b Fr(2)e Fo(W)1138 683 y Fn(1)p Fp(;)p Fn(2)1125 756 y(lo)r(c)1232 727 y Fs(\()p Fq(R)1327 689 y Fp(n)1380 727 y Fo(=)10 b Fr(\030)p Fs(\))72 b(for)30 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Fi(,)33 b(and)168 608 y Fs(\(55\))1088 b Fr(j)p Fo(w)23 b Fr(\000)d Fo(u)1672 622 y Fn(+)1731 608 y Fr(j)26 b(\024)f Fs(2)p Fo(\016)1963 622 y Fn(0)2003 608 y Fo(e)2045 570 y Fl(\000)p Fp(c)2131 579 y Fe(1)2165 570 y Fn(\()p Fp(x)p Fl(\001)p Fp(!)r(=)p Fl(j)p Fp(!)r Fl(j\000)p Fp(M)7 b Fn(\))168 755 y Fi(for)33 b(any)g Fo(x)g Fi(so)g(that)h Fo(x)20 b Fr(\001)g Fo(!)s(=)p Fr(j)p Fo(!)s Fr(j)26 b(\024)f(\000)p Fo(M)10 b Fi(,)32 b(for)i(a)f(suitable)f(universal)i Fo(c)2540 769 y Fn(1)2605 755 y Fo(>)25 b Fs(0)p Fi(.)168 863 y(Mor)-5 b(e)g(over,)168 1015 y Fs(\(56\))1091 b Fr(jr)p Fs(\()p Fo(w)23 b Fr(\000)d Fo(u)1786 1029 y Fl(\006)1845 1015 y Fs(\))p Fr(j)26 b(\024)f Fo(c)2066 1029 y Fn(2)2106 1015 y Fo(e)2148 978 y Fl(\000)p Fp(c)2234 987 y Fe(3)2268 978 y Fl(j)p Fp(x)p Fl(\001)p Fp(!)r(=)p Fl(j)p Fp(!)r Fl(jj)2553 1015 y Fo(;)168 1163 y Fi(for)33 b(suitable)g(universal)g Fo(c)1071 1177 y Fn(2)1111 1163 y Fo(;)d(c)1205 1177 y Fn(3)1270 1163 y Fo(>)25 b Fs(0)168 1322 y Fi(Pr)-5 b(o)g(of.)43 b Fs(W)-8 b(e)39 b(only)e(pro)m(v)m(e)1034 1287 y Fn(2)1108 1322 y Fs(the)h(claim)e(in)h(\(54\))r(,)i(since)f(the) g(one)g(in)e(\(55\))k(is)d(analogous)h(and)f(then)h(\(56\))168 1430 y(follo)m(ws)29 b(from)h(elliptic)e(estimates)j(\(see,)h(e.g.,)g (Theorem)e(8.32)i(of)e([GT83)q(]\).)42 b(Let)30 b Fo(v)f Fs(:=)c Fr(\006)p Fo(w)e Fr(\007)d Fo(u)3496 1444 y Fl(\000)3585 1430 y Fs(and)1085 1641 y Fo(\015)5 b Fs(\()p Fo(x)p Fs(\))26 b(:=)1406 1517 y Fm(Z)1497 1543 y Fn(1)1457 1723 y(0)1552 1641 y Fo(F)1610 1655 y Fp(uu)1696 1540 y Fm(\020)1750 1641 y Fo(x;)15 b(\034)10 b(w)r Fs(\()p Fo(x)p Fs(\))22 b(+)e(\(1)h Fr(\000)f Fo(\034)10 b Fs(\))p Fo(u)2523 1655 y Fl(\000)2582 1641 y Fs(\()p Fo(x)p Fs(\))2704 1540 y Fm(\021)2774 1641 y Fo(d\034)26 b(:)168 1841 y Fs(Note)37 b(that,)h(b)m(y)g(\(53\))q(,)g(if)d Fo(u)g Fr(2)f Fs([)p Fr(\000)p Fs(1)25 b Fr(\000)e Fs(2)p Fo(\016)1596 1855 y Fn(0)1637 1841 y Fo(;)15 b Fr(\000)p Fs(1)24 b(+)g(2)p Fo(\016)1997 1855 y Fn(0)2037 1841 y Fs(],)38 b(w)m(e)f(ha)m(v)m(e)g (that)g Fo(F)2741 1855 y Fp(uu)2827 1841 y Fs(\()p Fo(x;)15 b(u)p Fs(\))36 b Fr(2)e Fs([)p Fo(C)q(;)15 b(C)3375 1808 y Fl(0)3399 1841 y Fs(],)38 b(for)e(some)168 1949 y Fo(C)240 1916 y Fl(0)290 1949 y Fr(\025)28 b Fo(C)34 b(>)28 b Fs(0,)33 b(as)f(long)f(as)h Fo(\016)1154 1963 y Fn(0)1226 1949 y Fs(is)f(small)f(enough.)45 b(Since)31 b Fr(j)p Fo(v)s Fs(\()p Fo(x)p Fs(\))p Fr(j)f(\024)d Fs(2)p Fo(\016)2582 1963 y Fn(0)2654 1949 y Fs(if)k Fo(x)21 b Fr(\001)h Fo(!)s(=)p Fr(j)p Fo(!)s Fr(j)28 b(\025)g Fo(M)10 b Fs(,)32 b(due)g(to)g(\(38\))r (,)168 2056 y(w)m(e)f(gather)g(that)g Fo(\015)5 b Fs(\()p Fo(x)p Fs(\))26 b Fr(2)e Fs([)p Fo(C)q(;)15 b(C)1270 2023 y Fl(0)1294 2056 y Fs(])31 b(if)e Fo(x)20 b Fr(\001)h Fo(!)s(=)p Fr(j)p Fo(!)s Fr(j)26 b(\025)f Fo(M)10 b Fs(.)168 2164 y(Let)31 b Fo(a)25 b(>)g Fs(0)31 b(and)558 2372 y Fo(\014)5 b Fs(\()p Fo(x)p Fs(\))26 b(:=)899 2311 y(2)p Fo(\016)984 2325 y Fn(0)1024 2311 y Fs(\()p Fo(e)1101 2221 y Fl(p)p 1161 2221 55 3 v 1161 2278 a Fp(C)5 b(a)1278 2311 y Fr(\000)19 b Fs(1\))p 893 2351 562 4 v 893 2451 a Fo(e)935 2368 y Fl(p)p 994 2368 55 3 v 56 x Fp(C)5 b(a)1111 2451 y Fr(\000)20 b Fo(e)1244 2424 y Fl(\000)1299 2368 y(p)p 1358 2368 V 56 x Fp(C)5 b(a)1465 2372 y Fo(e)1507 2335 y Fl(\000)1562 2278 y(p)p 1621 2278 V 57 x Fp(C)g Fn(\()p Fp(x)p Fl(\001)p Fp(!)r(=)p Fl(j)p Fp(!)r Fl(j\000)p Fp(M)i Fn(\))2111 2372 y Fs(+)2212 2311 y(2)p Fo(\016)2297 2325 y Fn(0)2337 2311 y Fs(\(1)21 b Fr(\000)f Fo(e)2571 2278 y Fl(\000)2626 2221 y(p)p 2685 2221 V 57 x Fp(C)5 b(a)2782 2311 y Fs(\))p 2212 2351 606 4 v 2234 2451 a Fo(e)2276 2368 y Fl(p)p 2335 2368 55 3 v 56 x Fp(C)g(a)2452 2451 y Fr(\000)20 b Fo(e)2585 2424 y Fl(\000)2640 2368 y(p)p 2699 2368 V 56 x Fp(C)5 b(a)2827 2372 y Fo(e)2869 2278 y Fl(p)p 2928 2278 V 57 x Fp(C)g Fn(\()p Fp(x)p Fl(\001)p Fp(!)r(=)p Fl(j)p Fp(!)r Fl(j\000)p Fp(M)i Fn(\))3414 2372 y Fo(:)168 2574 y Fs(Then,)29 b(if)h Fo(x)20 b Fr(\001)g Fo(!)s(=)p Fr(j)p Fo(!)s Fr(j)26 b(2)f(f)p Fo(M)5 b(;)15 b(M)31 b Fs(+)20 b Fo(a)p Fr(g)p Fs(,)31 b Fo(\014)5 b Fs(\()p Fo(x)p Fs(\))26 b(=)f(2)p Fo(\016)1880 2588 y Fn(0)1946 2574 y Fr(\025)g Fo(v)s Fs(\()p Fo(x)p Fs(\),)31 b(while,)e(if)g Fo(x)21 b Fr(\001)f Fo(!)s(=)p Fr(j)p Fo(!)s Fr(j)26 b(2)f Fs([)p Fo(M)5 b(;)15 b(M)31 b Fs(+)20 b Fo(a)p Fs(])1250 2721 y(\001)p Fo(\014)26 b Fr(\000)19 b Fo(\015)5 b(\014)31 b Fs(=)25 b(\()p Fo(C)i Fr(\000)20 b Fo(\015)5 b Fs(\))p Fo(\014)31 b Fr(\024)25 b Fs(0)h(=)e(\001)p Fo(v)g Fr(\000)c Fo(\015)5 b(v)18 b(:)168 2868 y Fs(Hence,)30 b(b)m(y)f(the)g(elliptic)d (comparison)i(principle)e(\(see,)k(e.g.,)g Fr(x)g Fs(8.7)f(of)g([GT83)r (]\),)h Fo(v)s Fs(\()p Fo(x)p Fs(\))c Fr(\024)f Fo(\014)5 b Fs(\()p Fo(x)p Fs(\))29 b(for)g(an)m(y)g Fo(x)168 2976 y Fs(so)h(that)h Fo(x)20 b Fr(\001)h Fo(!)s(=)p Fr(j)p Fo(!)s Fr(j)26 b(2)f Fs([)p Fo(M)5 b(;)15 b(M)31 b Fs(+)20 b Fo(a)p Fs(].)41 b(In)29 b(particular,)g(if)g Fo(x)21 b Fr(\001)f Fo(!)s(=)p Fr(j)p Fo(!)s Fr(j)26 b(2)f Fs([)p Fo(M)5 b(;)15 b(M)31 b Fs(+)20 b(\()p Fo(a=)p Fs(2\)],)727 3203 y Fo(v)s Fs(\()p Fo(x)p Fs(\))26 b Fr(\024)1033 3142 y Fs(2)p Fo(\016)1118 3156 y Fn(0)1159 3142 y Fs(\()p Fo(e)1236 3053 y Fl(p)p 1295 3053 V 56 x Fp(C)5 b(a)1412 3142 y Fr(\000)20 b Fs(1\))p 1028 3182 562 4 v 1028 3282 a Fo(e)1070 3199 y Fl(p)p 1129 3199 55 3 v 57 x Fp(C)5 b(a)1246 3282 y Fr(\000)20 b Fo(e)1379 3256 y Fl(\000)1434 3199 y(p)p 1493 3199 V 57 x Fp(C)5 b(a)1599 3203 y Fo(e)1641 3166 y Fl(\000)1696 3110 y(p)p 1755 3110 V 56 x Fp(C)g Fn(\()p Fp(x)p Fl(\001)p Fp(!)r(=)p Fl(j)p Fp(!)r Fl(j\000)p Fp(M)i Fn(\))2246 3203 y Fs(+)2346 3142 y(2)p Fo(\016)2431 3156 y Fn(0)2472 3142 y Fs(\(1)21 b Fr(\000)f Fo(e)2706 3109 y Fl(\000)2761 3053 y(p)p 2820 3053 V 56 x Fp(C)5 b(a)2916 3142 y Fs(\))p 2346 3182 606 4 v 2368 3282 a Fo(e)2410 3199 y Fl(p)p 2469 3199 55 3 v 57 x Fp(C)g(a)2586 3282 y Fr(\000)20 b Fo(e)2719 3256 y Fl(\000)2774 3199 y(p)p 2833 3199 V 57 x Fp(C)5 b(a)2962 3203 y Fo(e)3004 3110 y Fl(p)p 3063 3110 V 56 x Fp(C)g(a=)p Fn(2)3245 3203 y Fo(:)168 3400 y Fs(By)30 b(letting)g Fo(a)c Fr(!)f Fs(+)p Fr(1)p Fs(,)30 b(it)g(follo)m(ws)f(that)1457 3565 y Fo(v)s Fs(\()p Fo(x)p Fs(\))d Fr(\024)f Fs(2)p Fo(\016)1833 3579 y Fn(0)1873 3565 y Fo(e)1915 3528 y Fl(\000)1970 3471 y(p)p 2029 3471 V 57 x Fp(C)5 b Fn(\()p Fp(x)p Fl(\001)p Fp(!)r(=)p Fl(j)p Fp(!)r Fl(j\000)p Fp(M)i Fn(\))2515 3565 y Fo(;)168 3712 y Fs(as)30 b(desired.)3174 b Ff(\003)168 3872 y Fs(W)-8 b(e)31 b(are)g(no)m(w)f(in)g(p)s(osition)e(to)j (minimize)d Fr(G)1672 3890 y Fk(R)1720 3872 y Fj(n)1763 3890 y Fp(=)-12 b Fl(\030)1875 3872 y Fs(in)29 b Fr(Y)2049 3839 y Fp(!)2042 3899 y(M)2121 3872 y Fs(:)168 4003 y Fw(Lemma)j(12.)43 b Fi(Assume)c Fs(\(53\))q Fi(.)j(The)33 b(functional)g Fr(G)1958 4022 y Fk(R)2006 4003 y Fj(n)2049 4022 y Fp(=)-12 b Fl(\030)2164 4003 y Fi(attains)34 b(its)f(minimum)g (on)g Fr(Y)3203 3970 y Fp(!)3196 4030 y(M)3275 4003 y Fi(.)168 4176 y(Pr)-5 b(o)g(of.)43 b Fs(Giv)m(en)f Fo(\027)52 b Fr(2)46 b Fq(N)6 b Fs(,)52 b(b)m(y)43 b(arguing)f(as)h(in)1745 4141 y Fn(3)1822 4176 y Fs(the)g(pro)s(of)f(of)h(Lemma)g(7,)j(one)d (\014nds)f Fo(v)3264 4190 y Fp(\027)3350 4176 y Fs(whic)m(h)f(mini-)168 4289 y(mizes)26 b Fr(G)466 4303 y Fl(R)526 4285 y Fj(!)526 4320 y(\027)575 4289 y Fs(\()p Fo(u)p Fs(\))i(among)f(all)f(the)g (functions)g Fo(u)g Fs(so)h(that)h Fo(u)13 b Fr(\000)18 b Fs(\026)-50 b Fo(u)24 b Fr(2)h Fo(W)2468 4245 y Fn(1)p Fp(;)p Fn(2)2455 4315 y(0)2562 4289 y Fs(\()p Fr(R)2674 4256 y Fp(!)2674 4312 y(\027)2725 4289 y Fs(\).)40 b(F)-8 b(urther,)27 b(b)m(y)g(the)g(argumen)m(t)168 4397 y(on)33 b(page)h(14,)h(w)m(e)f(ha)m(v)m(e)g(that)g Fr(j)p Fo(v)1280 4411 y Fp(\027)1323 4397 y Fs(\()p Fo(x)p Fs(\))p Fr(j)d(\024)f Fs(1)23 b(+)e Fo(\016)1802 4411 y Fn(0)1842 4397 y Fs(.)50 b(Then,)33 b(b)m(y)g(in)m(terior)f(regularit)m(y)g(estimates)i(\(see,)i (e.g.,)168 4505 y(Theorem)30 b(8.32)i(in)d([GT83)q(]\),)i(w)m(e)g (deduce)f(that,)h(up)f(to)h(subsequences,)168 4652 y(\(57\))822 b Fo(v)1194 4666 y Fp(\027)1268 4652 y Fs(con)m(v)m(erges)32 b(in)d Fo(C)1853 4619 y Fn(1)1846 4680 y(lo)r(c)1939 4652 y Fs(\()p Fq(R)2034 4619 y Fp(n)2087 4652 y Fo(=)10 b Fr(\030)p Fs(\))30 b(to)i(a)e(suitable)f Fo(v)s Fs(.)168 4800 y(By)c(construction,)g Fo(v)j Fs(is)c(a)h(lo)s(cal)f(minimizer)f (of)i Fr(G)1855 4814 y Fn(\012)1935 4800 y Fs(in)e(an)m(y)j(b)s(ounded) c(subset)j(\012)f(of)h Fq(R)3089 4767 y Fp(n)3142 4800 y Fo(=)10 b Fr(\030)p Fs(,)27 b(therefore)e Fo(v)k Fr(2)168 4908 y(Y)236 4875 y Fp(!)229 4934 y(M)307 4908 y Fs(,)24 b(thanks)d(to)i(Lemma)e(11.)39 b(W)-8 b(e)23 b(no)m(w)e(sho)m(w)h (that,)i(indeed,)c Fo(v)25 b Fs(minimizes)19 b Fr(G)2818 4926 y Fk(R)2866 4907 y Fj(n)2909 4926 y Fp(=)-12 b Fl(\030)3013 4908 y Fs(in)20 b Fr(Y)3178 4875 y Fp(!)3171 4934 y(M)3250 4908 y Fs(.)38 b(F)-8 b(or)22 b(this,)g(tak)m(e)168 5027 y(an)m(y)g Fo(u)k Fr(2)e(Y)562 4994 y Fp(!)555 5054 y(M)634 5027 y Fs(.)38 b(Then,)23 b Fo(u)t Fr(\000)t Fo(v)i Fs(b)s(elongs)c(to) i Fo(W)1673 4994 y Fn(1)p Fp(;)p Fn(1)1766 5027 y Fs(\()p Fq(R)1861 4994 y Fp(n)1914 5027 y Fo(=)10 b Fr(\030)q Fs(\))t Fr(\\)t Fo(W)2244 4994 y Fn(1)p Fp(;)p Fl(1)2372 5027 y Fs(\()p Fq(R)2467 4994 y Fp(n)2520 5027 y Fo(=)g Fr(\030)p Fs(\))23 b(since)e(the)i(same)f(holds)f(for)h Fo(u)t Fr(\000)10 b Fs(\026)-51 b Fo(u)p 168 5088 499 4 v 267 5159 a Fn(2)303 5188 y Fv(A)25 b(di\013eren)n(t)g(pro)r(of)i (ma)n(y)d(b)r(e)i(also)h(obtained)e(using)h(the)g(ring-shap)r(ed)f (barrier)h(of)h(Lemma)d(3.3)j(in)e([GG03)r(].)267 5258 y Fn(3)303 5287 y Fv(Though)30 b(the)h(energy)f(is)i(b)r(ounded)d(b)n (y)h(b)r(elo)n(w)i(due)e(to)g(\(50\))i(and)e(an)h(upp)r(er)f(b)r(ound)f (for)j(the)e(minimizing)g(energy)h(is)168 5378 y(giv)n(en)d(b)n(y)g (\(51\))q(,)h(standard)f(direct)h(metho)r(ds)e(do)h(not)g(su\016ce)h (to)f(pro)n(v)n(e)g(Lemma)f(12,)j(since,)g(in)e(principle,)h(the)f (minimizer)168 5469 y(could)e(jump)e(out)h(of)i Fc(Y)852 5438 y Fj(!)846 5484 y(M)913 5469 y Fv(.)35 b(Lemma)24 b(11)i(prev)n(en)n(ts)f(this)h(to)g(o)r(ccur.)p eop %%Page: 21 21 21 20 bop 587 251 a Fn(THE)30 b(GEOMETR)-6 b(Y)31 b(OF)f(MESOSCOPIC)h (PHASE)f(TRANSITION)h(INTERF)-8 b(A)n(CES)517 b(21)0 450 y Fs(and)24 b Fo(v)11 b Fr(\000)j Fs(\026)-51 b Fo(u)p Fs(.)38 b(Hence,)27 b(w)m(e)e(ma)m(y)f(consider)g(a)g(molli\014ed)e (sequence)j(,)g(sa)m(y)g Fo(u)2434 464 y Fp(j)2471 450 y Fs(,)g(so)g(that)g Fo(u)2870 464 y Fp(j)2914 450 y Fr(\000)8 b Fo(v)29 b Fr(2)24 b Fo(C)3223 417 y Fl(1)3216 474 y Fn(0)3298 450 y Fs(\()p Fr(R)3410 417 y Fp(!)3410 477 y(R)3463 487 y Fj(j)3500 450 y Fo(=)10 b Fr(\030)p Fs(\))0 568 y(for)30 b(suitable)f Fo(R)547 582 y Fp(j)609 568 y Fo(>)c Fs(0,)31 b(in)e(suc)m(h)h(a)h(w)m(a)m(y)754 764 y Fo(u)806 778 y Fp(j)873 764 y Fs(approac)m(hes)g Fo(u)f Fs(in)f Fo(W)1628 731 y Fn(1)p Fp(;)p Fn(1)1722 764 y Fs(\()p Fq(R)1817 731 y Fp(n)1870 764 y Fo(=)10 b Fr(\030)p Fs(\))754 912 y(with)29 b Fo(W)1060 879 y Fn(1)p Fp(;)p Fl(1)1189 912 y Fs(\()p Fq(R)1284 879 y Fp(n)1337 912 y Fo(=)10 b Fr(\030)p Fs(\)-norm)30 b(b)s(ounded)f(indep) s(enden)m(tly)e(of)k Fo(j)5 b Fs(.)0 841 y(\(58\))0 1110 y(W)-8 b(e)33 b(also)f(set)g Fo(u)537 1124 y Fp(j;\027)656 1110 y Fs(:=)27 b Fo(u)831 1124 y Fp(j)889 1110 y Fr(\000)21 b Fo(v)k Fs(+)20 b Fo(v)1185 1124 y Fp(\027)1229 1110 y Fs(.)45 b(Since)30 b Fo(u)1589 1124 y Fp(j;\027)1712 1110 y Fs(agrees)j(with)e Fo(v)2240 1124 y Fp(\027)2315 1110 y Fs(outside)g Fr(R)2707 1077 y Fp(!)2707 1137 y(R)2760 1147 y Fj(j)2797 1110 y Fo(=)12 b Fr(\030)p Fs(,)32 b(when)f Fo(\027)j(>)27 b(R)3467 1124 y Fp(j)3535 1110 y Fs(the)0 1228 y(minimizing)g(prop)s(ert)m(y)i(of)i Fo(v)989 1242 y Fp(\027)1063 1228 y Fs(yields)d(that)983 1421 y Fr(G)1037 1435 y Fl(R)1097 1412 y Fj(!)1097 1458 y(R)1143 1474 y(j)1184 1421 y Fs(\()p Fo(u)1271 1435 y Fp(j;\027)1363 1421 y Fs(\))83 b(=)g Fr(G)1689 1435 y Fl(R)1749 1416 y Fj(!)1749 1452 y(\027)1798 1421 y Fs(\()p Fo(u)1885 1435 y Fp(j;\027)1977 1421 y Fs(\))20 b Fr(\000)g(G)2177 1440 y Fl(R)2237 1421 y Fj(!)2237 1456 y(\027)2282 1440 y Fl(nR)2377 1417 y Fj(!)2377 1463 y(R)2423 1479 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b(al)g(y\))168 4484 y Fb(E-mail)27 b(addr)l(ess)6 b Fv(:)36 b Fa (valdinoci@mat.uniroma2.it)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0603130738169--