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%!PS-Adobe-2.0 %%Creator: dvips(k) 5.996 Copyright 2016 Radical Eye Software %%Title: arXiv:1810.05834v1 [math.AP] 13 Oct 2018 %%CreationDate: Thu Dec 28 18:34:37 2023 %%Pages: 9 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z local_uniqueness.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2023.12.28:1834 %%BeginProcSet: texc.pro 0 0 %! /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 /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}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 0 0 %! 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Fr(\(\012\),)29 b(the)f(question)e(of)i(uniqueness)f(from)f(data)h(of)h(t)n(yp)r(e)456 1947 y Fm(C)521 1916 y Fn(N)515 1967 y(q)614 1947 y Fr(with)h(\000)e(=) f(\000)1027 1916 y Fn(D)1114 1947 y Fr(=)h(\000)1258 1916 y Fn(N)1351 1947 y Fr(is)i(equiv)-5 b(alen)n(t)27 b(to)j(the)g(question)f(of)h(uniqueness)e(from)h(the)h(lo)r(cal)456 2049 y(NtD)e(op)r(erator)e(\003)1034 2061 y Fk(\000)1079 2049 y Fr(\()p Fm(q)s Fr(\).)456 2201 y(Hereafter,)h(w)n(e)h(list)e (some)h(recen)n(t)g(results.)37 b(Let)28 b(us)g(also)e(refer)h(to)h (the)h(article)c(of)j(Kenig)f(and)456 2300 y(Salo)e([)658 2300 y SDict begin H.S end 658 2300 a 0 TeXcolorgray Fr(21)p 0 TeXcolorgray 742 2247 a SDict begin H.R end 742 2247 a 742 2300 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.kenig2014recent) cvn H.B /ANN pdfmark end 742 2300 a Fr(])i(for)g(a)h(further)f (o)n(v)n(erview.)456 2452 y(F)-7 b(or)37 b(dimension)d Fm(n)40 b Fr(=)g(2,)g(Iman)n(uvilo)n(v,)c(Uhlmann)g(and)h(Y)-7 b(amamoto)35 b(sho)n(w)n(ed)i(uniqueness)456 2552 y(from)30 b(data)h(of)g(t)n(yp)r(e)h Fm(C)1203 2522 y Fn(D)1197 2572 y(q)1295 2552 y Fr(in)f([)1419 2552 y SDict begin H.S end 1419 2552 a 0 TeXcolorgray Fr(16)p 0 TeXcolorgray 1502 2498 a SDict begin H.R end 1502 2498 a 1502 2552 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.imanuvilov2010calderon) cvn H.B /ANN pdfmark end 1502 2552 a Fr(],)i(where)e(\000)1877 2522 y Fn(D)1967 2552 y Fr(=)e(\000)2113 2522 y Fn(N)2208 2552 y Fr(is)h(an)h(arbitrary)e(op)r(en)i(subset)h(in)f Fm(@)5 b Fr(\012)456 2665 y(and)27 b(the)h(p)r(oten)n(tials)d(are)i(in) g Fp(C)1428 2635 y Fk(2+)p Fn(\013)1573 2598 y Fi(\000)p 1611 2599 60 4 v 67 x Fr(\012)1671 2598 y Fi(\001)1736 2665 y Fr(for)g Fm(\013)d(>)e Fr(0.)456 2817 y(F)-7 b(or)26 b(dimension)d Fm(n)g Fp(\025)g Fr(3,)j(Kenig,)f(Sj\177)-42 b(ostrand)26 b(and)g(Uhlmann)f(pro)n(v)n(ed)g(uniqueness)h(from)f(data) 456 2917 y(of)38 b(t)n(yp)r(e)h Fm(C)824 2887 y Fn(D)818 2937 y(q)923 2917 y Fr(for)f Fm(q)44 b Fp(2)d Fm(L)1295 2887 y Fl(1)1365 2917 y Fr(\(\012\))e(in)f([)1659 2917 y SDict begin H.S end 1659 2917 a 0 TeXcolorgray Fr(20)p 0 TeXcolorgray 1742 2863 a SDict begin H.R end 1742 2863 a 1742 2917 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.kenig2007calderon) cvn H.B /ANN pdfmark end 1742 2917 a Fr(],)k(where)c(\000)2133 2887 y Fn(D)2231 2917 y Fr(and)h(\000)2456 2887 y Fn(N)2557 2917 y Fr(are)e(op)r(en)i(neigh)n(b)r(orho)r(o)r(ds)456 3016 y(sligh)n(tly)24 b(larger)g(than)k(a)f(fron)n(t)g(face)h(and)f(a)g (bac)n(k)g(face)h(of)f Fm(@)5 b Fr(\012,)28 b(resp)r(ectiv)n(ely)-7 b(.)34 b(Nac)n(hman)26 b(and)456 3116 y(Street)31 b(presen)n(ted)h(a)f (constructiv)n(e)f(pro)r(of)h(of)g(this)g(result)g(in)g([)2470 3116 y SDict begin H.S end 2470 3116 a 0 TeXcolorgray Fr(24)p 0 TeXcolorgray 2553 3062 a SDict begin H.R end 2553 3062 a 2553 3116 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.nachman2010reconstruction) cvn H.B /ANN pdfmark end 2553 3116 a Fr(].)49 b(In)32 b([)2779 3116 y SDict begin H.S end 2779 3116 a 0 TeXcolorgray Fr(17)p 0 TeXcolorgray 2862 3062 a SDict begin H.R end 2862 3062 a 2862 3116 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.isakov2007uniqueness) cvn H.B /ANN pdfmark end 2862 3116 a Fr(],)h(Isak)n(o)n(v)d(pro)n(v)n(ed)456 3216 y(uniqueness)d (from)g(data)i(of)f(t)n(yp)r(e)h Fm(C)1611 3185 y Fn(D)1605 3236 y(q)1700 3216 y Fr(for)g Fm(q)e Fp(2)f Fm(L)2031 3185 y Fl(1)2100 3216 y Fr(\(\012\))k(assuming)c(\000)2668 3185 y Fn(D)2753 3216 y Fr(=)e(\000)2894 3185 y Fn(N)2986 3216 y Fr(and)k(that)h(the)456 3318 y(remaining)f(b)r(oundary)k(part)g (is)g(con)n(tained)f(in)g(a)i(plane)e(or)h(a)g(sphere.)51 b(In)33 b([)2903 3318 y SDict begin H.S end 2903 3318 a 0 TeXcolorgray Fr(19)p 0 TeXcolorgray 2986 3264 a SDict begin H.R end 2986 3264 a 2986 3318 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.kenig2012calderon) cvn H.B /ANN pdfmark end 2986 3318 a Fr(],)h(Kenig)d(and)456 3418 y(Salo)e(presen)n(ted)i(a)h(result) e(that)i(uni\014es)f(and)g(impro)n(v)n(es)d(the)k(approac)n(hes)d(of)j ([)2989 3418 y SDict begin H.S end 2989 3418 a 0 TeXcolorgray Fr(20)p 0 TeXcolorgray 3072 3364 a SDict begin H.R end 3072 3364 a 3072 3418 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.kenig2007calderon) cvn H.B /ANN pdfmark end 3072 3418 a Fr(])g(and)f([)3315 3418 y SDict begin H.S end 3315 3418 a 0 TeXcolorgray Fr(17)p 0 TeXcolorgray 3398 3364 a SDict begin H.R end 3398 3364 a 3398 3418 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.isakov2007uniqueness) cvn H.B /ANN pdfmark end 3398 3418 a Fr(].)456 3517 y(In)g(particular,)e(they)j(reduced)f(the)h(assumptions)d (regarding)f(the)k(sets)f(\000)2839 3487 y Fn(D)2899 3517 y Fr(,)i(\000)3007 3487 y Fn(N)3101 3517 y Fr(and,)f(if)f(so)456 3617 y(\(in)f([)611 3617 y SDict begin H.S end 611 3617 a 0 TeXcolorgray Fr(17)p 0 TeXcolorgray 694 3563 a SDict begin H.R end 694 3563 a 694 3617 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.isakov2007uniqueness) cvn H.B /ANN pdfmark end 694 3617 a Fr(]\),)h(the)g(remaining)c(b)r(oundary)j(part.)45 b(Let)30 b(us)h(refer)f(to)g(their)g(article)d(for)j(a)g(detailed)456 3716 y(description.)456 3868 y(Theorem)819 3868 y SDict begin H.S end 819 3868 a 0 TeXcolorgray Fr(1.1)p 0 TeXcolorgray 925 3815 a SDict begin H.R end 925 3815 a 925 3868 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.1) cvn H.B /ANN pdfmark end 925 3868 a 41 w Fr(is,)42 b(to)e(the)h(kno)n (wledge)d(of)i(the)h(authors,)h(the)f(\014rst)f(result)f(that)h(presen) n(ts)456 3968 y(a)h(uniqueness)f(result)g(for)h(partial)d(data)j(on)g (an)g(arbitrary)d(non-empt)n(y)i(relativ)n(ely)d(op)r(en)456 4068 y(b)r(oundary)d(part)g(\000)h Fp(\022)g Fm(@)5 b Fr(\012)35 b(\(with)f(\000)h(=)g(\000)1822 4037 y Fn(D)1917 4068 y Fr(=)g(\000)2069 4037 y Fn(N)2132 4068 y Fr(\))g(for)g (dimension)c Fm(n)36 b Fp(\025)e Fr(3.)59 b(Except)34 b(the)456 4167 y(assumption)24 b(that)j(\012)f(has)g(to)h(b)r(e)g(a)f (Lipsc)n(hitz-domain,)c(there)k(are)f(no)i(further)f(assumptions)456 4267 y(to)34 b(the)h(b)r(oundary)f(required:)50 b(neither)34 b(to)g(the)h(b)r(oundary)f(part)h(\000)g(nor)f(to)g(the)h(remaining)456 4366 y(b)r(oundary)26 b(part.)456 4518 y(This)k(pap)r(er)h(is)f (organized)f(as)i(follo)n(ws.)45 b(In)31 b(Section)2175 4518 y SDict begin H.S end 2175 4518 a 0 TeXcolorgray Fr(2)p 0 TeXcolorgray 2216 4465 a SDict begin H.R end 2216 4465 a 2216 4518 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.2) cvn H.B /ANN pdfmark end 2216 4518 a Fr(,)i(w)n(e)e(pro)n(v) n(e)e(Theorem)2980 4518 y SDict begin H.S end 2980 4518 a 0 TeXcolorgray Fr(1.1)p 0 TeXcolorgray 3086 4465 a SDict begin H.R end 3086 4465 a 3086 4518 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.1) cvn H.B /ANN pdfmark end 3086 4518 a Fr(.)48 b(F)-7 b(or)31 b(this)456 4618 y(purp)r(ose,)21 b(w)n(e)f(presen)n(t)g(and)g(com)n(bine)e(a)i Fj(monotonicity)k(r)l(elation)k Fr(for)20 b(the)h(lo)r(cal)d(NtD)j(op)r (erator)456 4717 y(\(Lemma)785 4717 y SDict begin H.S end 785 4717 a 0 TeXcolorgray Fr(2.1)p 0 TeXcolorgray 892 4664 a SDict begin H.R end 892 4664 a 892 4717 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 892 4717 a Fr(\))29 b(and)f(a)h(new)g(v)-5 b(arian)n(t)27 b(of)i(the)g(concept)f(of)h Fj(lo)l(c)l(alize)l(d)k(p)l(otentials)j Fr(\(Lemma)3315 4717 y SDict begin H.S end 3315 4717 a 0 TeXcolorgray Fr(2.2)p 0 TeXcolorgray 3421 4664 a SDict begin H.R end 3421 4664 a 3421 4717 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 3421 4717 a Fr(,)456 4817 y(cf.)29 b([)593 4817 y SDict begin H.S end 593 4817 a 0 TeXcolorgray Fr(8)p 0 TeXcolorgray 635 4764 a SDict begin H.R end 635 4764 a 635 4817 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.gebauer2008localized) cvn H.B /ANN pdfmark end 635 4817 a Fr(])g(for)f(the)i(initial)25 b(concept\).)42 b(Lemma)1872 4817 y SDict begin H.S end 1872 4817 a 0 TeXcolorgray Fr(2.1)p 0 TeXcolorgray 1978 4764 a SDict begin H.R end 1978 4764 a 1978 4817 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 1978 4817 a 30 w Fr(presen)n(ts)28 b(a)g(monotonicit)n(y)e (inequalit)n(y)f(that)456 4917 y(yields)j(a)i(lo)n(w)n(er)d(b)r(ound)k (for)f(the)g(c)n(hange)f(of)i(the)f(lo)r(cal)e(NtD)j(op)r(erator)e (\(represen)n(ted)g(b)n(y)h(its)456 5016 y(corresp)r(onding)j (quadratic)i(form\))h(caused)g(b)n(y)h(a)f(p)r(oten)n(tial)f(c)n (hange.)63 b(This)36 b(lo)n(w)n(er)e(b)r(ound)456 5116 y(dep)r(ends)39 b(on)g(the)h(spatial)c(c)n(hange)i(of)h(the)h(p)r(oten) n(tial)d(w)n(eigh)n(ted)h(b)n(y)h(the)g(solution)e(of)i(the)456 5216 y(Sc)n(hr\177)-42 b(odinger)32 b(equation)h(for)h(the)i(initial)31 b(p)r(oten)n(tial.)57 b(Lemma)2505 5216 y SDict begin H.S end 2505 5216 a 0 TeXcolorgray Fr(2.2)p 0 TeXcolorgray 2611 5162 a SDict begin H.R end 2611 5162 a 2611 5216 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 2611 5216 a 35 w Fr(sho)n(ws)33 b(a)i(p)r(ossibilit) n(y)c(to)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 3 3 TeXDict begin HPSdict begin 3 2 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 447 176 a SDict begin H.S end 447 176 a 447 176 a SDict begin H.R end 447 176 a 447 176 a SDict begin [/View [/XYZ H.V]/Dest (page.3) cvn /DEST pdfmark end 447 176 a 0 TeXcolorgray 834 251 a Fk(LOCAL)29 b(UNIQUENESS)f(F)n(OR)h(AN)g(INVERSE)f(BVP)i(WITH)f(P)-5 b(AR)g(TIAL)28 b(D)n(A)-5 b(T)g(A)345 b(3)p 0 TeXcolorgray 456 450 a Fr(con)n(trol)17 b(the)j(lo)n(w)n(er)c(b)r(ound)k(of)g(the)f (monotonicit)n(y)d(inequalit)n(y)-7 b(.)2411 450 y SDict begin H.S end 2411 450 a -32 x Fk(1)2441 450 y SDict begin 12 H.L end 2441 450 a 2441 450 a SDict begin [/Subtype /Link/Dest (Hfootnote.1) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [1 0 0] H.B /ANN pdfmark end 2441 450 a 34 w Fr(The)19 b(approac)n(h)f(of)h(com)n(bining)456 550 y(a)j(monotonicit)n(y)d(relation)g(with)j(the)h(concept)g(of)f(lo)r (calized)d(p)r(oten)n(tials)i(has)h(previously)d(b)r(een)456 649 y(used)31 b(in)g([)773 649 y SDict begin H.S end 773 649 a 0 TeXcolorgray Fr(13)p 0 TeXcolorgray 856 596 a SDict begin H.R end 856 596 a 856 649 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.harrach2013monotonicity) cvn H.B /ANN pdfmark end 856 649 a Fr(,)911 649 y SDict begin H.S end 911 649 a 0 TeXcolorgray Fr(11)p 0 TeXcolorgray 994 596 a SDict begin H.R end 994 596 a 994 649 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.harrach2009uniqueness) cvn H.B /ANN pdfmark end 994 649 a Fr(,)1049 649 y SDict begin H.S end 1049 649 a 0 TeXcolorgray Fr(12)p 0 TeXcolorgray 1132 596 a SDict begin H.R end 1132 596 a 1132 649 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.harrach2012simultaneous) cvn H.B /ANN pdfmark end 1132 649 a Fr(].)49 b(The)32 b(pro)r(ofs)e(of)i(the)g(Lemmas)2234 649 y SDict begin H.S end 2234 649 a 0 TeXcolorgray Fr(2.1)p 0 TeXcolorgray 2340 596 a SDict begin H.R end 2340 596 a 2340 649 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 2340 649 a 32 w Fr(and)2537 649 y SDict begin H.S end 2537 649 a 0 TeXcolorgray Fr(2.2)p 0 TeXcolorgray 2643 596 a SDict begin H.R end 2643 596 a 2643 649 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 2643 649 a 32 w Fr(are)f(p)r(ostp)r (oned)h(to)f(the)456 749 y(Sections)779 749 y SDict begin H.S end 779 749 a 0 TeXcolorgray Fr(3)p 0 TeXcolorgray 820 695 a SDict begin H.R end 820 695 a 820 749 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.3) cvn H.B /ANN pdfmark end 820 749 a 28 w Fr(and)1010 749 y SDict begin H.S end 1010 749 a 0 TeXcolorgray Fr(4)p 0 TeXcolorgray 1051 695 a SDict begin H.R end 1051 695 a 1051 749 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.4) cvn H.B /ANN pdfmark end 1051 749 a Fr(,)d(resp)r(ectiv)n(ely)-7 b(.)456 903 y SDict begin H.S end 456 903 a 456 903 a SDict begin 12 H.A end 456 903 a 456 903 a SDict begin [/View [/XYZ H.V]/Dest (section.2) cvn /DEST pdfmark end 456 903 a 1336 1034 a Fr(2.)41 b Fq(The)32 b(pr)n(oof)g(of)g(Theorem) 2445 1034 y SDict begin H.S end 2445 1034 a 0 TeXcolorgray Fq(1.1)p 0 TeXcolorgray 2564 981 a SDict begin H.R end 2564 981 a 2564 1034 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.1) cvn H.B /ANN pdfmark end 2564 1034 a 456 1232 a Fr(Let)25 b Fm(q)639 1244 y Fk(1)676 1232 y Fm(;)14 b(q)750 1244 y Fk(2)811 1232 y Fp(2)23 b Fm(L)946 1201 y Fl(1)946 1252 y Fk(+)1016 1232 y Fr(\(\012\),)j(let)f Fm(V)42 b Fp(\022)22 b Fo(R)1538 1201 y Fn(n)1615 1232 y Fr(b)r(e)j(an)g(op)r(en)g(connected)g(set)h(and)f(let)f(\000)f(:=)g Fm(@)5 b Fr(\012)14 b Fp(\\)g Fm(V)41 b Fp(6)p Fr(=)23 b Fp(;)p Fr(.)456 1379 y(T)-7 b(o)39 b(pro)n(v)n(e)f(Theorem)1189 1379 y SDict begin H.S end 1189 1379 a 0 TeXcolorgray Fr(1.1)p 0 TeXcolorgray 1295 1326 a SDict begin H.R end 1295 1326 a 1295 1379 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.1) cvn H.B /ANN pdfmark end 1295 1379 a Fr(,)43 b(w)n(e)c(com)n(bine)f(a)h(monotonicit)n(y)d(inequalit)n(y)g(\(Lemma) 3167 1379 y SDict begin H.S end 3167 1379 a 0 TeXcolorgray Fr(2.1)p 0 TeXcolorgray 3273 1326 a SDict begin H.R end 3273 1326 a 3273 1379 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 3273 1379 a Fr(\))k(for)456 1479 y(Neumann-to-Diric)n(hlet)25 b(op)r(erators)k(with)g(a)h(result)f (ab)r(out)h(the)h(existence)e(of)h(lo)r(calized)d(p)r(o-)456 1578 y(ten)n(tials)e(\(Lemma)1078 1578 y SDict begin H.S end 1078 1578 a 0 TeXcolorgray Fr(2.2)p 0 TeXcolorgray 1184 1525 a SDict begin H.R end 1184 1525 a 1184 1578 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 1184 1578 a Fr(\).)456 1757 y SDict begin H.S end 456 1757 a 456 1757 a SDict begin 12 H.A end 456 1757 a 456 1757 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.1) cvn /DEST pdfmark end 456 1757 a 0 TeXcolorgray Fy(Lemma)k(2.1.)p 0 TeXcolorgray 40 w Fj(L)l(et)g Fm(g)d Fp(2)d Fm(L)1328 1727 y Fk(2)1365 1757 y Fr(\(\000\))30 b Fj(and)g Fm(u)1720 1769 y Fk(1)1780 1757 y Fr(:=)22 b Fm(u)1938 1714 y Fk(\()p Fn(g)r Fk(\))1938 1767 y Fn(q)1968 1775 y Fd(1)2051 1757 y Fp(2)i Fm(H)2206 1727 y Fk(1)2243 1757 y Fr(\(\012\))30 b Fj(b)l(e)f(the)h(c)l(orr)l(esp)l(onding)h (solution)456 1857 y(of)48 b Fr(\()603 1857 y SDict begin H.S end 603 1857 a 0 TeXcolorgray Fr(1.3)p 0 TeXcolorgray 709 1804 a SDict begin H.R end 709 1804 a 709 1857 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 709 1857 a Fr(\))p Fj(.)39 b(Then,)456 2058 y Fr(\(2.1\))1004 2058 y SDict begin H.S end 1004 2058 a 1004 2058 a SDict begin 12 H.A end 1004 2058 a 1004 2058 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.1) cvn /DEST pdfmark end 1004 2058 a 13 w Fr(\()q Fm(g)s(;)14 b Fr(\(\003)1220 2070 y Fk(\000)1264 2058 y Fr(\()p Fm(q)1333 2070 y Fk(2)1371 2058 y Fr(\))19 b Fp(\000)f Fr(\003)1563 2070 y Fk(\000)1608 2058 y Fr(\()p Fm(q)1677 2070 y Fk(1)1714 2058 y Fr(\)\))d Fm(g)s Fr(\))1867 2083 y Fn(L)1913 2066 y Fd(2)1945 2083 y Fk(\(\000\))2066 2058 y Fp(\025)22 b(\000)2232 1945 y Fi(Z)2278 2133 y Fk(\012)2329 2058 y Fr(\()p Fm(q)2398 2070 y Fk(2)2454 2058 y Fp(\000)c Fm(q)2574 2070 y Fk(1)2612 2058 y Fr(\))p Fm(u)2692 2070 y Fk(1)2729 2024 y(2)2780 2058 y Fr(d)p Fm(x:)456 2310 y Fr(Lemma)752 2310 y SDict begin H.S end 752 2310 a 0 TeXcolorgray Fr(2.1)p 0 TeXcolorgray 858 2256 a SDict begin H.R end 858 2256 a 858 2310 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 858 2310 a 28 w Fr(is)26 b(pro)n(v)n(en)g(in)h(Section)1627 2310 y SDict begin H.S end 1627 2310 a 0 TeXcolorgray Fr(3)p 0 TeXcolorgray 1668 2256 a SDict begin H.R end 1668 2256 a 1668 2310 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.3) cvn H.B /ANN pdfmark end 1668 2310 a Fr(.)456 2473 y SDict begin H.S end 456 2473 a 456 2473 a SDict begin 12 H.A end 456 2473 a 456 2473 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.2) cvn /DEST pdfmark end 456 2473 a 0 TeXcolorgray Fy(Lemma)i(2.2.)p 0 TeXcolorgray 40 w Fj(L)l(et)g Fm(q)1164 2485 y Fk(1)1224 2473 y Fo(\015)23 b Fm(q)1349 2485 y Fk(2)1415 2473 y Fj(on)30 b Fr(\012)18 b Fp(\\)g Fm(V)48 b Fj(\(i.e.,)32 b Fm(q)2022 2485 y Fk(1)2059 2473 y Fp(j)2082 2485 y Fk(\012)p Fl(\\)p Fn(V)2255 2473 y Fp(\025)22 b Fm(q)2379 2485 y Fk(2)2417 2473 y Fp(j)2440 2485 y Fk(\012)p Fl(\\)p Fn(V)2619 2473 y Fj(and)30 b Fm(q)2817 2485 y Fk(1)2854 2473 y Fp(j)2877 2485 y Fk(\012)p Fl(\\)p Fn(V)3050 2473 y Fp(6\021)23 b Fm(q)3175 2485 y Fk(2)3212 2473 y Fp(j)3235 2485 y Fk(\012)p Fl(\\)p Fn(V)3385 2473 y Fj(\).)456 2572 y(Then,)34 b(ther)l(e)f(exists)g(a)g (se)l(quenc)l(e)g Fr(\()p Fm(g)1630 2584 y Fn(m)1692 2572 y Fr(\))1724 2584 y Fn(m)p Fl(2)p Fc(N)1903 2572 y Fp(\032)c Fm(L)2054 2542 y Fk(2)2090 2572 y Fr(\(\000\))34 b Fj(such)e(that)h(the)g(c)l(orr)l(esp)l(onding)i(solu-)456 2697 y(tions)29 b Fr(\()q Fm(u)742 2709 y Fn(m)804 2697 y Fr(\))837 2722 y Fn(m)p Fl(2)p Fc(N)1009 2697 y Fr(:=)1120 2605 y Fi(\020)1170 2697 y Fm(u)1218 2654 y Fk(\()p Fn(g)1276 2662 y Fe(m)1331 2654 y Fk(\))1218 2707 y Fn(q)1248 2715 y Fd(1)1361 2605 y Fi(\021)1410 2755 y Fn(m)p Fl(2)p Fc(N)1583 2697 y Fp(\032)23 b Fm(H)1747 2667 y Fk(1)1784 2697 y Fr(\(\012\))30 b Fj(of)48 b Fr(\()2085 2697 y SDict begin H.S end 2085 2697 a 0 TeXcolorgray Fr(1.3)p 0 TeXcolorgray 2192 2643 a SDict begin H.R end 2192 2643 a 2192 2697 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 2192 2697 a Fr(\))30 b Fj(ful\014l)t(l)456 2939 y Fr(\(2.2\))721 2939 y SDict begin H.S end 721 2939 a 721 2939 a SDict begin 12 H.A end 721 2939 a 721 2939 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.2) cvn /DEST pdfmark end 721 2939 a 52 w Fr(lim)735 2989 y Fn(m)p Fl(!1)940 2826 y Fi(Z)986 3015 y Fn(V)14 b Fl(\\)p Fk(\012)1136 2939 y Fr(\()p Fm(q)1205 2951 y Fk(1)1261 2939 y Fp(\000)k Fm(q)1381 2951 y Fk(2)1418 2939 y Fr(\))p Fm(u)1498 2951 y Fn(m)1561 2905 y Fk(2)1612 2939 y Fr(d)p Fm(x)24 b Fr(=)e Fp(1)85 b Fj(and)137 b Fr(lim)2214 2989 y Fn(m)p Fl(!1)2420 2826 y Fi(Z)2466 3015 y Fk(\012)p Fl(n)p Fn(V)2605 2939 y Fr(\()p Fm(q)2674 2951 y Fk(1)2730 2939 y Fp(\000)18 b Fm(q)2850 2951 y Fk(2)2887 2939 y Fr(\))p Fm(u)2967 2951 y Fn(m)3030 2905 y Fk(2)3081 2939 y Fr(d)p Fm(x)24 b Fr(=)e(0)p Fm(:)456 3206 y Fr(Lemma)752 3206 y SDict begin H.S end 752 3206 a 0 TeXcolorgray Fr(2.2)p 0 TeXcolorgray 858 3152 a SDict begin H.R end 858 3152 a 858 3206 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 858 3206 a 28 w Fr(is)k(pro)n(v)n(en)g(in)h (Section)1627 3206 y SDict begin H.S end 1627 3206 a 0 TeXcolorgray Fr(4)p 0 TeXcolorgray 1668 3152 a SDict begin H.R end 1668 3152 a 1668 3206 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (section.4) cvn H.B /ANN pdfmark end 1668 3206 a Fr(.)p 0 TeXcolorgray 456 3437 a Fj(Pr)l(o)l(of)j(of)h(The)l(or) l(em)1121 3437 y SDict begin H.S end 1121 3437 a 0 TeXcolorgray Fj(1.1)p 0 TeXcolorgray 1231 3384 a SDict begin H.R end 1231 3384 a 1231 3437 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1.1) cvn H.B /ANN pdfmark end 1231 3437 a Fj(.)p 0 TeXcolorgray 42 w Fr(First,)e(w)n(e)h(apply)e(Lemma) 2179 3437 y SDict begin H.S end 2179 3437 a 0 TeXcolorgray Fr(2.2)p 0 TeXcolorgray 2285 3384 a SDict begin H.R end 2285 3384 a 2285 3437 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 2285 3437 a Fr(:)41 b(There)30 b(exists)e(a)h Fm(g)g Fp(2)e Fm(L)3102 3407 y Fk(2)3139 3437 y Fr(\(\000\))j(suc)n(h)456 3553 y(that)d(the)h (corresp)r(onding)d(solution)g Fm(u)e Fr(:=)g Fm(u)1856 3510 y Fk(\()p Fn(g)r Fk(\))1856 3563 y Fn(q)1886 3571 y Fd(1)1973 3553 y Fr(of)34 b(\()2106 3553 y SDict begin H.S end 2106 3553 a 0 TeXcolorgray Fr(1.3)p 0 TeXcolorgray 2213 3500 a SDict begin H.R end 2213 3500 a 2213 3553 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 2213 3553 a Fr(\))28 b(ful\014lls)900 3641 y Fi(Z)946 3830 y Fn(V)14 b Fl(\\)p Fk(\012)1096 3754 y Fr(\()p Fm(q)1165 3766 y Fk(1)1221 3754 y Fp(\000)k Fm(q)1341 3766 y Fk(2)1378 3754 y Fr(\))p Fm(u)1458 3720 y Fk(2)1509 3754 y Fr(d)p Fm(x)24 b(>)e Fr(1)83 b(and)2068 3641 y Fi(Z)2114 3830 y Fk(\012)p Fl(n)p Fn(V)2254 3754 y Fr(\()p Fm(q)2323 3766 y Fk(1)2379 3754 y Fp(\000)18 b Fm(q)2499 3766 y Fk(2)2536 3754 y Fr(\))p Fm(u)2616 3720 y Fk(2)2667 3754 y Fr(d)p Fm(x)23 b(>)g Fp(\000)p Fr(1)p Fm(:)456 3969 y Fr(No)n(w,)k(w)n(e)g(apply)f(Lemma)1315 3969 y SDict begin H.S end 1315 3969 a 0 TeXcolorgray Fr(2.1)p 0 TeXcolorgray 1421 3915 a SDict begin H.R end 1421 3915 a 1421 3969 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 1421 3969 a 27 w Fr(and)i(obtain)634 4165 y(\()p Fm(g)s(;)14 b Fr(\(\003)836 4177 y Fk(\000)881 4165 y Fr(\()p Fm(q)950 4177 y Fk(2)988 4165 y Fr(\))k Fp(\000)g Fr(\003)1179 4177 y Fk(\000)1224 4165 y Fr(\()p Fm(q)1293 4177 y Fk(1)1331 4165 y Fr(\)\))c Fm(g)s Fr(\))1484 4190 y Fn(L)1530 4173 y Fd(2)1562 4190 y Fk(\(\000\))1682 4165 y Fp(\025)23 b(\000)1849 4052 y Fi(Z)1894 4240 y Fk(\012)1946 4165 y Fr(\()p Fm(q)2015 4177 y Fk(2)2071 4165 y Fp(\000)18 b Fm(q)2191 4177 y Fk(1)2228 4165 y Fr(\))p Fm(u)2308 4131 y Fk(2)2359 4165 y Fr(d)p Fm(x)1682 4387 y Fr(=)1770 4274 y Fi(Z)1816 4462 y Fn(V)c Fl(\\)p Fk(\012)1966 4387 y Fr(\()p Fm(q)2035 4399 y Fk(1)2091 4387 y Fp(\000)k Fm(q)2211 4399 y Fk(2)2248 4387 y Fr(\))p Fm(u)2328 4352 y Fk(2)2379 4387 y Fr(d)p Fm(x)h Fr(+)2574 4274 y Fi(Z)2620 4462 y Fk(\012)p Fl(n)p Fn(V)2759 4387 y Fr(\()p Fm(q)2828 4399 y Fk(1)2884 4387 y Fp(\000)f Fm(q)3004 4399 y Fk(2)3042 4387 y Fr(\))p Fm(u)3122 4352 y Fk(2)3172 4387 y Fr(d)p Fm(x)1682 4567 y(>)23 b Fr(1)18 b Fp(\000)g Fr(1)k(=)h(0)p Fm(:)456 4731 y Fr(This)c(sho)n(ws)h(that)h(\003)1101 4743 y Fk(\000)1146 4731 y Fr(\()p Fm(q)1215 4743 y Fk(2)1253 4731 y Fr(\))5 b Fp(\000)g Fr(\003)1418 4743 y Fk(\000)1462 4731 y Fr(\()p Fm(q)1531 4743 y Fk(1)1568 4731 y Fr(\))22 b(is)d(not)i(semi)e(negativ) n(e)f(de\014nite)j(and)f(th)n(us)h(has)g(a)f(p)r(ositiv)n(e)456 4830 y(eigen)n(v)-5 b(alue.)2529 b Fb(\003)p 0 TeXcolorgray 456 4951 499 4 v 555 5024 a Fk(1)588 5050 y SDict begin H.S end 588 5050 a 588 5050 a SDict begin H.R end 588 5050 a 588 5050 a SDict begin [/View [/XYZ H.V]/Dest (Hfootnote.1) cvn /DEST pdfmark end 588 5050 a Fx(Originally)-6 b(,)26 b(the)e(concept)h(of)e(lo)r(calized)j(p)r(oten)n(tials)g(w)n(as) d(used)h(to)f(lo)r(cally)j(con)n(trol)f(electrical)h(p)r(oten)n(tials) 456 5133 y(for)21 b(the)j(in)n(v)n(erse)f(conductivit)n(y)k(problem.)k (Since)24 b(in)f(this)h(w)n(ork)e(it)i(is)e(used)h(to)g(lo)r(cally)j(w) n(eigh)n(t)e(the)f(p)r(oten)n(tials)456 5216 y(of)g(the)i(Sc)n(hr\177) -35 b(odinger)25 b(equation,)h(it)e(seems)g(appropriate)h(to)g(k)n(eep) f(with)h(the)g(name)f(\\lo)r(calized)j(p)r(oten)n(tials".)p 0 TeXcolorgray 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 447 176 a SDict begin H.S end 447 176 a 447 176 a SDict begin H.R end 447 176 a 447 176 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end 447 176 a 0 TeXcolorgray 456 251 a Fk(4)729 b(BASTIAN)22 b(HARRA)n(CH)i(AND)e(MAR)n(CEL)h(ULLRICH)p 0 TeXcolorgray 456 367 a SDict begin H.S end 456 367 a 456 367 a SDict begin 12 H.A end 456 367 a 456 367 a SDict begin [/View [/XYZ H.V]/Dest (section.3) cvn /DEST pdfmark end 456 367 a 908 450 a Fr(3.)41 b Fq(Monotonicity)31 b(f)n(or)h (Neumann-to-Dirichlet)f(maps)456 640 y Fr(Again,)24 b(let)h Fm(q)874 652 y Fk(1)911 640 y Fm(;)14 b(q)985 652 y Fk(2)1045 640 y Fp(2)23 b Fm(L)1180 609 y Fl(1)1180 660 y Fk(+)1250 640 y Fr(\(\012\),)k Fm(V)42 b Fp(\022)22 b Fo(R)1655 609 y Fn(n)1732 640 y Fr(b)r(e)j(an)g(op)r(en)h(connected)f(set)g(and)g (\000)e(:=)g Fm(@)5 b Fr(\012)14 b Fp(\\)g Fm(V)41 b Fp(6)p Fr(=)23 b Fp(;)p Fr(.)456 779 y(Suc)n(h)28 b(monotonicit)n(y)d (estimates)h(are)i(w)n(ell-kno)n(wn)d(for)j(the)h(in)n(v)n(erse)e (conductivit)n(y)f(problem,)456 879 y(cf.,)i(e.g.,)f(Ik)n(ehata,)f (Kang,)h(Seo,)g(and)h(Sheen)f([)1931 879 y SDict begin H.S end 1931 879 a 0 TeXcolorgray Fr(18)p 0 TeXcolorgray 2015 825 a SDict begin H.R end 2015 825 a 2015 879 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Kan97) cvn H.B /ANN pdfmark end 2015 879 a Fr(,)2065 879 y SDict begin H.S end 2065 879 a 0 TeXcolorgray Fr(15)p 0 TeXcolorgray 2148 825 a SDict begin H.R end 2148 825 a 2148 879 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.ikehata1998size) cvn H.B /ANN pdfmark end 2148 879 a Fr(].)456 1019 y(Lemma)754 1019 y SDict begin H.S end 754 1019 a 0 TeXcolorgray Fr(2.1)p 0 TeXcolorgray 860 965 a SDict begin H.R end 860 965 a 860 1019 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 860 1019 a 30 w Fr(follo)n(ws)g(from)h ([)1386 1019 y SDict begin H.S end 1386 1019 a 0 TeXcolorgray Fr(11)p 0 TeXcolorgray 1470 965 a SDict begin H.R end 1470 965 a 1470 1019 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.harrach2009uniqueness) cvn H.B /ANN pdfmark end 1470 1019 a Fr(,)i(Lemma)e(4.1].)42 b(Since)29 b(the)i(pro)r(of)e(is)g(simple)d(and)k(short,)g(w)n(e)456 1118 y(include)c(it)h(for)g(the)h(sak)n(e)e(of)i(completeness.)p 0 TeXcolorgray 456 1331 a Fj(Pr)l(o)l(of)i(of)h(L)l(emma)1070 1331 y SDict begin H.S end 1070 1331 a 0 TeXcolorgray Fj(2.1)p 0 TeXcolorgray 1180 1278 a SDict begin H.R end 1180 1278 a 1180 1331 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.1) cvn H.B /ANN pdfmark end 1180 1331 a Fj(.)p 0 TeXcolorgray 42 w Fr(Let)f Fm(g)f Fp(2)d Fm(L)1605 1301 y Fk(2)1642 1331 y Fr(\(\000\))k(and)f Fm(u)1999 1343 y Fn(i)2052 1331 y Fr(:=)d Fm(u)2214 1288 y Fk(\()p Fn(g)r Fk(\))2214 1341 y Fn(q)2244 1349 y Fe(i)2330 1331 y Fp(2)h Fm(H)2488 1301 y Fk(1)2525 1331 y Fr(\(\012\))j(b)r(e)g(the)f (corresp)r(onding)456 1431 y(solutions)24 b(of)34 b(\()935 1431 y SDict begin H.S end 935 1431 a 0 TeXcolorgray Fr(1.3)p 0 TeXcolorgray 1042 1377 a SDict begin H.R end 1042 1377 a 1042 1431 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 1042 1431 a Fr(\))28 b(for)f Fm(i)c Fp(2)g(f)p Fr(1)p Fm(;)14 b Fr(2)p Fp(g)p Fr(.)35 b(Then,)486 1616 y Fm(b)522 1628 y Fn(i)549 1616 y Fr(\()p Fm(u)629 1628 y Fn(i)656 1616 y Fm(;)14 b(w)r Fr(\))24 b(:=)921 1503 y Fi(Z)967 1692 y Fk(\012)1032 1616 y Fp(r)p Fm(u)1149 1628 y Fn(i)1177 1616 y Fp(r)p Fm(w)d Fr(+)d Fm(q)1446 1628 y Fn(i)1474 1616 y Fm(uw)e Fr(d)p Fm(x)23 b Fr(=)1801 1503 y Fi(Z)1847 1692 y Fk(\000)1906 1616 y Fm(g)s(w)r Fp(j)2033 1628 y Fk(\000)2093 1616 y Fr(d)p Fm(s)g Fr(=:)f Fm(l)r Fr(\()p Fm(w)r Fr(\))84 b Fp(8)p Fm(w)25 b Fp(2)e Fm(H)2832 1582 y Fk(1)2869 1616 y Fr(\(\012\))p Fm(;)42 b(i)23 b Fp(2)g(f)p Fr(1)p Fm(;)14 b Fr(2)p Fp(g)p Fm(:)456 1841 y Fr(No)n(w,)27 b(w)n(e)g(use)g(this)g(and)h(consider)894 1984 y(\()p Fm(g)s(;)14 b Fr(\(\003)1096 1996 y Fk(\000)1141 1984 y Fr(\()p Fm(q)1210 1996 y Fk(2)1248 1984 y Fr(\))k Fp(\000)g Fr(\003)1439 1996 y Fk(\000)1484 1984 y Fr(\()p Fm(q)1553 1996 y Fk(1)1591 1984 y Fr(\)\))c Fm(g)s Fr(\))1744 2009 y Fn(L)1790 1992 y Fd(2)1822 2009 y Fk(\(\000\))1069 2119 y Fr(=)23 b Fm(l)r Fr(\()p Fm(u)1264 2131 y Fk(2)1301 2119 y Fr(\))18 b Fp(\000)g Fm(l)r Fr(\()p Fm(u)1541 2131 y Fk(1)1578 2119 y Fr(\))23 b(=)g Fm(b)1757 2131 y Fk(2)1794 2119 y Fr(\()p Fm(u)1874 2131 y Fk(2)1911 2119 y Fm(;)14 b(u)1996 2131 y Fk(2)2032 2119 y Fr(\))19 b Fp(\000)f Fr(2)p Fm(b)2244 2131 y Fk(2)2280 2119 y Fr(\()p Fm(u)2360 2131 y Fk(2)2398 2119 y Fm(;)c(u)2483 2131 y Fk(1)2519 2119 y Fr(\))19 b(+)f Fm(b)2689 2131 y Fk(1)2726 2119 y Fr(\()p Fm(u)2806 2131 y Fk(1)2843 2119 y Fm(;)c(u)2928 2131 y Fk(1)2965 2119 y Fr(\))1069 2299 y(=)23 b Fp(\000)1236 2186 y Fi(Z)1282 2375 y Fk(\012)1333 2299 y Fr(\()p Fm(q)1402 2311 y Fk(2)1458 2299 y Fp(\000)18 b Fm(q)1578 2311 y Fk(1)1615 2299 y Fr(\))p Fm(u)1695 2265 y Fk(2)1695 2320 y(1)1751 2299 y Fp(\000)g Fr(\()p Fp(r)p Fr(\()p Fm(u)2015 2311 y Fk(2)2071 2299 y Fp(\000)g Fm(u)2202 2311 y Fk(1)2239 2299 y Fr(\)\))2303 2258 y Fk(2)2359 2299 y Fp(\000)g Fm(q)2479 2311 y Fk(2)2516 2299 y Fr(\()p Fm(u)2596 2311 y Fk(1)2652 2299 y Fp(\000)g Fm(u)2783 2311 y Fk(2)2820 2299 y Fr(\))2852 2265 y Fk(2)2903 2299 y Fr(d)p Fm(x:)456 2484 y Fr(Since)26 b Fm(q)709 2496 y Fk(2)770 2484 y Fp(\025)c Fr(0,)28 b(the)g(assertion)d(follo)n (ws.)1669 b Fb(\003)456 2594 y SDict begin H.S end 456 2594 a 456 2594 a SDict begin 12 H.A end 456 2594 a 456 2594 a SDict begin [/View [/XYZ H.V]/Dest (section.4) cvn /DEST pdfmark end 456 2594 a 1440 2718 a Fr(4.)41 b Fq(Localized)30 b(Potentials)456 2907 y Fr(Again,)24 b(let)h Fm(q)874 2919 y Fk(1)911 2907 y Fm(;)14 b(q)985 2919 y Fk(2)1045 2907 y Fp(2)23 b Fm(L)1180 2877 y Fl(1)1180 2928 y Fk(+)1250 2907 y Fr(\(\012\),)k Fm(V)42 b Fp(\022)22 b Fo(R)1655 2877 y Fn(n)1732 2907 y Fr(b)r(e)j(an)g(op)r(en)h (connected)f(set)g(and)g(\000)e(:=)g Fm(@)5 b Fr(\012)14 b Fp(\\)g Fm(V)41 b Fp(6)p Fr(=)23 b Fp(;)p Fr(.)456 3007 y(In)k(addition,)d(as)i(assumed)g(in)f(Lemma)1733 3007 y SDict begin H.S end 1733 3007 a 0 TeXcolorgray Fr(2.2)p 0 TeXcolorgray 1839 2953 a SDict begin H.R end 1839 2953 a 1839 3007 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 1839 3007 a Fr(,)i(let)f Fm(q)2045 3019 y Fk(1)2105 3007 y Fo(\015)d Fm(q)2230 3019 y Fk(2)2294 3007 y Fr(on)k(\012)16 b Fp(\\)h Fm(V)46 b Fr(\(i.e.,)26 b Fm(q)2876 3019 y Fk(1)2914 3007 y Fp(j)2937 3019 y Fk(\012)p Fl(\\)p Fn(V)3110 3007 y Fp(\025)c Fm(q)3234 3019 y Fk(2)3272 3007 y Fp(j)3295 3019 y Fk(\012)p Fl(\\)p Fn(V)456 3107 y Fr(and)27 b Fm(q)654 3119 y Fk(1)691 3107 y Fp(j)714 3119 y Fk(\012)p Fl(\\)p Fn(V)887 3107 y Fp(6\021)c Fm(q)1012 3119 y Fk(2)1049 3107 y Fp(j)1072 3119 y Fk(\012)p Fl(\\)p Fn(V)1222 3107 y Fr(\).)456 3246 y(Since)h(the)i(op)r(en)g(set)f Fm(V)34 b Fp(\\)14 b Fr(\012)26 b(is)f(a)g(coun)n(table)f(union)g(of)i(closed)e(balls)f (and)i Fm(q)2849 3258 y Fk(1)2910 3246 y Fo(\015)d Fm(q)3034 3258 y Fk(2)3097 3246 y Fr(on)j(\012)15 b Fp(\\)f Fm(V)19 b Fr(,)456 3346 y(there)27 b(exists)f(a)h(closed)f(ball)456 3489 y(\(4.1\))1433 3489 y SDict begin H.S end 1433 3489 a 1433 3489 a SDict begin 12 H.A end 1433 3489 a 1433 3489 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.1) cvn /DEST pdfmark end 1433 3489 a Fm(B)h Fp(\022)22 b Fm(V)38 b Fp(\\)19 b Fr(\012)83 b(where)f Fm(q)2245 3501 y Fk(1)2305 3489 y Fo(\015)23 b Fm(q)2430 3501 y Fk(2)456 3632 y Fr(and)k Fm(V)37 b Fp(n)18 b Fm(B)32 b Fr(is)27 b(connected.)456 3771 y(T)-7 b(o)38 b(pro)n(v)n(e)f(Lemma)1131 3771 y SDict begin H.S end 1131 3771 a 0 TeXcolorgray Fr(2.2)p 0 TeXcolorgray 1237 3718 a SDict begin H.R end 1237 3718 a 1237 3771 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 1237 3771 a Fr(,)k(w)n(e)e(in)n(tro)r (duce)e(t)n(w)n(o)g(op)r(erators)g(in)h(De\014nition)2866 3771 y SDict begin H.S end 2866 3771 a 0 TeXcolorgray Fr(4.3)p 0 TeXcolorgray 2972 3718 a SDict begin H.R end 2972 3718 a 2972 3771 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.3) cvn H.B /ANN pdfmark end 2972 3771 a 39 w Fr(and)g(presen)n(t)456 3871 y(some)29 b(prop)r(erties)f(of)j(these)g (op)r(erators)d(and)j(their)e(adjoin)n(ts)h(in)g(Lemma)2827 3871 y SDict begin H.S end 2827 3871 a 0 TeXcolorgray Fr(4.4)p 0 TeXcolorgray 2933 3818 a SDict begin H.R end 2933 3818 a 2933 3871 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.4) cvn H.B /ANN pdfmark end 2933 3871 a Fr(.)46 b(In)31 b(the)g(pro)r(of)456 3971 y(of)c(Lemma)847 3971 y SDict begin H.S end 847 3971 a 0 TeXcolorgray Fr(4.4)p 0 TeXcolorgray 953 3917 a SDict begin H.R end 953 3917 a 953 3971 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.4) cvn H.B /ANN pdfmark end 953 3971 a Fr(,)g(the)h(follo)n(wing)c(t)n (w)n(o)j(theorems)f(pla)n(y)f(a)j(k)n(ey)e(role.)456 4125 y SDict begin H.S end 456 4125 a 456 4125 a SDict begin 12 H.A end 456 4125 a 456 4125 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.1) cvn /DEST pdfmark end 456 4125 a 0 TeXcolorgray Fy(Theorem)42 b(4.1.)p 0 TeXcolorgray 46 w Fj(L)l(et)d Fm(H)1291 4137 y Fk(1)1328 4125 y Fm(;)14 b(H)1434 4137 y Fk(2)1510 4125 y Fj(b)l(e)40 b(two)f(Hilb)l(ert)g(sp)l (ac)l(es,)k Fm(L)d Fp(2)g(L)p Fr(\()p Fm(H)2706 4137 y Fk(1)2744 4125 y Fm(;)14 b(H)2850 4137 y Fk(2)2887 4125 y Fr(\))39 b Fj(and)h Fm(h)g Fp(2)h Fm(H)3382 4137 y Fk(2)3419 4125 y Fj(.)456 4225 y(Then,)456 4367 y Fr(\(4.2\))840 4367 y SDict begin H.S end 840 4367 a 840 4367 a SDict begin 12 H.A end 840 4367 a 840 4367 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.2) cvn /DEST pdfmark end 840 4367 a Fm(h)23 b Fp(2)g(R)p Fr(\()p Fm(L)p Fr(\))108 b Fp(,)g(9)p Fm(C)30 b(>)22 b Fr(0)37 b(:)g Fp(j)p Fr(\()p Fm(h;)14 b(g)s Fr(\))2055 4379 y Fn(H)2109 4387 y Fd(2)2146 4367 y Fp(j)23 b(\024)f Fm(C)6 b Fp(k)p Fm(L)2443 4333 y Fl(\003)2481 4367 y Fm(g)s Fp(k)2566 4379 y Fn(H)2620 4387 y Fd(1)2741 4367 y Fp(8)p Fm(g)25 b Fp(2)e Fm(H)3000 4379 y Fk(2)3037 4367 y Fm(:)p 0 TeXcolorgray 456 4569 a Fj(Pr)l(o)l(of.)p 0 TeXcolorgray 43 w Fr(This)30 b(is)g(a)h(w)n (ell-kno)n(wn)d(result)i(from)g(functional)f(analysis)f(\(see,)k(e.g.,) g(the)g(b)r(o)r(ok)f(of)456 4669 y(Bourbarki)24 b([)872 4669 y SDict begin H.S end 872 4669 a 0 TeXcolorgray Fr(2)p 0 TeXcolorgray 914 4615 a SDict begin H.R end 914 4615 a 914 4669 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.bourbaki2003elements) cvn H.B /ANN pdfmark end 914 4669 a Fr(]\).)37 b(F)-7 b(or)27 b(Banac)n(h)f(spaces,)h(a)g(pro)r(of)g(is)g(giv)n(en)f (in)g([)2458 4669 y SDict begin H.S end 2458 4669 a 0 TeXcolorgray Fr(7)p 0 TeXcolorgray 2500 4615 a SDict begin H.R end 2500 4615 a 2500 4669 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.fruhauf2007detecting) cvn H.B /ANN pdfmark end 2500 4669 a Fr(,)i(Lemma)d(3.4].)380 b Fb(\003)456 4830 y SDict begin H.S end 456 4830 a 456 4830 a SDict begin 12 H.A end 456 4830 a 456 4830 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.2) cvn /DEST pdfmark end 456 4830 a 0 TeXcolorgray Fy(Theorem)30 b(4.2)h Fr(\(Unique)c(con)n(tin)n (uation)d(from)i(sets)h(of)h(p)r(ositiv)n(e)d(measure\))p Fy(.)p 0 TeXcolorgray 39 w Fj(L)l(et)k Fr(\012)3171 4800 y Fl(0)3218 4830 y Fp(2)23 b Fo(R)3350 4800 y Fn(m)3419 4830 y Fj(,)456 4930 y Fm(m)g Fp(\025)f Fr(3)p Fj(,)30 b(b)l(e)g(a)g(c)l(onne)l(cte)l(d)f(op)l(en)i(set)e(and)h Fm(q)c Fp(2)e Fm(L)1970 4900 y Fl(1)2040 4930 y Fr(\(\012)2132 4900 y Fl(0)2155 4930 y Fr(\))p Fj(.)39 b(The)31 b(trivial)g(solution)f (of)456 5073 y Fr(\(4.3\))1670 5073 y SDict begin H.S end 1670 5073 a 1670 5073 a SDict begin 12 H.A end 1670 5073 a 1670 5073 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.3) cvn /DEST pdfmark end 1670 5073 a 19 w Fp(\000)18 b Fr(\001)p Fm(u)g Fr(+)g Fm(q)s(u)23 b Fr(=)f(0)456 5216 y Fj(is)30 b(the)g(only)g Fm(H)939 5185 y Fk(1)976 5216 y Fr(\(\012)1068 5185 y Fl(0)1092 5216 y Fr(\))p Fj(-solution)f(vanishing)j(on)d(a)i(me)l(asur)l(able)f(set)f(of)i(p)l (ositive)g(me)l(asur)l(e.)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 447 176 a SDict begin H.S end 447 176 a 447 176 a SDict begin H.R end 447 176 a 447 176 a SDict begin [/View [/XYZ H.V]/Dest (page.5) cvn /DEST pdfmark end 447 176 a 0 TeXcolorgray 834 251 a Fk(LOCAL)29 b(UNIQUENESS)f(F)n(OR)h(AN)g(INVERSE)f(BVP)i(WITH)f(P)-5 b(AR)g(TIAL)28 b(D)n(A)-5 b(T)g(A)345 b(5)p 0 TeXcolorgray 0 TeXcolorgray 456 450 a Fj(Pr)l(o)l(of.)p 0 TeXcolorgray 43 w Fr(Theorem)1064 450 y SDict begin H.S end 1064 450 a 0 TeXcolorgray Fr(4.2)p 0 TeXcolorgray 1170 397 a SDict begin H.R end 1170 397 a 1170 450 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.2) cvn H.B /ANN pdfmark end 1170 450 a 25 w Fr(is)24 b(the)h(com)n(bination)d(of)j(the)g(follo)n (wing)c(t)n(w)n(o)j(results)g(\(cf.)36 b(the)26 b(w)n(ork)e(of)456 550 y(Rac)n(hid)i(Regbaoui)f([)1124 550 y SDict begin H.S end 1124 550 a 0 TeXcolorgray Fr(26)p 0 TeXcolorgray 1207 496 a SDict begin H.R end 1207 496 a 1207 550 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.regbaoui2001unique) cvn H.B /ANN pdfmark end 1207 550 a Fr(,)j(pro)r(of)f(of)h(Theorem)d(2.1]\).)p 0 TeXcolorgray 597 723 a(\(a\))p 0 TeXcolorgray 41 w Fm(H)820 693 y Fk(1)857 723 y Fr(\(\012)949 693 y Fl(0)973 723 y Fr(\)-solutions)g(of)34 b(\()1513 723 y SDict begin H.S end 1513 723 a 0 TeXcolorgray Fr(4.3)p 0 TeXcolorgray 1620 669 a SDict begin H.R end 1620 669 a 1620 723 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.3) cvn H.B /ANN pdfmark end 1620 723 a Fr(\))28 b(that)g(v)-5 b(anish)27 b(on)h(a)f(measurable)e(set)j(of)g(p)r(ositiv)n(e)d(mea-)744 822 y(sure)31 b(ha)n(v)n(e)e(zeros)h(of)g(in\014nite)g(order)f(\(see,)j (e.g.,)f(the)h(result)d(of)i(de)g(Figueiredo)d(and)744 922 y(Gossez)35 b([)1049 922 y SDict begin H.S end 1049 922 a 0 TeXcolorgray Fr(6)p 0 TeXcolorgray 1091 868 a SDict begin H.R end 1091 868 a 1091 922 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.de1992strict) cvn H.B /ANN pdfmark end 1091 922 a Fr(,)j(Prop)r(osition)33 b(3])i(or)g(the)i(result)e(of)g (Hadi)h(and)f(Tsouli)f([)2977 922 y SDict begin H.S end 2977 922 a 0 TeXcolorgray Fr(10)p 0 TeXcolorgray 3060 868 a SDict begin H.R end 3060 868 a 3060 922 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.hadi2001strong) cvn H.B /ANN pdfmark end 3060 922 a Fr(,)k(Theorem)744 1022 y(2.1]\).)p 0 TeXcolorgray 592 1121 a(\(b\))p 0 TeXcolorgray 42 w(The)32 b(trivial)c(solution)h Fm(u)g Fr(=)g(0)j(is)e(the)i(only)e Fm(H)2229 1091 y Fk(1)2266 1121 y Fr(\(\012)2358 1091 y Fl(0)2382 1121 y Fr(\)-solution)f(of)38 b(\()2897 1121 y SDict begin H.S end 2897 1121 a 0 TeXcolorgray Fr(4.3)p 0 TeXcolorgray 3003 1068 a SDict begin H.R end 3003 1068 a 3003 1121 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.3) cvn H.B /ANN pdfmark end 3003 1121 a Fr(\))32 b(that)g(has)f(a)744 1221 y(zero)19 b(of)h(in\014nite)f(order)f(\(see,)k(e.g,)f(the)f(b)r(o) r(ok)g(of)f(H\177)-42 b(ormander)18 b([)2683 1221 y SDict begin H.S end 2683 1221 a 0 TeXcolorgray Fr(14)p 0 TeXcolorgray 2766 1167 a SDict begin H.R end 2766 1167 a 2766 1221 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.hormander1983analysis) cvn H.B /ANN pdfmark end 2766 1221 a Fr(,)k(Theorem)c(17.2.6]\).)3380 1394 y Fb(\003)456 1572 y SDict begin H.S end 456 1572 a 456 1572 a SDict begin 12 H.A end 456 1572 a 456 1572 a SDict begin [/View [/XYZ H.V]/Dest (theorem.4.3) cvn /DEST pdfmark end 456 1572 a 0 TeXcolorgray Fy(De\014nition)29 b(4.3)g Fr(\(Virtual)24 b(measuremen)n(t)f(op)r(erators\))p Fy(.)p 0 TeXcolorgray 38 w Fr(Let)j Fm(B)h Fp(\022)c Fm(V)34 b Fp(\\)16 b Fr(\012)26 b(b)r(e)g(a)g(non-empt)n(y)456 1671 y(closed)g(ball)g(with)h Fm(q)1089 1683 y Fk(1)1150 1671 y Fo(\015)c Fm(q)1275 1683 y Fk(2)1340 1671 y Fr(on)28 b Fm(B)k Fr(as)27 b(in)g(\()1782 1671 y SDict begin H.S end 1782 1671 a 0 TeXcolorgray Fr(4.1)p 0 TeXcolorgray 1888 1618 a SDict begin H.R end 1888 1618 a 1888 1671 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.1) cvn H.B /ANN pdfmark end 1888 1671 a Fr(\))q(.)38 b(The)28 b(op)r(erators)d Fm(L)2577 1683 y Fn(B)2662 1671 y Fr(and)j Fm(L)2881 1686 y Fk(\012)p Fl(n)p Fn(V)3047 1671 y Fr(are)f(de\014ned) 456 1771 y(b)n(y)1326 1925 y SDict begin H.S end 1326 1925 a 1326 1925 a SDict begin 12 H.A end 1326 1925 a 1326 1925 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.4) cvn /DEST pdfmark end 1326 1925 a Fm(L)1383 1937 y Fn(B)1463 1925 y Fr(:)c Fm(L)1566 1891 y Fk(2)1603 1925 y Fr(\()p Fm(B)t Fr(\))h Fp(!)f Fm(L)1921 1891 y Fk(2)1958 1925 y Fr(\(\000\))p Fm(;)249 b(f)32 b Fp(7!)23 b Fm(v)2565 1937 y Fn(B)2622 1925 y Fp(j)2645 1937 y Fk(\000)2690 1925 y Fm(;)p 0 TeXcolorgray -2257 w Fr(\(4.4\))p 0 TeXcolorgray 1106 2051 a Fm(L)1163 2066 y Fk(\012)p Fl(n)p Fn(V)1325 2051 y Fr(:)g Fm(L)1428 2017 y Fk(2)1465 2051 y Fr(\(\012)18 b Fp(n)g Fm(V)h Fr(\))24 b Fp(!)f Fm(L)1921 2017 y Fk(2)1958 2051 y Fr(\(\000\))p Fm(;)249 b(h)23 b Fp(7!)g Fm(v)2563 2066 y Fk(\012)p Fl(n)p Fn(V)2702 2051 y Fp(j)2725 2063 y Fk(\000)2771 2051 y Fm(;)p 0 TeXcolorgray -2338 w Fr(\(4.5\))p 0 TeXcolorgray 456 2211 a(where)k Fm(v)736 2223 y Fn(B)793 2211 y Fm(;)14 b(v)870 2226 y Fk(\012)p Fl(n)p Fn(V)1032 2211 y Fp(2)24 b Fm(H)1187 2181 y Fk(1)1224 2211 y Fr(\(\012\))k(are)f (the)h(unique)e(solutions)f(of)952 2381 y SDict begin H.S end 952 2381 a 952 2381 a SDict begin 12 H.A end 952 2381 a 952 2381 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.6) cvn /DEST pdfmark end 952 2381 a Fp(\000)p Fr(\001)p Fm(v)1126 2393 y Fn(B)1202 2381 y Fr(+)18 b Fm(q)1322 2393 y Fk(1)1359 2381 y Fm(v)1399 2393 y Fn(B)1479 2381 y Fr(=)23 b Fp(j)p Fm(q)1627 2393 y Fk(1)1683 2381 y Fp(\000)18 b Fm(q)1803 2393 y Fk(2)1840 2381 y Fp(j)1863 2347 y Fk(1)p Fn(=)p Fk(2)1968 2381 y Fm(f)9 b(\037)2070 2393 y Fn(B)2154 2381 y Fr(in)27 b(\012)83 b(with)f Fm(@)2682 2393 y Fn(\027)2724 2381 y Fm(v)2764 2393 y Fn(B)2821 2381 y Fp(j)2844 2393 y Fn(@)t Fk(\012)2958 2381 y Fr(=)22 b(0)p Fm(;)p 0 TeXcolorgray -2654 w Fr(\(4.6\))p 0 TeXcolorgray 708 2513 a Fp(\000)p Fr(\001)p Fm(v)882 2528 y Fk(\012)p Fl(n)p Fn(V)1040 2513 y Fr(+)c Fm(q)1160 2525 y Fk(1)1197 2513 y Fm(v)1237 2528 y Fk(\012)p Fl(n)p Fn(V)1399 2513 y Fr(=)23 b Fp(j)p Fm(q)1547 2525 y Fk(1)1603 2513 y Fp(\000)18 b Fm(q)1723 2525 y Fk(2)1760 2513 y Fp(j)1783 2479 y Fk(1)p Fn(=)p Fk(2)1887 2513 y Fm(h\037)1987 2528 y Fk(\012)p Fl(n)p Fn(V)2154 2513 y Fr(in)27 b(\012)83 b(with)f Fm(@)2682 2525 y Fn(\027)2724 2513 y Fm(v)2764 2528 y Fk(\012)p Fl(n)p Fn(V)2903 2513 y Fp(j)2926 2525 y Fn(@)t Fk(\012)3040 2513 y Fr(=)22 b(0)p Fm(;)p 0 TeXcolorgray -2736 w Fr(\(4.7\))p 0 TeXcolorgray 456 2668 a(or)k(equiv)-5 b(alen)n(tly)922 2747 y Fi(Z)968 2935 y Fk(\012)1033 2860 y Fp(r)p Fm(v)1142 2872 y Fn(B)1218 2860 y Fp(\001)19 b(r)p Fm(w)i Fr(+)d Fm(q)1529 2872 y Fk(1)1566 2860 y Fm(v)1606 2872 y Fn(B)1664 2860 y Fm(w)e Fr(d)p Fm(x)24 b Fr(=)1943 2747 y Fi(Z)1990 2935 y Fn(B)2061 2860 y Fp(j)p Fm(q)2121 2872 y Fk(1)2176 2860 y Fp(\000)18 b Fm(q)2296 2872 y Fk(2)2334 2860 y Fp(j)2357 2826 y Fk(1)p Fn(=)p Fk(2)2461 2860 y Fm(w)r(f)23 b Fr(d)p Fm(x)84 b Fp(8)p Fm(w)25 b Fp(2)e Fm(H)3048 2826 y Fk(1)3085 2860 y Fr(\(\012\))p Fm(;)456 2860 y SDict begin H.S end 456 2860 a 456 2860 a SDict begin 15 H.A end 456 2860 a 456 2860 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.8) cvn /DEST pdfmark end 456 2860 a Fr(\(4.8\))758 2969 y Fi(Z)804 3157 y Fk(\012)869 3082 y Fp(r)p Fm(v)978 3097 y Fk(\012)p Fl(n)p Fn(V)1136 3082 y Fp(\001)c(r)p Fm(w)i Fr(+)d Fm(q)1447 3094 y Fk(1)1484 3082 y Fm(v)1524 3097 y Fk(\012)p Fl(n)p Fn(V)1664 3082 y Fm(w)e Fr(d)p Fm(x)24 b Fr(=)1943 2969 y Fi(Z)1990 3157 y Fk(\012)p Fl(n)p Fn(V)2142 3082 y Fp(j)p Fm(q)2202 3094 y Fk(1)2258 3082 y Fp(\000)18 b Fm(q)2378 3094 y Fk(2)2416 3082 y Fp(j)2439 3047 y Fk(1)p Fn(=)p Fk(2)2543 3082 y Fm(w)r(h)c Fr(d)p Fm(x)84 b Fp(8)p Fm(w)25 b Fp(2)e Fm(H)3128 3047 y Fk(1)3165 3082 y Fr(\(\012\))p Fm(:)456 3082 y SDict begin H.S end 456 3082 a 456 3082 a SDict begin 15 H.A end 456 3082 a 456 3082 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.9) cvn /DEST pdfmark end 456 3082 a Fr(\(4.9\))p 0 TeXcolorgray 456 3304 a Fy(Lemma)29 b(4.4.)p 0 TeXcolorgray 0 TeXcolorgray 181 w Fr(\(a\))p 0 TeXcolorgray 42 w Fj(The)i(adjoint)g (op)l(er)l(ators)456 3459 y Fr(\(4.10\))942 3459 y SDict begin H.S end 942 3459 a 942 3459 a SDict begin 12 H.A end 942 3459 a 942 3459 a SDict begin [/View [/XYZ H.V]/Dest (Item.1) cvn /DEST pdfmark end 942 3459 a Fm(L)999 3425 y Fl(\003)999 3480 y Fn(B)1078 3459 y Fr(:)24 b Fm(L)1182 3425 y Fk(2)1218 3459 y Fr(\(\000\))g Fp(!)f Fm(L)1521 3425 y Fk(2)1558 3459 y Fr(\()p Fm(B)t Fr(\))85 b Fj(and)h Fm(L)2048 3425 y Fl(\003)2048 3482 y Fk(\012)p Fl(n)p Fn(V)2209 3459 y Fr(:)23 b Fm(L)2312 3425 y Fk(2)2349 3459 y Fr(\(\000\))h Fp(!)f Fm(L)2652 3425 y Fk(2)2688 3459 y Fr(\(\012)c Fp(n)f Fm(V)h Fr(\))744 3617 y Fj(ful\014l)t(l)456 3792 y Fr(\(4.11\))889 3792 y SDict begin H.S end 889 3792 a 889 3792 a SDict begin 12 H.A end 889 3792 a 889 3792 a SDict begin [/View [/XYZ H.V]/Dest (Item.2) cvn /DEST pdfmark end 889 3792 a Fm(L)946 3757 y Fl(\003)946 3812 y Fn(B)1003 3792 y Fm(g)25 b Fr(=)1166 3699 y Fi(\020)1216 3792 y Fp(j)p Fm(q)1276 3804 y Fk(1)1331 3792 y Fp(\000)19 b Fm(q)1452 3804 y Fk(2)1489 3792 y Fp(j)1512 3757 y Fk(1)p Fn(=)p Fk(2)1616 3792 y Fm(u)1664 3699 y Fi(\021)1713 3696 y(\014)1713 3746 y(\014)1713 3796 y(\014)1741 3850 y Fn(B)1897 3792 y Fj(and)85 b Fm(L)2170 3757 y Fl(\003)2170 3814 y Fk(\012)p Fl(n)p Fn(V)2309 3792 y Fm(g)25 b Fr(=)2472 3699 y Fi(\020)2522 3792 y Fp(j)p Fm(q)2582 3804 y Fk(1)2637 3792 y Fp(\000)18 b Fm(q)2757 3804 y Fk(2)2795 3792 y Fp(j)2818 3757 y Fk(1)p Fn(=)p Fk(2)2922 3792 y Fm(u)2970 3699 y Fi(\021)3019 3696 y(\014)3019 3746 y(\014)3019 3796 y(\014)3047 3850 y Fk(\012)p Fl(n)p Fn(V)3200 3792 y Fm(;)744 4014 y Fj(wher)l(e)31 b Fm(u)22 b Fr(:=)h Fm(u)1208 3971 y Fk(\()p Fn(g)r Fk(\))1208 4024 y Fn(q)1238 4032 y Fd(1)1321 4014 y Fp(2)h Fm(H)1476 3984 y Fk(1)1513 4014 y Fr(\(\012\))30 b Fj(is)g(the)g(c)l(orr)l(esp)l(onding)h (solution)f(of)48 b Fr(\()2879 4014 y SDict begin H.S end 2879 4014 a 0 TeXcolorgray Fr(1.3)p 0 TeXcolorgray 2985 3961 a SDict begin H.R end 2985 3961 a 2985 4014 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 2985 4014 a Fr(\))q Fj(.)p 0 TeXcolorgray 592 4122 a Fr(\(b\))p 0 TeXcolorgray 42 w Fj(The)31 b(adjoint)g(op)l(er)l (ator)g Fm(L)1580 4092 y Fl(\003)1580 4145 y Fn(B)1666 4122 y Fj(is)f(inje)l(ctive)h(and)p 2246 4050 249 4 v 31 w Fp(R)p Fr(\()p Fm(L)2405 4134 y Fn(B)2462 4122 y Fr(\))23 b(=)g Fm(L)2662 4092 y Fk(2)2699 4122 y Fr(\(\000\))p Fj(.)p 0 TeXcolorgray 601 4222 a Fr(\(c\))p 0 TeXcolorgray 42 w Fp(R)p Fr(\()p Fm(L)903 4234 y Fn(B)961 4222 y Fr(\))18 b Fp(\\)h(R)p Fr(\()p Fm(L)1244 4237 y Fk(\012)p Fl(n)p Fn(V)1383 4222 y Fr(\))24 b(=)e Fp(f)p Fr(0)p Fp(g)p Fj(.)p 0 TeXcolorgray 592 4322 a Fr(\(d\))p 0 TeXcolorgray 42 w Fp(R)p Fr(\()p Fm(L)903 4334 y Fn(B)961 4322 y Fr(\))h Fp(6\022)g(R)p Fr(\()p Fm(L)1263 4337 y Fk(\012)p Fl(n)p Fn(V)1402 4322 y Fr(\))p Fj(.)p 0 TeXcolorgray 601 4427 a Fr(\(e\))p 0 TeXcolorgray 42 w Fp(6)g(9)p Fm(C)30 b(>)22 b Fr(0)37 b(:)g Fp(k)p Fm(L)1227 4397 y Fl(\003)1227 4450 y Fn(B)1283 4427 y Fm(g)s Fp(k)22 b(\024)h Fm(C)6 b Fp(k)p Fm(L)1642 4397 y Fl(\003)1642 4454 y Fk(\012)p Fl(n)p Fn(V)1780 4427 y Fm(g)s Fp(k)84 b(8)p Fm(g)25 b Fp(2)e Fm(L)2196 4397 y Fk(2)2233 4427 y Fr(\(\000\))p Fj(.)p 0 TeXcolorgray 456 4688 a(Pr)l(o)l(of.)p 0 TeXcolorgray 0 TeXcolorgray 184 w Fr(\(a\))p 0 TeXcolorgray 41 w(F)-7 b(or)27 b Fm(f)32 b Fp(2)24 b Fm(L)1362 4658 y Fk(2)1398 4688 y Fr(\()p Fm(B)t Fr(\),)29 b(let)e Fm(v)1744 4645 y Fk(\()p Fn(f)7 b Fk(\))1741 4712 y Fn(B)1862 4688 y Fp(2)23 b Fm(H)2016 4658 y Fk(1)2053 4688 y Fr(\(\012\))28 b(b)r(e)g(the)g(solution)e(of)754 4775 y Fi(Z)800 4964 y Fk(\012)865 4888 y Fp(r)p Fm(v)977 4845 y Fk(\()p Fn(f)7 b Fk(\))974 4913 y Fn(B)1091 4888 y Fp(\001)19 b(r)p Fm(w)i Fr(+)d Fm(q)1402 4900 y Fk(1)1439 4888 y Fm(v)1482 4845 y Fk(\()p Fn(f)7 b Fk(\))1479 4913 y Fn(B)1577 4888 y Fm(w)17 b Fr(d)p Fm(x)23 b Fr(=)1857 4775 y Fi(Z)1903 4964 y Fn(B)1974 4888 y Fp(j)p Fm(q)2034 4900 y Fk(1)2090 4888 y Fp(\000)18 b Fm(q)2210 4900 y Fk(2)2247 4888 y Fp(j)2270 4854 y Fk(1)p Fn(=)p Fk(2)2375 4888 y Fm(w)r(f)23 b Fr(d)p Fm(x)83 b Fp(8)p Fm(w)25 b Fp(2)f Fm(H)2962 4854 y Fk(1)2999 4888 y Fr(\(\012\))p Fm(:)744 5085 y Fr(Then,)k(in)f(particular,)1698 5216 y Fm(L)1755 5228 y Fn(B)1812 5216 y Fm(f)32 b Fr(=)22 b Fm(v)2015 5172 y Fk(\()p Fn(f)7 b Fk(\))2012 5240 y Fn(B)2110 5216 y Fp(j)2133 5228 y Fk(\000)2179 5216 y Fm(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 447 176 a SDict begin H.S end 447 176 a 447 176 a SDict begin H.R end 447 176 a 447 176 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end 447 176 a 0 TeXcolorgray 456 251 a Fk(6)729 b(BASTIAN)22 b(HARRA)n(CH)i(AND)e(MAR)n(CEL)h(ULLRICH)p 0 TeXcolorgray 744 454 a Fr(F)-7 b(urthermore,)24 b(for)h Fm(g)h Fp(2)d Fm(L)1573 424 y Fk(2)1610 454 y Fr(\(\000\),)j(let)f Fm(u)1941 411 y Fk(\()p Fn(g)r Fk(\))1941 464 y Fn(q)1971 472 y Fd(1)2057 454 y Fr(b)r(e)h(the)f(corresp)r(onding)e(solution)g (of)32 b(\()3283 454 y SDict begin H.S end 3283 454 a 0 TeXcolorgray Fr(1.3)p 0 TeXcolorgray 3389 400 a SDict begin H.R end 3389 400 a 3389 454 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 3389 454 a Fr(\).)744 570 y(Then,)c Fm(u)1032 527 y Fk(\()p Fn(g)r Fk(\))1032 580 y Fn(q)1062 588 y Fd(1)1150 570 y Fr(also)d(solv)n(es)h(the)i(equiv)-5 b(alen)n(t)25 b(v)-5 b(ariational)23 b(form)n(ulation)934 669 y Fi(Z)980 857 y Fk(\012)1045 782 y Fp(r)p Fm(w)f Fp(\001)c(r)p Fm(u)1353 748 y Fk(\()p Fn(g)r Fk(\))1353 802 y Fn(q)1383 810 y Fd(1)1462 782 y Fr(+)g Fm(q)1582 794 y Fk(1)1619 782 y Fm(w)r(u)1728 748 y Fk(\()p Fn(g)r Fk(\))1728 802 y Fn(q)1758 810 y Fd(1)1833 782 y Fr(d)p Fm(x)23 b Fr(=)2037 669 y Fi(Z)2083 857 y Fk(\000)2142 782 y Fm(g)s(w)r Fp(j)2269 794 y Fk(\000)2328 782 y Fr(d)p Fm(s)83 b Fp(8)p Fm(w)25 b Fp(2)f Fm(H)2782 748 y Fk(1)2819 782 y Fr(\(\012\))p Fm(:)744 1003 y Fr(Hence,)k(for)f(arbitrary)e Fm(f)31 b Fp(2)24 b Fm(L)1703 973 y Fk(2)1740 1003 y Fr(\()p Fm(B)t Fr(\))k(and)f Fm(g)f Fp(2)d Fm(L)2261 973 y Fk(2)2298 1003 y Fr(\(\000\),)1077 1215 y(\()p Fm(f)t(;)14 b(L)1248 1181 y Fl(\003)1248 1235 y Fn(B)1305 1215 y Fm(g)s Fr(\))1380 1230 y Fn(L)1426 1213 y Fd(2)1458 1230 y Fk(\()p Fn(B)s Fk(\))1590 1215 y Fr(=)22 b(\()p Fm(L)1766 1227 y Fn(B)1823 1215 y Fm(f)t(;)14 b(g)s Fr(\))1980 1230 y Fn(L)2026 1213 y Fd(2)2058 1230 y Fk(\(\000\))2178 1215 y Fr(=)2266 1102 y Fi(Z)2312 1290 y Fk(\000)2371 1215 y Fm(g)s(v)2457 1172 y Fk(\()p Fn(f)7 b Fk(\))2454 1239 y Fn(B)2552 1215 y Fp(j)2575 1227 y Fk(\000)2634 1215 y Fr(d)p Fm(s)1590 1437 y Fr(=)1677 1324 y Fi(Z)1724 1512 y Fk(\012)1789 1437 y Fp(r)p Fm(v)1901 1394 y Fk(\()p Fn(f)g Fk(\))1898 1461 y Fn(B)2015 1437 y Fp(\001)18 b(r)p Fm(u)2173 1402 y Fk(\()p Fn(g)r Fk(\))2173 1457 y Fn(q)2203 1465 y Fd(1)2282 1437 y Fr(+)g Fm(q)2402 1449 y Fk(1)2439 1437 y Fm(v)2482 1394 y Fk(\()p Fn(f)7 b Fk(\))2479 1461 y Fn(B)2577 1437 y Fm(u)2625 1402 y Fk(\()p Fn(g)r Fk(\))2625 1457 y Fn(q)2655 1465 y Fd(1)2729 1437 y Fr(d)p Fm(x)1590 1659 y Fr(=)1677 1546 y Fi(Z)1724 1734 y Fn(B)1795 1659 y Fp(j)p Fm(q)1855 1671 y Fk(1)1910 1659 y Fp(\000)18 b Fm(q)2030 1671 y Fk(2)2068 1659 y Fp(j)2091 1624 y Fk(1)p Fn(=)p Fk(2)2195 1659 y Fm(u)2243 1624 y Fk(\()p Fn(g)r Fk(\))2243 1679 y Fn(q)2273 1687 y Fd(1)2333 1659 y Fm(f)23 b Fr(d)p Fm(x)1590 1863 y Fr(=)1677 1771 y Fi(\020)1727 1863 y Fm(f)t(;)1809 1771 y Fi(\020)1859 1863 y Fp(j)p Fm(q)1919 1875 y Fk(1)1974 1863 y Fp(\000)18 b Fm(q)2094 1875 y Fk(2)2132 1863 y Fp(j)2155 1829 y Fk(1)p Fn(=)p Fk(2)2259 1863 y Fm(u)2307 1829 y Fk(\()p Fn(g)r Fk(\))2307 1883 y Fn(q)2337 1891 y Fd(1)2397 1771 y Fi(\021)2461 1863 y Fp(j)2484 1875 y Fn(B)2541 1771 y Fi(\021)2590 1921 y Fn(L)2636 1904 y Fd(2)2668 1921 y Fk(\()p Fn(B)s Fk(\))2791 1863 y Fm(:)744 2109 y Fr(This)27 b(yields)e Fm(L)1224 2079 y Fl(\003)1224 2132 y Fn(B)1281 2109 y Fm(g)h Fr(=)1444 2017 y Fi(\020)1494 2109 y Fp(j)p Fm(q)1554 2121 y Fk(1)1610 2109 y Fp(\000)18 b Fm(q)1730 2121 y Fk(2)1767 2109 y Fp(j)1790 2079 y Fk(1)p Fn(=)p Fk(2)1894 2109 y Fm(u)1942 2066 y Fk(\()p Fn(g)r Fk(\))1942 2119 y Fn(q)1972 2127 y Fd(1)2032 2017 y Fi(\021)2082 2013 y(\014)2082 2063 y(\014)2082 2113 y(\014)2110 2167 y Fn(B)2167 2109 y Fr(.)844 2271 y(Analogously)-7 b(,)24 b(it)j(follo)n(ws)e Fm(L)1744 2241 y Fl(\003)1744 2298 y Fk(\012)p Fl(n)p Fn(V)1882 2271 y Fm(g)h Fr(=)2045 2179 y Fi(\020)2095 2271 y Fp(j)p Fm(q)2155 2283 y Fk(1)2211 2271 y Fp(\000)18 b Fm(q)2331 2283 y Fk(2)2368 2271 y Fp(j)2391 2241 y Fk(1)p Fn(=)p Fk(2)2496 2271 y Fm(u)2544 2228 y Fk(\()p Fn(g)r Fk(\))2544 2281 y Fn(q)2574 2289 y Fd(1)2634 2179 y Fi(\021)2683 2175 y(\014)2683 2225 y(\014)2683 2275 y(\014)2711 2329 y Fk(\012)p Fl(n)p Fn(V)2850 2271 y Fr(.)p 0 TeXcolorgray 592 2419 a(\(b\))p 0 TeXcolorgray 42 w(First,)35 b(w)n(e)f(pro)n(v)n(e)f(the)i(injectivit)n(y)d(of)i Fm(L)2045 2389 y Fl(\003)2045 2442 y Fn(B)2102 2419 y Fr(.)57 b(Let)35 b Fm(g)i Fp(2)d Fm(L)2561 2389 y Fk(2)2598 2419 y Fr(\(\000\))h(with)f Fm(L)3002 2389 y Fl(\003)3002 2442 y Fn(B)3059 2419 y Fm(g)j Fr(=)d(0)g(and)744 2537 y Fm(u)h Fr(:=)h Fm(u)999 2494 y Fk(\()p Fn(g)r Fk(\))999 2547 y Fn(q)1029 2555 y Fd(1)1124 2537 y Fp(2)g Fm(H)1291 2507 y Fk(1)1328 2537 y Fr(\(\012\))g(b)r(e)f(the)h(corresp)r(onding)c (solution)g(of)42 b(\()2762 2537 y SDict begin H.S end 2762 2537 a 0 TeXcolorgray Fr(1.3)p 0 TeXcolorgray 2868 2484 a SDict begin H.R end 2868 2484 a 2868 2537 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 2868 2537 a Fr(\))q(.)59 b(>F)-7 b(rom)34 b(\(a\))h(it)744 2640 y(follo)n(ws)c Fm(L)1080 2610 y Fl(\003)1080 2663 y Fn(B)1137 2640 y Fm(g)37 b Fr(=)1312 2573 y Fi(\000)1350 2640 y Fp(j)p Fm(q)1410 2652 y Fk(1)1466 2640 y Fp(\000)18 b Fm(q)1586 2652 y Fk(2)1623 2640 y Fp(j)1646 2610 y Fk(1)p Fn(=)p Fk(2)1751 2640 y Fm(u)1799 2573 y Fi(\001)1850 2640 y Fp(j)1873 2652 y Fn(B)1930 2640 y Fr(.)57 b(Since)33 b Fm(q)2270 2652 y Fk(1)2330 2640 y Fp(\000)23 b Fm(q)2455 2652 y Fk(2)2526 2640 y Fo(\015)34 b Fr(0)f(on)h Fm(B)t Fr(,)i(there)e(exists)f(a)744 2751 y(measurable)27 b(set)j Fm(E)i Fp(\022)27 b Fm(B)34 b Fr(of)c(p)r(ositiv)n(e)e(measure)g(where) i Fp(j)p Fm(q)2631 2763 y Fk(1)2688 2751 y Fp(\000)20 b Fm(q)2810 2763 y Fk(2)2847 2751 y Fp(j)2870 2721 y Fk(1)p Fn(=)p Fk(2)3002 2751 y Fp(6)p Fr(=)26 b(0.)44 b(Hence,)744 2787 y Fi(\000)782 2854 y Fp(j)p Fm(q)842 2866 y Fk(1)898 2854 y Fp(\000)18 b Fm(q)1018 2866 y Fk(2)1056 2854 y Fp(j)1079 2824 y Fk(1)p Fn(=)p Fk(2)1183 2854 y Fm(u)1231 2787 y Fi(\001)1282 2854 y Fp(j)1305 2866 y Fn(E)1388 2854 y Fp(\021)26 b Fr(0)j(implies)d Fm(u)p Fp(j)1905 2866 y Fn(E)1987 2854 y Fp(\021)g Fr(0.)42 b(>F)-7 b(rom)28 b(Theorem)2797 2854 y SDict begin H.S end 2797 2854 a 0 TeXcolorgray Fr(4.2)p 0 TeXcolorgray 2903 2801 a SDict begin H.R end 2903 2801 a 2903 2854 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.2) cvn H.B /ANN pdfmark end 2903 2854 a 30 w Fr(it)h(follo)n(ws)d(that)744 2954 y Fm(u)f Fp(\021)f Fr(0)k(on)h(\012)g(and)f(th)n(us)h Fm(g)e Fr(=)e Fm(@)1729 2966 y Fn(\027)1770 2954 y Fm(u)p Fp(j)1841 2966 y Fk(\000)1911 2954 y Fr(=)f(0.)40 b(This)28 b(sho)n(ws)f(the)i(injectivit)n(y)e(of)h Fm(L)3225 2924 y Fl(\003)3225 2977 y Fn(B)3311 2954 y Fr(and)744 3064 y(th)n(us)p 927 2992 249 4 v 28 w Fp(R)p Fr(\()p Fm(L)1086 3076 y Fn(B)1143 3064 y Fr(\))c(=)e Fp(N)12 b Fr(\()p Fm(L)1455 3034 y Fl(\003)1455 3087 y Fn(B)1513 3064 y Fr(\))1545 3034 y Fl(?)1624 3064 y Fr(=)23 b Fm(L)1769 3034 y Fk(2)1805 3064 y Fr(\(\000\).)p 0 TeXcolorgray 601 3163 a(\(c\))p 0 TeXcolorgray 42 w(Recall)j(that)h Fm(B)32 b Fr(and)c(\012)18 b Fp(n)g Fm(V)47 b Fr(are)26 b(closed)g(in)h(\012)h(and)f(that)h Fm(V)37 b Fp(n)18 b Fm(B)32 b Fr(is)27 b(connected.)36 b(Let)1225 3333 y Fm(\036)23 b Fr(=)g Fm(L)1442 3345 y Fn(B)1499 3333 y Fm(f)31 b Fr(=)23 b Fm(L)1716 3348 y Fk(\012)p Fl(n)p Fn(V)1855 3333 y Fm(h)f Fp(2)i(R)p Fr(\()p Fm(L)2163 3345 y Fn(B)2220 3333 y Fr(\))19 b Fp(\\)g(R)p Fr(\()p Fm(L)2504 3348 y Fk(\012)p Fl(n)p Fn(V)2643 3333 y Fr(\))744 3508 y(and)38 b Fm(v)956 3520 y Fn(B)1013 3508 y Fm(;)14 b(v)1090 3523 y Fk(\012)p Fl(n)p Fn(V)1269 3508 y Fp(2)40 b Fm(H)1440 3478 y Fk(1)1477 3508 y Fr(\(\012\))e(b)r(e)g(the)f (corresp)r(onding)e(solutions)g(of)i(De\014nition)3315 3508 y SDict begin H.S end 3315 3508 a 0 TeXcolorgray Fr(4.3)p 0 TeXcolorgray 3421 3454 a SDict begin H.R end 3421 3454 a 3421 3508 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.3) cvn H.B /ANN pdfmark end 3421 3508 a Fr(.)744 3607 y(First,)27 b(w)n(e)g(sho)n(w)g(that)456 3797 y(\(4.12\))1035 3797 y SDict begin H.S end 1035 3797 a 1035 3797 a SDict begin 12 H.A end 1035 3797 a 1035 3797 a SDict begin [/View [/XYZ H.V]/Dest (Item.3) cvn /DEST pdfmark end 1035 3797 a Fm(v)1075 3809 y Fn(B)1156 3797 y Fr(=)22 b Fm(v)1283 3812 y Fk(\012)p Fl(n)p Fn(V)1506 3797 y Fr(on)82 b(\012)19 b Fp(n)1815 3705 y Fi(\020)p 1864 3730 68 4 v 1864 3797 a Fm(B)k Fp([)p 2024 3725 206 4 v 19 w Fr(\012)18 b Fp(n)g Fm(V)2229 3705 y Fi(\021)2302 3797 y Fr(=)k(\(\012)d Fp(\\)g Fm(V)g Fr(\))f Fp(n)g Fm(B)28 b Fr(:)744 3987 y(On)g(\012)18 b Fp([)h Fm(V)g Fr(,)28 b(w)n(e)f(de\014ne)h(the)g(con)n(tin)n(uations)1563 4275 y Fm(q)e Fr(:=)1736 4133 y Fi(\()1803 4218 y Fm(q)1840 4230 y Fk(1)1878 4218 y Fm(;)83 b Fr(on)27 b(\012)p Fm(;)1803 4338 y Fr(1)p Fm(;)116 b Fr(on)27 b Fm(V)37 b Fp(n)18 b Fr(\012)p Fm(;)975 4619 y Fr(~)-45 b Fm(v)26 b Fr(:=)1149 4477 y Fi(\()1216 4563 y Fm(v)s(;)83 b Fr(on)27 b(\012)p Fm(;)1216 4682 y Fr(0)p Fm(;)84 b Fr(on)27 b Fm(V)38 b Fp(n)18 b Fr(\012)p Fm(;)1816 4619 y Fr(and)86 b(~)-46 b Fm(v)2072 4631 y Fn(j)2131 4619 y Fr(:=)2241 4477 y Fi(\()2308 4563 y Fm(@)2352 4575 y Fn(x)2390 4583 y Fe(j)2425 4563 y Fm(v)s(;)83 b Fr(on)28 b(\012)p Fm(;)2308 4682 y Fr(0)p Fm(;)201 b Fr(on)28 b Fm(V)37 b Fp(n)18 b Fr(\012)p Fm(;)744 4841 y Fr(where)27 b Fm(v)g Fr(:=)22 b Fm(v)1201 4853 y Fn(B)1277 4841 y Fp(\000)c Fm(v)1400 4856 y Fk(\012)p Fl(n)p Fn(V)1540 4841 y Fr(.)844 4947 y(Ob)n(viously)-7 b(,)27 b(~)-45 b Fm(v)s(;)17 b Fr(~)-45 b Fm(v)1370 4959 y Fn(j)1428 4947 y Fp(2)24 b Fm(L)1564 4916 y Fk(2)1600 4947 y Fr(\(\012)18 b Fp([)f Fm(V)i Fr(\).)37 b(T)-7 b(o)27 b(v)n(erify)e(that)30 b(~)-45 b Fm(v)26 b Fp(2)e Fm(H)2693 4916 y Fk(1)2730 4947 y Fr(\(\012)17 b Fp([)g Fm(V)i Fr(\),)28 b(it)e(is)g(left)h(to)744 5046 y(sho)n(w)g Fm(@)994 5058 y Fn(x)1032 5066 y Fe(j)1070 5046 y Fr(~)-45 b Fm(v)26 b Fr(=)g(~)-45 b Fm(v)1261 5058 y Fn(j)1296 5046 y Fr(.)37 b(This)27 b(can)g(b)r(e)h(sho)n(wn)f(b)n(y)g(using)1338 5216 y Fm(v)s Fp(j)1404 5228 y Fk(\000)1473 5216 y Fr(=)c Fm(v)1601 5228 y Fn(B)1658 5216 y Fp(j)1681 5228 y Fk(\000)1745 5216 y Fp(\000)18 b Fm(v)1868 5231 y Fk(\012)p Fl(n)p Fn(V)2007 5216 y Fp(j)2030 5228 y Fk(\000)2098 5216 y Fr(=)23 b Fm(\036)c Fp(\000)f Fm(\036)23 b Fr(=)g(0)p Fm(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 447 176 a SDict begin H.S end 447 176 a 447 176 a SDict begin H.R end 447 176 a 447 176 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end 447 176 a 0 TeXcolorgray 834 251 a Fk(LOCAL)29 b(UNIQUENESS)f(F)n(OR)h(AN)g(INVERSE)f(BVP)i(WITH)f(P)-5 b(AR)g(TIAL)28 b(D)n(A)-5 b(T)g(A)345 b(7)p 0 TeXcolorgray 744 450 a Fr(Let)28 b Fm(')23 b Fp(2)h(D)r Fr(\(\012)19 b Fp([)g Fm(V)g Fr(\))p Fm(;)28 b Fr(then,)1051 528 y Fi(Z)1097 716 y Fk(\012)p Fl([)p Fn(V)1264 641 y Fr(~)-45 b Fm(v)s(@)1348 653 y Fn(x)1386 661 y Fe(j)1421 641 y Fm(')14 b Fr(d)p Fm(x)24 b Fr(=)1694 528 y Fi(Z)1740 716 y Fk(\012)1805 641 y Fm(v)s(@)1892 653 y Fn(x)1930 661 y Fe(j)1965 641 y Fm(')14 b Fr(d)p Fm(x)1606 863 y Fr(=)23 b Fp(\000)1773 750 y Fi(Z)1818 938 y Fk(\012)1883 863 y Fm('@)1981 875 y Fn(x)2019 883 y Fe(j)2055 863 y Fm(v)17 b Fr(d)p Fm(x)i Fr(+)2307 750 y Fi(Z)2353 938 y Fn(@)t Fk(\012)2443 863 y Fr(\()p Fm('v)s Fr(\))p Fp(j)2627 875 y Fk(\000)2674 863 y Fm(\027)2715 875 y Fn(j)2764 863 y Fr(d)p Fm(s)1606 1084 y Fr(=)k Fp(\000)1773 971 y Fi(Z)1818 1160 y Fk(\012)1883 1084 y Fm(v)1923 1096 y Fn(j)1959 1084 y Fm(')14 b Fr(d)p Fm(x)23 b Fr(=)g Fp(\000)2310 971 y Fi(Z)2356 1160 y Fk(\012)p Fl([)p Fn(V)2522 1084 y Fr(~)-45 b Fm(v)2559 1096 y Fn(j)2594 1084 y Fm(')14 b Fr(d)p Fm(x:)844 1279 y Fr(No)n(w,)27 b(w)n(e)g(go)g(on)g(sho)n(wing)g(\()1754 1279 y SDict begin H.S end 1754 1279 a 0 TeXcolorgray Fr(4.12)p 0 TeXcolorgray 1902 1226 a SDict begin H.R end 1902 1226 a 1902 1279 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.3) cvn H.B /ANN pdfmark end 1902 1279 a Fr(\).)37 b(Since)27 b Fm(v)f Fr(=)g(~)-45 b Fm(v)s Fp(j)2431 1291 y Fk(\012)2510 1279 y Fr(ful\014lls)927 1357 y Fi(Z)973 1546 y Fk(\012)1038 1470 y Fp(r)p Fm(v)22 b Fp(\001)d(r)p Fm(w)i Fr(+)d Fm(q)1480 1482 y Fk(1)1517 1470 y Fm(v)s(w)f Fr(d)p Fm(x)1102 1692 y Fr(=)1190 1579 y Fi(Z)1236 1768 y Fk(\012)1301 1692 y Fp(j)p Fm(q)1361 1704 y Fk(1)1417 1692 y Fp(\000)h Fm(q)1537 1704 y Fk(2)1574 1692 y Fp(j)1597 1658 y Fk(1)p Fn(=)p Fk(2)1702 1692 y Fm(w)r Fr(\()p Fm(f)9 b(\037)1897 1704 y Fn(B)1973 1692 y Fp(\000)18 b Fm(h\037)2156 1707 y Fk(\012)p Fl(n)p Fn(V)2295 1692 y Fr(\))c(d)p Fm(x)83 b Fp(8)p Fm(w)25 b Fp(2)f Fm(H)2803 1658 y Fk(1)2840 1692 y Fr(\(\012\))p Fm(;)744 1882 y Fr(it)j(holds)1111 1939 y Fi(Z)1157 2128 y Fn(V)15 b Fl(n)p Fn(B)1316 2052 y Fp(r)s Fr(~)-45 b Fm(v)21 b Fp(\001)e(r)p Fm(')g Fr(+)f Fm(q)6 b Fr(~)-45 b Fm(v)s(')14 b Fr(d)p Fm(x)24 b Fr(=)f(0)82 b Fp(8)p Fm(')23 b Fp(2)g(D)16 b Fr(\()p Fm(V)38 b Fp(n)18 b Fm(B)t Fr(\))c Fm(:)744 2267 y Fr(W)-7 b(e)28 b(obtain)f(that)k(~)-46 b Fm(v)31 b Fr(\(as)d(a)f(function)g(in)g Fm(H)2098 2237 y Fk(1)2149 2267 y Fr(\()p Fm(V)37 b Fp(n)18 b Fm(B)t Fr(\))q(\))28 b(solv)n(es)1460 2415 y Fp(\000)p Fr(\001)s(~)-45 b Fm(v)21 b Fr(+)d Fm(q)6 b Fr(~)-45 b Fm(v)27 b Fr(=)22 b(0)83 b(on)g Fm(V)37 b Fp(n)18 b Fm(B)744 2563 y Fr(and)25 b(v)-5 b(anishes)23 b(on)i Fm(V)31 b Fp(n)13 b Fr(\012.)36 b(Since)24 b Fm(V)31 b Fp(n)13 b Fr(\012)24 b(is)g(a)g(non-empt)n(y)g (op)r(en)g(set)h(\()p Fm(V)44 b Fr(is)24 b(op)r(en)h(and)744 2663 y(has)h(a)g(non-empt)n(y)f(in)n(tersection)f(\000)i(with)g(the)h (Lipsc)n(hitz-domain)22 b(\012\))27 b(and)f Fm(V)35 b Fp(n)16 b Fm(B)31 b Fr(is)744 2762 y(connected,)d(Theorem)1506 2762 y SDict begin H.S end 1506 2762 a 0 TeXcolorgray Fr(4.2)p 0 TeXcolorgray 1612 2709 a SDict begin H.R end 1612 2709 a 1612 2762 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.2) cvn H.B /ANN pdfmark end 1612 2762 a 27 w Fr(sho)n(ws)f(that)k(~)-45 b Fm(v)26 b Fp(\021)d Fr(0)k(on)g Fm(V)37 b Fp(n)18 b Fm(B)32 b Fr(and)c(th)n(us)1403 2911 y Fm(v)1443 2923 y Fn(B)1524 2911 y Fr(=)22 b Fm(v)1651 2926 y Fk(\012)p Fl(n)p Fn(V)1874 2911 y Fr(on)82 b(\()p Fm(V)38 b Fp(\\)19 b Fr(\012\))g Fp(n)e Fm(B)t(:)844 3059 y Fr(T)-7 b(o)27 b(\014nally)f(sho)n(w)g Fm(\036)e Fr(=)e(0,)28 b(w)n(e)f(de\014ne)1518 3277 y Fm(u)22 b Fr(:=)1699 3135 y Fi(\()1766 3221 y Fm(v)1806 3233 y Fn(B)2028 3221 y Fr(on)28 b(\012)18 b Fp(n)g Fm(B)t(;)1766 3340 y(v)1806 3355 y Fk(\012)p Fl(n)p Fn(V)2028 3340 y Fr(on)28 b Fm(B)t(:)744 3506 y Fr(W)-7 b(e)38 b(can)f(partition)e (test)j(functions)f(\(in)g Fp(D)r Fr(\(\012\))h(and)g Fm(H)2570 3475 y Fk(1)2607 3506 y Fr(\(\012\)\),)i(b)n(y)e(using)e(smo) r(oth)744 3605 y(partitions)c(of)h(unit)n(y)-7 b(,)35 b(to)f(pro)n(v)n(e)e(that)j Fm(u)e Fr(is)g(an)g Fm(H)2364 3575 y Fk(1)2401 3605 y Fr(\(\012\)-function)h(and)g(the)g(unique)744 3705 y(solution)25 b(of)1503 3853 y Fp(\000)p Fr(\001)p Fm(u)18 b Fr(+)g Fm(q)1823 3865 y Fk(1)1860 3853 y Fm(u)23 b Fr(=)g(0)82 b(on)h(\012)p Fm(;)1661 3983 y(@)1705 3995 y Fn(\027)1747 3983 y Fm(u)p Fp(j)1818 3995 y Fn(@)t Fk(\012)1931 3983 y Fr(=)23 b(0)p Fm(:)744 4131 y Fr(Hence,)28 b Fm(u)f Fr(has)g(to)h(b)r(e)g(equal)e(to)h(the)h(trivial)c(solution)i (and)h(th)n(us)1586 4279 y Fm(\036)d Fr(=)e Fm(v)1786 4291 y Fn(B)1844 4279 y Fp(j)1867 4291 y Fk(\000)1935 4279 y Fr(=)h Fm(u)p Fp(j)2094 4291 y Fk(\000)2162 4279 y Fp(\021)f Fr(0)p Fm(:)p 0 TeXcolorgray 592 4428 a Fr(\(d\))p 0 TeXcolorgray 42 w(This)27 b(simply)e(follo)n(ws)f(from)i(\(b\))j(and) e(\(c\).)p 0 TeXcolorgray 601 4527 a(\(e\))p 0 TeXcolorgray 42 w(Let)h(us)g(assume)e(there)h(exists)f(a)h(constan)n(t)g Fm(C)j(>)22 b Fr(0)27 b(suc)n(h)h(that)1329 4677 y Fp(k)p Fm(L)1428 4643 y Fl(\003)1428 4697 y Fn(B)1484 4677 y Fm(g)s Fp(k)22 b(\024)h Fm(C)6 b Fp(k)p Fm(L)1843 4643 y Fl(\003)1843 4699 y Fk(\012)p Fl(n)p Fn(V)1981 4677 y Fm(g)s Fp(k)82 b(8)p Fm(g)25 b Fp(2)e Fm(L)2395 4643 y Fk(2)2432 4677 y Fr(\(\000\))p Fm(:)744 4828 y Fr(Then,)1605 4942 y Fp(R)p Fr(\()p Fm(L)1764 4954 y Fn(B)1821 4942 y Fr(\))h Fp(\022)e(R)p Fr(\()p Fm(L)2123 4957 y Fk(\012)p Fl(n)p Fn(V)2263 4942 y Fr(\))744 5073 y(immediately)h(follo)n(ws)i (from)h(Theorem)2037 5073 y SDict begin H.S end 2037 5073 a 0 TeXcolorgray Fr(4.1)p 0 TeXcolorgray 2143 5020 a SDict begin H.R end 2143 5020 a 2143 5073 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.1) cvn H.B /ANN pdfmark end 2143 5073 a 27 w Fr(and)i(this)f(is)f(a)h(con)n(tradiction)e(to) i(\(d\).)3380 5216 y Fb(\003)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 8 8 TeXDict begin HPSdict begin 8 7 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 447 176 a SDict begin H.S end 447 176 a 447 176 a SDict begin H.R end 447 176 a 447 176 a SDict begin [/View [/XYZ H.V]/Dest (page.8) cvn /DEST pdfmark end 447 176 a 0 TeXcolorgray 456 251 a Fk(8)729 b(BASTIAN)22 b(HARRA)n(CH)i(AND)e(MAR)n(CEL)h(ULLRICH)p 0 TeXcolorgray 0 TeXcolorgray 456 450 a Fj(Pr)l(o)l(of)30 b(of)h(L)l(emma)1070 450 y SDict begin H.S end 1070 450 a 0 TeXcolorgray Fj(2.2)p 0 TeXcolorgray 1180 397 a SDict begin H.R end 1180 397 a 1180 450 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2.2) cvn H.B /ANN pdfmark end 1180 450 a Fj(.)p 0 TeXcolorgray 42 w Fr(The)d(assertion)d(follo)n(ws)g(from)h(Lemma)2530 450 y SDict begin H.S end 2530 450 a 0 TeXcolorgray Fr(4.4)p 0 TeXcolorgray 2636 397 a SDict begin H.R end 2636 397 a 2636 450 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.4) cvn H.B /ANN pdfmark end 2636 450 a Fr(:)456 585 y(The)j(trivial)c(case)j(is)g(that)h(when)h Fm(L)1595 555 y Fl(\003)1595 612 y Fk(\012)p Fl(n)p Fn(V)1762 585 y Fr(is)e(not)h(injectiv)n(e.)40 b(Then,)29 b(there)g(exists)f(an)g (elemen)n(t)456 703 y Fm(g)34 b Fp(2)f Fm(L)675 672 y Fk(2)711 703 y Fr(\(\000\))23 b Fp(n)e(f)p Fr(0)p Fp(g)32 b Fr(with)g Fp(k)p Fm(L)1364 672 y Fl(\003)1364 729 y Fk(\012)p Fl(n)p Fn(V)1503 703 y Fm(g)s Fp(k)f Fr(=)g(0.)53 b(By)33 b(the)g(injectivit)n(y)d(of)j Fm(L)2671 672 y Fl(\003)2671 725 y Fn(B)2761 703 y Fr(w)n(e)g(ha)n(v)n(e)e Fp(k)p Fm(L)3184 672 y Fl(\003)3184 725 y Fn(B)3241 703 y Fm(g)s Fp(k)g Fr(=:)456 810 y Fm(c)492 822 y Fn(g)553 810 y Fp(\025)23 b Fr(0.)36 b(In)28 b(this)f(case,)g(w)n(e)g(can)g(set) h Fm(g)1651 822 y Fn(m)1737 810 y Fr(:=)22 b Fm(mg)30 b Fr(for)d(all)f Fm(m)d Fp(2)g Fo(N)t Fr(.)456 945 y(When)h Fm(L)751 915 y Fl(\003)751 972 y Fk(\012)p Fl(n)p Fn(V)914 945 y Fr(is)f(injectiv)n(e,)g(w)n(e)h(deriv)n(e)e(a)h(suitable)f (sequence)i(\()p Fm(g)2488 957 y Fn(m)2551 945 y Fr(\))2583 957 y Fn(m)p Fl(2)p Fc(N)2756 945 y Fp(\022)f Fm(L)2901 915 y Fk(2)2937 945 y Fr(\(\000\))i(as)e(follo)n(ws.)456 1098 y(Let)38 b Fm(C)674 1110 y Fn(m)780 1098 y Fr(=)j Fm(m)959 1068 y Fk(2)1035 1098 y Fr(for)d Fm(m)j Fp(2)h Fo(N)t Fr(.)77 b(Lemma)1845 1098 y SDict begin H.S end 1845 1098 a 0 TeXcolorgray Fr(4.4)p 0 TeXcolorgray 1951 1044 a SDict begin H.R end 1951 1044 a 1951 1098 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.4.4) cvn H.B /ANN pdfmark end 1951 1098 a 39 w Fr(\(e\))39 b(implies)c(the)40 b(existence)d(of)i(a)f(sequence)456 1197 y(\()p Fm(g)531 1167 y Fl(0)528 1218 y Fn(m)591 1197 y Fr(\))623 1209 y Fn(m)p Fl(2)p Fc(N)796 1197 y Fp(\022)22 b Fm(L)940 1167 y Fk(2)977 1197 y Fr(\(\000\))28 b(with)456 1332 y(\(4.13\))1300 1332 y SDict begin H.S end 1300 1332 a 1300 1332 a SDict begin 12 H.A end 1300 1332 a 1300 1332 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.13) cvn /DEST pdfmark end 1300 1332 a Fp(k)p Fm(L)1399 1298 y Fl(\003)1399 1352 y Fn(B)1455 1332 y Fm(g)1498 1298 y Fl(0)1495 1352 y Fn(m)1558 1332 y Fp(k)22 b Fm(>)h(C)1769 1344 y Fn(m)1832 1332 y Fp(k)p Fm(L)1931 1298 y Fl(\003)1931 1355 y Fk(\012)p Fl(n)p Fn(V)2069 1332 y Fm(g)2112 1298 y Fl(0)2109 1352 y Fn(m)2172 1332 y Fp(k)83 b(8)p Fm(m)22 b Fp(2)h Fo(N)t Fm(:)456 1510 y Fr(In)i(particular,)d(this)i(implies)d Fm(g)1442 1480 y Fl(0)1439 1531 y Fn(m)1524 1510 y Fp(6)p Fr(=)i(0)h(for)g(all)f Fm(m)g Fp(2)g Fo(N)t Fr(.)42 b(Since)24 b Fm(L)2479 1480 y Fl(\003)2479 1537 y Fk(\012)p Fl(n)p Fn(V)2643 1510 y Fr(is)f(injectiv)n(e,)h(w)n(e)g(can)g(set)456 1650 y Fm(g)496 1662 y Fn(m)581 1650 y Fr(:=)848 1608 y Fn(g)882 1583 y Fa(0)880 1625 y Fe(m)p 702 1631 381 4 v 702 1679 a Fn(m)p Fl(k)p Fn(L)841 1659 y Fa(\003)841 1701 y Fd(\012)p Fa(n)p Fe(V)961 1679 y Fn(g)995 1662 y Fa(0)993 1696 y Fe(m)1048 1679 y Fl(k)1092 1650 y Fr(.)37 b(By)28 b(m)n(ultiplying)c(\()1759 1650 y SDict begin H.S end 1759 1650 a 0 TeXcolorgray Fr(4.13)p 0 TeXcolorgray 1906 1597 a SDict begin H.R end 1906 1597 a 1906 1650 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.13) cvn H.B /ANN pdfmark end 1906 1650 a Fr(\))k(with)2339 1618 y Fk(1)p 2165 1632 381 4 v 2165 1679 a Fn(m)p Fl(k)p Fn(L)2304 1659 y Fa(\003)2304 1701 y Fd(\012)p Fa(n)p Fe(V)2424 1679 y Fn(g)2458 1662 y Fa(0)2456 1696 y Fe(m)2512 1679 y Fl(k)2556 1650 y Fr(,)g(w)n(e)f(obtain)1515 1817 y Fp(k)p Fm(L)1614 1783 y Fl(\003)1614 1838 y Fn(B)1670 1817 y Fm(g)1710 1829 y Fn(m)1773 1817 y Fp(k)c Fm(>)f(m)83 b Fp(8)p Fm(m)22 b Fp(2)i Fo(N)t Fm(:)456 1952 y Fr(F)-7 b(urthermore,)25 b(it)i(holds)1464 2086 y Fp(k)p Fm(L)1563 2051 y Fl(\003)1563 2108 y Fk(\012)p Fl(n)p Fn(V)1701 2086 y Fm(g)1741 2098 y Fn(m)1804 2086 y Fp(k)c Fr(=)1982 2030 y(1)p 1966 2067 73 4 v 1966 2143 a Fm(m)2132 2086 y Fp(8)p Fm(m)f Fp(2)i Fo(N)t Fm(:)456 2266 y Fr(F)-7 b(or)27 b(b)r(oth)h(cases,)e(w)n(e)h(obtain)g(a)g(sequence)g(\()p Fm(g)1898 2278 y Fn(m)1961 2266 y Fr(\))1993 2278 y Fn(m)p Fl(2)p Fc(N)2171 2266 y Fr(suc)n(h)g(that)1104 2443 y(lim)1066 2493 y Fn(m)p Fl(!1)1271 2330 y Fi(Z)1317 2519 y Fn(V)14 b Fl(\\)p Fk(\012)1467 2443 y Fr(\()p Fm(q)1536 2455 y Fk(1)1592 2443 y Fp(\000)k Fm(q)1712 2455 y Fk(2)1749 2443 y Fr(\))p Fm(u)1829 2455 y Fn(m)1892 2409 y Fk(2)1943 2443 y Fr(d)p Fm(x)24 b Fr(=)60 b(lim)2147 2493 y Fn(m)p Fl(!1)2352 2443 y Fp(k)p Fm(L)2451 2409 y Fl(\003)2451 2464 y Fn(B)2507 2443 y Fm(g)2547 2455 y Fn(m)2610 2443 y Fp(k)2652 2409 y Fk(2)2712 2443 y Fr(=)23 b Fp(1)p Fm(;)1032 2665 y Fr(lim)994 2715 y Fn(m)p Fl(!1)1199 2552 y Fi(Z)1246 2741 y Fk(\012)p Fl(n)p Fn(V)1385 2665 y Fr(\()p Fm(q)1454 2677 y Fk(1)1510 2665 y Fp(\000)18 b Fm(q)1630 2677 y Fk(2)1667 2665 y Fr(\))p Fm(u)1747 2677 y Fn(m)1810 2631 y Fk(2)1861 2665 y Fr(d)p Fm(x)24 b Fr(=)60 b(lim)2065 2715 y Fn(m)p Fl(!1)2270 2665 y Fp(k)p Fm(L)2369 2631 y Fl(\003)2369 2688 y Fk(\012)p Fl(n)p Fn(V)2507 2665 y Fm(g)2547 2677 y Fn(m)2610 2665 y Fp(k)2652 2631 y Fk(2)2712 2665 y Fr(=)23 b(0)456 2885 y(where)k Fm(u)744 2897 y Fn(m)829 2885 y Fr(:=)c Fm(u)988 2842 y Fk(\()p Fn(g)1046 2850 y Fe(m)1101 2842 y Fk(\))988 2895 y Fn(q)1018 2903 y Fd(1)1154 2885 y Fp(2)h Fm(H)1309 2855 y Fk(1)1346 2885 y Fr(\(\012\))k(is)e(the)i(corresp)r(onding)d (solution)g(of)34 b(\()2705 2885 y SDict begin H.S end 2705 2885 a 0 TeXcolorgray Fr(1.3)p 0 TeXcolorgray 2811 2832 a SDict begin H.R end 2811 2832 a 2811 2885 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 2811 2885 a Fr(\))q(.)513 b Fb(\003)1708 3090 y SDict begin H.S end 1708 3090 a 1708 3090 a SDict begin 12 H.A end 1708 3090 a 1708 3090 a SDict begin [/View [/XYZ H.V]/Dest (section*.1) cvn /DEST pdfmark end 1708 3090 a Fq(References)456 3140 y SDict begin H.S end 456 3140 a 456 3140 a SDict begin 10 H.A end 456 3140 a 456 3140 a SDict begin [/View [/XYZ H.V]/Dest (cite.astala2006calderon) cvn /DEST pdfmark end 456 3140 a 0 TeXcolorgray 491 3223 a Fx(1.)p 0 TeXcolorgray 35 w(Kari)22 b(Astala)g(and)g(Lassi)g(P\177) -35 b(aiv\177)g(arin)n(ta,)24 b Fh(Calder\023)-36 b(on)-7 b('s)25 b(inverse)e(c)l(onductivity)g(pr)l(oblem)i(in)f(the)f(plane)p Fx(,)g(Ann.)581 3306 y(of)g(Math.)h(\(2006\),)h(265{299.)456 3323 y SDict begin H.S end 456 3323 a 456 3323 a SDict begin 10 H.A end 456 3323 a 456 3323 a SDict begin [/View [/XYZ H.V]/Dest (cite.bourbaki2003elements) cvn /DEST pdfmark end 456 3323 a 0 TeXcolorgray 491 3389 a Fx(2.)p 0 TeXcolorgray 35 w(Nicolas)18 b(Bourbaki,)g Fh(Elements)g(of)h(mathematics:)30 b(T)-5 b(op)l(olo)l(gic)l(al)21 b(ve)l(ctor)d(sp)l(ac)l(es,)j(chapters) e(1-5)p Fx(,)e(Springer-)581 3472 y(V)-6 b(erlag,)24 b(2003.)456 3485 y SDict begin H.S end 456 3485 a 456 3485 a SDict begin 10 H.A end 456 3485 a 456 3485 a SDict begin [/View [/XYZ H.V]/Dest (cite.bukhgeim2008recovering) cvn /DEST pdfmark end 456 3485 a 0 TeXcolorgray 491 3555 a Fx(3.)p 0 TeXcolorgray 35 w(AL)37 b(Bukhgeim)j(and)e(AA)f(Bukhgeim,)42 b Fh(R)l(e)l(c)l(overing)d(a)g(p)l(otential)g(fr)l(om)h(c)l(auchy)f (data)h(in)e(the)g(two-)581 3638 y(dimensional)27 b(c)l(ase)p Fx(,)d(J.)f(In)n(v)n(erse)i(Ill-P)n(osed)g(Probl)f Fg(16)f Fx(\(2008\),)i(no.)f(1,)f(19{33.)456 3655 y SDict begin H.S end 456 3655 a 456 3655 a SDict begin 10 H.A end 456 3655 a 456 3655 a SDict begin [/View [/XYZ H.V]/Dest (cite.calderon1980inverse) cvn /DEST pdfmark end 456 3655 a 0 TeXcolorgray 491 3721 a Fx(4.)p 0 TeXcolorgray 35 w(Alb)r(erto)i(P)e(Calder\023)-35 b(on,)24 b Fh(On)h(an)h(inverse)f(b)l(oundary)i(value)f(pr)l(oblem)p Fx(,)e(Seminar)g(on)g(Numerical)h(Analysis)581 3804 y(and)e(its)g (Application)i(to)e(Con)n(tin)n(uum)h(Ph)n(ysics)f(\(W)g(H)f(Mey)n(er)g (and)g(M)g(A)g(Raupp,)h(eds.\),)f(Brasil.)g(Math.)581 3887 y(So)r(c.,)i(Rio)g(de)g(Janeiro,)h(1980,)f(pp.)f(65{73.)456 3900 y SDict begin H.S end 456 3900 a 456 3900 a SDict begin 10 H.A end 456 3900 a 456 3900 a SDict begin [/View [/XYZ H.V]/Dest (cite.calderon2006inverse) cvn /DEST pdfmark end 456 3900 a 0 TeXcolorgray 491 3970 a Fx(5.)p 0 TeXcolorgray 581 3970 212 4 v 259 w(,)29 b Fh(On)h(an)h(inverse)f(b)l(oundary)i (value)e(pr)l(oblem)p Fx(,)h(Comput.)f(Appl.)f(Math.)g Fg(25)f Fx(\(2006\),)j(no.)e(2{3,)581 4053 y(133{138.)456 4053 y SDict begin H.S end 456 4053 a 456 4053 a SDict begin 10 H.A end 456 4053 a 456 4053 a SDict begin [/View [/XYZ H.V]/Dest (cite.de1992strict) cvn /DEST pdfmark end 456 4053 a 0 TeXcolorgray 491 4136 a Fx(6.)p 0 TeXcolorgray 35 w(Djairo)18 b(G)g(de)g(Figueiredo)i(and)e(Jean-Pierre)h(Gossez,)g Fh(Strict)h(monotonicity)g(of)g(eigenvalues)g(and)h(unique)581 4219 y(c)l(ontinuation)p Fx(,)j(Comm.)f(P)n(artial)j(Di\013eren)n(tial) g(Equations)g Fg(17)d Fx(\(1992\),)i(no.)e(1-2,)h(339{346.)456 4236 y SDict begin H.S end 456 4236 a 456 4236 a SDict begin 10 H.A end 456 4236 a 456 4236 a SDict begin [/View [/XYZ H.V]/Dest (cite.fruhauf2007detecting) cvn /DEST pdfmark end 456 4236 a 0 TeXcolorgray 491 4302 a Fx(7.)p 0 TeXcolorgray 35 w(Florian)j(F)-6 b(r)r(\177)-37 b(uhauf,)26 b(Bastian)h(Gebauer,)f (and)h(Otmar)e(Sc)n(herzer,)h Fh(Dete)l(cting)g(interfac)l(es)h(in)h(a) f(p)l(ar)l(ab)l(olic-)581 4385 y(el)t(liptic)20 b(pr)l(oblem)i(fr)l(om) g(surfac)l(e)g(me)l(asur)l(ements)p Fx(,)e(SIAM)f(J.)f(Numer.)f(Anal.)i Fg(45)f Fx(\(2007\),)j(no.)e(2,)g(810{836.)456 4402 y SDict begin H.S end 456 4402 a 456 4402 a SDict begin 10 H.A end 456 4402 a 456 4402 a SDict begin [/View [/XYZ H.V]/Dest (cite.gebauer2008localized) cvn /DEST pdfmark end 456 4402 a 0 TeXcolorgray 491 4468 a Fx(8.)p 0 TeXcolorgray 35 w(Bastian)39 b(Gebauer,)j Fh(L)l(o)l(c)l(alize)l(d)f(p)l(otentials)f (in)e(ele)l(ctric)l(al)h(imp)l(e)l(danc)l(e)i(tomo)l(gr)l(aphy)p Fx(,)i(In)n(v)n(erse)38 b(Probl.)581 4551 y(Imaging)26 b Fg(2)e Fx(\(2008\),)h(no.)e(2,)h(251{269.)456 4568 y SDict begin H.S end 456 4568 a 456 4568 a SDict begin 10 H.A end 456 4568 a 456 4568 a SDict begin [/View [/XYZ H.V]/Dest (cite.haberman2013uniqueness) cvn /DEST pdfmark end 456 4568 a 0 TeXcolorgray 491 4634 a Fx(9.)p 0 TeXcolorgray 35 w(Boaz)k(Hab)r(erman,)g(Daniel)h(T)-6 b(ataru,)28 b(et)g(al.,)g Fh(Uniqueness)g(in)h(c)l(alder\023)-36 b(ons)31 b(pr)l(oblem)f(with)f (lipschitz)g(c)l(on-)581 4717 y(ductivities)p Fx(,)22 b(Duk)n(e)j(Mathematical)h(Journal)f Fg(162)e Fx(\(2013\),)i(no.)f(3,)f (497{516.)456 4734 y SDict begin H.S end 456 4734 a 456 4734 a SDict begin 10 H.A end 456 4734 a 456 4734 a SDict begin [/View [/XYZ H.V]/Dest (cite.hadi2001strong) cvn /DEST pdfmark end 456 4734 a 0 TeXcolorgray 66 x Fx(10.)p 0 TeXcolorgray 35 w(Islam)c(Eddine)h(Hadi)f(and)g(N)f(Tsouli,)i Fh(Str)l(ong)h(unique)f(c)l(ontinuation)i(of)f(eigenfunctions)f(for)h (p-laplacian)581 4883 y(op)l(er)l(ator)p Fx(,)k(In)n(t.)f(J.)g(Math.)g (Math.)f(Sci.)h Fg(25)f Fx(\(2001\),)j(no.)d(3,)h(213{216.)456 4900 y SDict begin H.S end 456 4900 a 456 4900 a SDict begin 10 H.A end 456 4900 a 456 4900 a SDict begin [/View [/XYZ H.V]/Dest (cite.harrach2009uniqueness) cvn /DEST pdfmark end 456 4900 a 0 TeXcolorgray 67 x Fx(11.)p 0 TeXcolorgray 35 w(Bastian)j(Harrac)n(h,)f Fh(On)h(uniqueness)h(in)f(di\013use)h(optic)l (al)g(tomo)l(gr)l(aphy)p Fx(,)g(In)n(v)n(erse)f(Problems)f Fg(25)f Fx(\(2009\),)581 5050 y(no.)f(5,)f(055010)i(\(14pp\).)456 5066 y SDict begin H.S end 456 5066 a 456 5066 a SDict begin 10 H.A end 456 5066 a 456 5066 a SDict begin [/View [/XYZ H.V]/Dest (cite.harrach2012simultaneous) cvn /DEST pdfmark end 456 5066 a 0 TeXcolorgray 67 x Fx(12.)p 0 TeXcolorgray 581 5133 212 4 v 259 w(,)17 b Fh(Simultane)l(ous)k(determination)e(of)g (the)g(di\013usion)g(and)h(absorption)g(c)l(o)l(e\016cient)f(fr)l(om)h (b)l(oundary)581 5216 y(data)p Fx(,)k(In)n(v)n(erse)h(Probl.)e(Imaging) k Fg(6)c Fx(\(2012\),)i(no.)f(4,)f(663{679.)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 447 176 a SDict begin H.S end 447 176 a 447 176 a SDict begin H.R end 447 176 a 447 176 a SDict begin [/View [/XYZ H.V]/Dest (page.9) cvn /DEST pdfmark end 447 176 a 0 TeXcolorgray 834 251 a Fk(LOCAL)29 b(UNIQUENESS)f(F)n(OR)h(AN)g(INVERSE)f(BVP)i(WITH)f(P)-5 b(AR)g(TIAL)28 b(D)n(A)-5 b(T)g(A)345 b(9)p 0 TeXcolorgray 456 367 a SDict begin H.S end 456 367 a 456 367 a SDict begin 10 H.A end 456 367 a 456 367 a SDict begin [/View [/XYZ H.V]/Dest (cite.harrach2013monotonicity) cvn /DEST pdfmark end 456 367 a 0 TeXcolorgray 83 x Fx(13.)p 0 TeXcolorgray 35 w(Bastian)33 b(Harrac)n(h)f(and)h(Marcel)f(Ullric)n(h,)k Fh(Monotonicity-b)l(ase)l(d)e(shap)l(e)g(r)l(e)l(c)l(onstruction)h(in)e (ele)l(ctric)l(al)581 533 y(imp)l(e)l(danc)l(e)28 b(tomo)l(gr)l(aphy)p Fx(,)d(SIAM)f(J.)f(Math.)h(Anal.)g Fg(45)f Fx(\(2013\),)i(no.)e(6,)h (3382{3403.)456 550 y SDict begin H.S end 456 550 a 456 550 a SDict begin 10 H.A end 456 550 a 456 550 a SDict begin [/View [/XYZ H.V]/Dest (cite.hormander1983analysis) cvn /DEST pdfmark end 456 550 a 0 TeXcolorgray 66 x Fx(14.)p 0 TeXcolorgray 35 w(Lars)i(H\177)-35 b(ormander,)26 b Fh(The)j(analysis)g(of)f(line)l 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