The Calibration Method for Free-Discontinuity Problems on Small Domains

on Small Domains

Candidate Supervisor

Maria Giovanna Mora Prof. Gianni Dal Maso

Thesis submitted for the degree of Do tor Philosophiae

Istruzione24 Aprile 1987 n. 419 tale diploma e equipollente al titolo di Dottoredi

Ri er ain Matemati a.

Trieste, annoa ademi o2000{2001.

Inottemperanzaaquantoprevistodall'art.1delDe retoLegislativoLuogotenenziale

31 Agosto 1945 n. 660 le pres ritte opie della presente pubbli azione sono state

depositate presso la Pro ura dellaRepubbli a di Trieste e presso il Commissariato

diri er adurantequestiquattroanni: lasuaprofonda onos enzadellamatemati a,

lasua ontinuadisponibilitaeil suoentusiasmoperlari er asono stati perme un

grandein entivoedinsegnamento.

Un ringraziamento molto spe iale al mio aro ami o e ollaboratore Massimiliano

Morini, on ui e stato un pia ere ondividere gioie e dolori del metodo delle

ali-brazioni.

Vorreiesprimerelamiagratitudinean heatuttiimieiami iinS.I.S.S.A., hehanno

ontribuitoaquestatesi on il loroaettoe lalorovi inanza, rendendolamia

per-manenza a Trieste un'esperienza indimenti abile. In parti olare, ringrazio Andrea,

Du io,Giovanni,MatteoeMassimiliano perlaloroami izia.

Ringraziotuttalamiafamigliaperavermisemprein oraggiatoesostenuto.

Inne un grazie di uore a Lu a: non s rivero i motivi per he sono os tanti he

questapaginanonpotrebbe ontenerlitutti,maliri orderosempre.

Introdu tion 1

1 Preliminary results 9

1.1 Fun tionsofboundedvariation . . . 9

1.2 TheEuler-LagrangeequationsfortheMumford-Shahfun tional. . . 10

1.3 The alibrationmethod fortheMumford-Shahfun tional . . . 11

2 Calibrations for minimizerswith a re tilinear dis ontinuity set 15 2.1 Amodel ase . . . 16

2.2 Thegeneral ase . . . 25

3 Calibrations for minimizerswith a regular dis ontinuity set 35 3.1 TheDiri hletminimality. . . 36

3.2 Thegraph-minimality . . . 49

3.2.1 Somepropertiesof K( ;U) . . . 57

4 Calibrations for minimizerswith a triple jun tion 61 4.1 Constru tionofthe alibration . . . 61

4.2 Estimatesfor t 1 and t 2 near u i 1 and u i . . . 68 4.3 Estimatesfor t 1 and t 2 near u 0 and u 2 . . . 75 4.4 Proofof ondition(b1) . . . 80

4.5 Theantisymmetri ase . . . 83

5 The alibrationmethodfor fun tionalson ve tor-valued maps 85 5.1 Calibrationsforfun tionalsonve tor-valuedmaps . . . 86

5.2 Anappli ationrelatedto lassi aleldtheory . . . 90

5.3 Somefurtherappli ations . . . 94

5.4 Calibrationsintermsof loseddierentialforms . . . 102

Manyvariationalproblemsarisingin severalbran hesofapplied analysis(asimage pro essing,fra ture

me hani s, theoryof nemati liquid rystals)leadto onsider minimumproblems forfun tionals whi h

oupleavolumeandasurfa eintegral,dependingona losedset K andafun tion usmoothoutsideK.

Following aterminologybyE. De Giorgi,variationalproblems ofthis kindare alled free-dis ontinuity

problems,and, intheweakformulationproposed byE.De GiorgiandL. Ambrosio in[13℄,theyappear

asminimumproblemsforfun tionalsoftheform

F(u)= Z f(x;u;ru)dx+ Z S u (x;u ;u + ; u )dH n 1 ; (1)

where isaboundedopensubsetof R n

,andtheunknownfun tion ubelongstothespa eSBV(;R N

)

of spe ial fun tions of bounded variation in with values in R N

. We re all that ru denotes the

approximate gradient of u, S

u

is the set of essentialdis ontinuity points of u, 

u

is the approximate

unitnormalve torto S

u

,and u ;u +

theapproximatelimitsof u onthetwosidesof S

u

(forapre ise

denitionsee Chapter1);nally, H n 1

denotesthe (n 1)-dimensionalHausdormeasure.

A typi al exampleis provided bythe so- alled Mumford-Shah fun tional, introdu ed in [31℄ in the

ontextofimagesegmentation,whi h an bewrittenas

MS ; (u):= Z jruj 2 dx+H n 1 (S u )+ Z ju gj 2 dx; (2) where g isafun tion in L 1 (;R N

),and >0 and 0 are onstants.

Oneofthemainfeatures offun tionalsoftheform(1) isthattheyareingeneralnot onvex;

there-fore, all theequilibrium onditionswhi h an be obtainedby innitesimal variationsare ne essaryfor

minimality,but ingeneralnotsuÆ ient.

G.Alberti,G.Bou hitte,andG.DalMasohaveproposedin[2℄asuÆ ient onditionforminimality,

whi his basedon the alibration methodand appliesfor fun tionalsof thegeneral form (1)dened on

s alarmaps.

In thisthesisweapplythis minimality riterionto identify awide lassof nontrivialminimizersfor

thehomogeneousversionoftheMumford-Shahfun tional(denedons alarmaps)

MS(u):= Z jruj 2 dx+H n 1 (S u ); (3)

whi ho ursinthetheoryofinnerregularityforminimizersof MS

;

andisobtainedbytaking =1

anddroppingthelowerordertermin(2). Inthelastpartwedevelopthetheoryof alibrationsformore

generalfun tionalswith freedis ontinuitiesonve tor-valuedmaps andwedes ribeseveralappli ations

ofthisresult.

All the appli ations and the examples shown throughout the thesis share the same purpose: we

by alibrationthat u is aminimizerof F in asuÆ ientlysmall domain; in other words, weshowthat

theequilibrium onditionsarealsosuÆ ientto guaranteetheminimalityonsmall domains,as inmany

lassi alproblemsoftheCal ulusofVariations.

Beforegivingthedetails ofthe results,letusdes ribethebasi ideabehindthe alibrationmethod

fo usingourattentiononDiri hletminimizersof(1),thatisminimizerswithpres ribedboundaryvalues.

Givena andidate u,ifweareableto onstru tafun tional Gwhi hisinvariantonthe lassoffun tions

havingthesameboundaryvaluesas u,andsatises

G(u)=F(u); and G(v)F(v) foreveryadmissiblev, (4)

then u is aDiri hlet minimizer of F. Indeed, if su h a fun tional exists, for every v with the same

boundaryvaluesas u wehavethat

F(u)=G(u)=G(v)F(v):

In[2℄theroleof G is arriedoutbythe ux ofasuitabledivergen e-freeve toreld ':R !R n

R

throughthe ompletegraph

v

ofv,whi hisdenedastheboundaryofthesubgraphofv (thesetofall

points (x;z)2R su hthat zv(x)),orientedbytheinnernormal 

v

. Sin e ' isdivergen e-free,

from the divergen e theorem the ux turns out to be invariant with respe t to the boundary values,

whilesuitablefurther onditionson ' guarantee(4). Considerforinstan ethe aseofthehomogeneous

Mumford-Shahfun tional, forsimpli ityin twodimensions,anddenotethevariablesin by (x;y) and

the\verti al"variablein R by z. Thenitisenoughtorequirethat '=(' xy

;' z

) isaboundedregular

ve toreldsatisfyingthefollowingassumptions:

(a1) ' z (x;y;z) 1 4 j' xy (x;y;z)j 2 for L 2

-a.e. (x;y)2 andeveryz2R;

(a2) ' xy

(x;y;u(x;y))=2ru(x;y) and ' z

(x;y;u(x;y))=jru(x;y)j 2 for L 2 -a.e. (x;y)2; (b1)     Z t2 t1 ' xy (x;y;z)dz     1for H 1

-a.e. (x;y)2 and everyt

1 <t 2 in R; (b2) Z u + (x;y) u (x;y) ' xy (x;y;z)dz= u (x;y) for H 1 -a.e. (x;y)2S u .

Indeed,the ux of ' through

v an beexpressedas Z [h' xy (x;y;v);rvi ' z (x;y;v)℄dxdy+ Z S v Z v + v h' xy (x;y;z); v idzdH 1 ; (5) where v, rv, v  ,and  v

are omputedat (x;y); sin e ondition(a1)impliesthat

h' xy (x;y;v);rvi ' z (x;y;v)jrvj 2 forL 2 -a.e. (x;y)2, (6)

while ondition(b1) implies

Z v + v h' xy (x;y;z); v idz1 forH 1 -a.e. (x;y)2S v ; (7)

by(5) we havethat theinequalityin (4) is satisedfor everyadmissible v. Moreover, onditions(a2)

and(b2)guaranteethattheequalityholdstruein(6)and(7),respe tively,sothat theequalityin(4)is

( 1) ' isdivergen e-freeon R.

Summarizing,ifthereexistsa alibration ' for u withrespe tto MS,then u isaDiri hletminimizer

of MS.

The rst appli ations of this minimality riterion are ontained in [2℄, where the authors provide

severalexamplesof nontrivialminimizersfortheMumford-Shahfun tional with shortand easyproofs.

Thesimpleexpressionofthe alibrationsinalltheseexamplesisrelatedtothefa tthatthey on ernonly

minimizershavingeither agradientvanishing almosteverywhereor anemptydis ontinuityset. In the

rstpartofthisthesiswedealwith andidateshavingamore ompli atedstru ture,that ispresenting

bothanonvanishinggradientand anonemptydis ontinuityset.

Were allfrom[6℄and[31℄thataDiri hletminimizeru for MS in R 2

mustsatisfythefollowing

equilibrium onditions(whi h an beglobally alledtheEuler-Lagrange onditionsfor(3)):

(i) u isharmoni on nS

u ;

(ii) thenormalderivativeof u vanishesonbothsidesof S

u

, where S

u

isaregular urve;

(iii) the urvature of S

u

(where dened) is equal to the dieren e of the squares of the tangential

derivativesof u onbothsidesof S

u ;

(iv) if S

u

is lo ally the union of nitely many regular ar s, then S

u

an present only two kinds of

singularities: either aregularar ending at somepoint,theso- alled\ ra k-tip",or threeregular

ar smeeting withequalanglesof 2=3,theso- alled\triplejun tion".

In Chapters 2 and 3 we onstru t alibrations for solutions of the Euler equations with a regular

dis ontinuityset,while inChapter4we onsiderthe aseofatriplejun tion. Allourresultsareintwo

dimensions. Theminimality ofthe ra k-tiphasbeenre entlyprovedbydierentmethods in[7℄,while

theproblemofndinga alibrationforitisstillopen.

Wepointoutthatwedonotknowofanygeneralmethodtond alibrations,butea htime,a ording

tothegeometryofthedis ontinuitysetofthe andidate,wehavetoperformadierent onstru tion. In

spiteofthela kofageneralformula,allour onstru tionspresentarathersimilarstru ture.

First ofall, in terms of alibrations thepresen e of both anon vanishinggradient anda nonempty

dis ontinuityset orrespondstoa on i tbetween onditions(a2)and(b2),sin e(a2)andtheNeumann

onditions(ii)implythat ' xy istangentialto S u at thepoints (x;y;u  (x)) for (x;y)2S u ,while(b2)

requiresthat itsaveragebetween u (x;y) and u + (x;y) is normalto S u for (x;y)2S u . Itis therefore

onvenient to onstru t the alibration ' by pie es in order to a t dierently on the regions around

the(usual)graphof u, where ' will besomehowdeterminedby ondition(a2),andan \intermediate"

region,whi h willgivethemain ontributionto theintegralin (b2). Morepre isely, wede ompose the

ylinder R inaniteunionofLips hitzopensets A

i

anddene ' insu hawaythatitagreeson A

i

withasuitabledivergen e-freeve toreld '

i

;inordertosatisfy ondition( 1)wehave learlytorequire

thattheve torelds '

i

satisfya ompatibility onditionalongtheboundaryofthesets A

i .

In a neighbourhood of the graph of u we haveto onstru t adivergen e-free ve toreld satisfying

(a2)and su hthat forevery (x;y)2S

u thereholds h' xy (x;y;z); u

(x;y)i>0 foru <z<u +"andforu + "<z<u + , h' xy (x;y;z); u (x;y)i<0 foru + <z<u + +"andforu "<z<u (8)

forasuitable ">0. These propertiesare ru ialinorderto obtain(b1) and(b2)simultaneously.

The aim of the denition of ' in the remainingregion is to make(b2) exa tlysatised, that is to

annihilate the tangential ontribution and to orre t the normal one due to the presen e of the eld

aroundthegraph. Of ourse, ' hastobe arefully hoseninordertopreserve onditions(a1)and(b1).

also ondition(b2). TheEuler onditionsareinvolvedintheproofin aratherte hni alway: ingeneral

they on ernthedenition of ' aroundthegraph,whi h an bethereforeregardedasthe ru ialpoint

ofthe onstru tion.

Therstexamplesof alibrationsfordis ontinuousfun tionswhi harenotlo ally onstant,are

pre-sentedinChapter2. Weprovethatif uisafun tionsatisfyingallEuler onditionsforthehomogeneous

Mumford-Shahfun tionalandwhosedis ontinuitysetisastraightlinesegment onne tingtwopointsof

,theneverypoint (x

0 ;y

0 ) in S

u

hasanopenneighbourhood U su hthat u isaDiri hletminimizer

of(3)in U,providedthetangentialderivatives 

 u and  2  u donotvanishat (x 0 ;y 0 ).

InTheorem2.1westudythespe ial ase

u(x;y):= (

x ify>0,

x ify<0,

whi h, evenif verysimple,involvesmost of themain diÆ ulties. The main ideaof theproofis in the

denitionof'nearthegraphof u: inordertoverify(a2)andtointrodu eanormal omponentsatisfying

(8)wetakeas ' xy

asuitable\rotation"oftheve tor2ru;inotherwords,weapplytotheve tors2e

1

asuitableorthogonalmatrix R dependingon x;y;z andsatisfying R (x;y;x)=I,andweset

'(x;y;z)=(2R (x;y;z)e

1 ;1):

This onstru tionisthenadaptedinTheorem2.4tothe aseofageneralfun tion u satisfyingtheEuler

onditionsandhavingare tilineardis ontinuityset. Nearthegraphof u wesimplytake

'(x;y;z)=(2R (u;v;z)ru;jruj 2

);

where v is the harmoni onjugate of u, while outsidea neighbourhoodof the graphwe are for edto

introdu esomeadditionalparameters. Wewill seethatit isa tually onvenienttoperforma hangeof

variablesthroughthemapping (x;y)7!(u(x;y);v(x;y)), whi his onformalnear (x

0 ;y 0 ),sin eweare assuming   u(x 0 ;y 0

)6=0. Theadditionalassumption  2  u(x 0 ;y 0

)6=0 isinsteadrelated tothe hoi e

of the eld in the region far from the graph and to the proof of (b1): indeed, it guarantees that the

parametersappearinginthedenitionof ' anbe hosenin su h awaythat thefun tion

I(x;y;t 1 ;t 2 ):=     Z t 2 t1 ' xy (x;y;z)dz    

hasastri tmaximumatthepoints (x;y;u (x;y);u +

(x;y)) with (x;y) rangingin S

u .

These rst examples are widely generalized in Chapter 3, where we onsider andidates u whose

dis ontinuity set an beany analyti urveand weprove theDiri hlet minimalityin auniform

neigh-bourhoodofS

u

,withoutadditionalte hni alassumptions. Morepre isely,inTheorem3.2weshowthat,

if u isafun tionsatisfyingallEuler onditionsfortheMumford-Shahfun tionaland S

u

isananalyti

urve onne tingtwopointsof ,then thereexists anopenneighbourhood U of S

u

\ su hthat u

isaDiri hletminimizerin U of(3).

Wenotethattheanalyti ityassumptionfor S

u

doesnotseemtoorestri tive,sin eithasbeenproved

thattheregularpartofthedis ontinuitysetofaminimizerisatleastof lass C 1

anditisa onje ture

thatitisin fa tanalyti (seeChapter1).

The original idea of the new onstru tion essentiallyrelies on the following remark: we an dene

divergen e-freeve toreldsonanopenset AR startingfromabrationof Abygraphsofharmoni

fun tions. Indeed, if fu

t g

t2R

isa familyof harmoni fun tions whose graphsare pairwise disjoint and

over A, thentheve toreld

'(x;y;z)=(2ru (x;y);jru(x;y)j 2

witht=t(x;y;z)satisfyingz=u

t

(x;y),turnsouttobedivergen e-freeonA;moreover,itautomati ally

fulls onditions(a1)and(a2).

Weusethiste hniqueto onstru tthe alibrationaroundthegraphof u: wetakeas fu

t

gthefamily

fu+tvg,where v isasuitableharmoni fun tion,anda ordingto formula(9) wedene

'(x;y;z)=(2ru+2 z u v rv;jru+ z u v rvj 2 );

thefun tion v is hoseninsu hawaythat rv isnormalto S

u

and(8)isveried.

Thismethod of onstru tionremindsofthe lassi almethodofWeierstrasselds,wheretheproofof

theminimalityofa andidate u isobtainedbythe onstru tionofaslopeeldstartingfromafamilyof

solutionsoftheEulerequation,whose graphsfoliateaneighbourhoodofthegraphof u.

InChapter 3wedealalsowithadierentnotionofminimality: in Theorem3.2we ompare u with

perturbationswhi h anbeverylarge,but on entratedinaxedsmalldomain;wewonderifaminimality

propertyispreservedalsoonalargedomain,whenweadmitas ompetitorsonlyperturbationsof uwith

L 1

-normverysmalloutsideasmallneighbourhood of S

u .

A ordingtothis idea,wewillsaythatafun tion u isaDiri hletgraph-minimizerofthe

Mumford-Shahfun tionalifthereexistsaneighbourhood A ofthe ompletegraphof usu hthatMS(u)MS(v)

forall v2SBV() havingthesametra eon  as u andwhose ompletegraphis ontainedin A.

As proved in [2, Example 4.10℄, any harmoni fun tion u: ! R is aDiri hlet graph-minimizer

of MS, whatever is. If we onsider instead a solution u of the Euler equations presenting some

dis ontinuities, what we dis over is that the Diri hlet graph-minimality of u may fail when is too

large, even in the ase of a re tilinear dis ontinuity set, as the ounterexample at the beginning of

Se tion 3.2 shows. Therefore, to a hievethis minimalitypropertywehaveto add somerestri tions on

thedomain . Tothis aimweintrodu easuitablequantitywhi h seemsusefultodes ribethe orre t

intera tionbetween S

u

and . Givenanopenset U (withLips hitzboundary)andaportion of U

(withnonemptyrelativeinteriorin U), wedene

K( ;U):=inf Z U jrv(x;y)j 2 dxdy: v2H 1 (U); Z v 2 dH 1 =1; andv=0onU n  :

As shown by thenotation, K( ;U) isaquantity depending onlyon and U, whi hdes ribesakind

of\ apa ity"ofthepres ribedportionoftheboundarywithrespe tto thewholeopenset. Notethat if

U 1 U 2 ,and 1  2 ,then K( 1 ;U 1 )K( 2 ;U 2

),whi hsuggeststhat if K( ;U) is verylarge,then

U isthininsomesense. Thequalitativepropertiesof K( ;U) arestudiedinthenalpartofSe tion3.2.

Theorem 3.5, whi h is the main result of Se tion 3.2, gives a suÆ ient ondition for the Diri hlet

graph-minimality in terms of K(S

u

;) and of the geometri al properties of S

u

. More pre isely, we

assumethat isagivenanalyti urvesu h that \ onne tstwopointsof ,and n hastwo

onne ted omponents

1 ,

2

withLips hitzboundary. Weprovethatthere existsapositive onstant

( )(dependingonlyonthelengthandonthe urvatureof )su hthat,if u isafun tionsatisfyingall

Euler onditionsin ,whosedis ontinuityset oin ides with \ andsu hthat

min i=1;2 K( \; i )> ( )  k  u k 2 C 1 ( \) +k  u + k 2 C 1 ( \)  ; (10)

then u isaDiri hletgraph-minimizerof MS.

Weremark that ondition (10)imposes arestri tion on the size of depending on the behaviour

of u along S

u

: if u has largeor veryos illating tangentialderivatives,wehaveto take quitesmall

to guarantee that (10)is satised. Inthe spe ial ase of alo ally onstant fun tion u, ondition (10)

isalwaysfullled whateverthedomainis; so u isalwaysaDiri hletgraph-minimizerwhatever is, in

agreementwitharesultprovedin[2℄.

graph- ompletegraphof u,andabounded ve toreld ' on A satisfying onditions(a1),(a2),(b1), (b2),and

( 1) (where now (x;y;z), (x;y;t

i

) range in A). Condition (10) guarantees that we an extend to a

neighbourhoodof

u

aslightlymodiedversionofthe alibrationofTheorem3.2.

InChapter4westudytheminimalityofsolutions uoftheEulerequationswhosedis ontinuitysetis

givenbythreeline segmentsmeeting at theoriginwithequalangles;in otherwords, S

u

is are tilinear

triplejun tion, generating apartition of in three se tors ofangle 2=3, that we all A

0 ;A 1 ;A 2 . In

Theorem4.1 weproveby alibrationthat, setting u

i :=uj A i andassuming u i 2C 2 (A i ), thereexists a

neighbourhood U oftheoriginsu hthat u isaDiri hletminimizerof MS in U. Thisresultgeneralizes

Example4in [1℄wherethefun tion u waspie ewise onstant.

Theproofisquitelongandte hni al,andissplitinseveralsteps. Thesymmetryduetothepresen e

of 2=3-anglesis exploited in thewhole onstru tionof the alibration. First ofall, sin ethefun tion

u

i

has tobeharmoni in A

i

with nullnormalderivativeat A

i

, applying S hwarzre e tion prin iple

we obtainthat u

i

an be harmoni allyextended to a neighbourhood of the origin, ut by ahalf-line;

moreover,from theEuler ondition(iii) itfollowsthattheextensionof u

i

oin ides,uptothesignand

to additive onstants,with u

j on A

j

forevery j 6=i. Usingthis remark itis easyto see that ea h u

i

mustbeeither symmetri or antisymmetri withrespe ttothebise tingline of A

i .

In Se tions 4.1 { 4.4 we dene ' in the symmetri ase and we prove that it is a alibration; in

Se tion4.5weadaptthe onstru tiontotheantisymmetri ase.

The ru ial pointof both onstru tionsis, as usual, thedenition ofthe eld near thegraph of u,

where we apply again the \bration" te hnique. Indeed, we brate a neighbourhood of the graph of

ea h u

i

by a family of harmoni fun tions of the form u

i +tv

i

. Unlike the onstru tion of ' in the

proofofTheorem3.2wherewe hoose rv orthogonalto S

u

,inthis aseitis onvenienttotakeas v

i a

linearfun tionwhosegradientisparalleltothebise tinglineof A

i

. Thankstothesymmetry,this hoi e

ensuresthatthetangential ontributionsto theintegralin (b2), givenbythe regionsnear u and u +

,

arealwaysofoppositesignsandannihilateea hother.

The assumption of C 2

-regularity for u

i

does not seem too restri tive: indeed, by the regularity

results for ellipti problems in non-smooth domains (see [22℄), it follows that u

i belongs at least to C 1 (A i ), sin e u i

solvestheLapla e equation with Neumann boundary onditionson ase torof angle

2=3. Moreover,sin e u

i

iseither symmetri orantisymmetri withrespe t tothebise tingline of A

i ,

one ansee u

i

asasolutionoftheLapla eequationona=3-se torwithNeumannboundary onditions

orrespe tivelymixedboundary onditions. Bytheregularityresultsin[22℄,itturnsoutthatintherst

ase u i belongsto C 2 (A i

), while in these ond one u

i

an be written (inpolar oordinates entredat

0)as u i (r;)=u~ i (r;)+ r 3=2 os 3 2 ,with u~ i 2C 2 (A i

) and 2R. So,onlythefun tion r 3=2

os 3

2  is

notre overedbyourtheorem: ifwewere ableto onstru ta alibrationalso forthisfun tion, thenwe

wouldre overallpossible ases.

Finallyweremarkthatthe asewhereS

u

isgivenbythreeregular urves(notne essarilyre tilinear)

meeting at a point with 2=3-angles, is at the moment an open problem and it doesnot seem to be

a hievablewith aplain arrangementof the alibration used forthere tilinear ase,essentiallybe ause

ofthela kofsymmetry.

The last part of the thesis orresponds to Chapter 5 where we generalize the alibration method

to fun tionals of theform (1) dened on ve tor-valued maps. The basi prin ipleis the samewe have

explained at the beginning: in order to provethe minimality of afun tion u, we wantto onstru t a

fun tional G satisfying onditions (4) and invariant on the lass of the admissible ompetitors for u.

When u is ave tor-valuedfun tion,itis onvenientto onsider adierentkindof invariantfun tional:

the alibration is no longer a ve toreld, but a pair of fun tions (S;S

0 ), where S : R N ! R n is

suitablyregular,while S

0

belongsto L 1

(); the omparisonfun tional for F isgivenby

G(v):= Z hS(x;v);  idH n 1 + Z S 0 (x)dx; (11)

where 



istheinnerunitnormalto. Itis learthatthefun tional(11)is onstantonthefun tions

havingthesamevaluesat . Moreover,bythedivergen etheoremwe anrewrite(11)as

Z d v + Z S 0 (x)dx; where  v

isthedivergen e(in thesenseof distributions)ofthe ompositefun tion S(
;v(
)). A

gener-alizedversionofthe hainrulein BV (whi hisprovedinLemma 5.2)impliesthat

 v =( [div x S℄(x;v)+h(D z S(x;v))  ;rvi)L n +hS(x;v + ) S(x;v ); v iH n 1 bS v ; where [div x

S℄ denotesthedivergen eof S withrespe ttothevariable x2,and(D

z S)

thetranspose

of theJa obianmatrix of S with respe t to thevariable z 2 R N

. Therefore thefun tional (11)turns

outtobeequalto Z ([div x S℄(x;v)+h(D z S(x;v))  ;rvi+S 0 (x))dx+ Z S v hS(x;v + ) S(x;v ); v idH n 1 : (12)

By omparing this expression with the fun tional (1), we nd pointwise onditions on S

0

, S, and the

derivativesof S, whi hguarantee(4), andthen theDiri hletminimalityof agiven u. For instan e, in

the aseoftheMumford-Shah fun tional(3) denedonve tor-valuedmaps, itis enoughto requirethe

following onditions: (a1) [div x S℄(x;z)+S 0 (x) 1 4 jD z S(x;z)j 2 for L n

-a.e. x2,and foreveryz2R N ; (a2) [div x S℄(x;u)+S 0 (x)= jru(x)j 2 and (D z S(x;u))  =2ru(x) for L n -a.e. x2; (b1) jS(x;z 1 ) S(x;z 2 )j1 for H n 1

-a.e. x2 andforevery z

1 ;z 2 2R N ; (b2) S(x;u + ) S(x;u )= u for H n 1 -a.e. x2S u .

For apre ise statementin the ase of ageneral fun tional of theform (1) werefer to Lemma 5.4 and

Lemma5.5in Se tion5.1.

The onne tion between the onditionsabove in the ase N = 1 and those ones of the s alar

for-mulationby Alberti,Bou hitte,Dal Maso,is studied in Remark 5.8. Here we onlyobservethat, while

in thes alar formulationweneed ondition ( 1) to ensure that the omparison fun tional is invariant

withrespe t totheboundaryvalues,in thisnewframeworkthisisguaranteedjust bytheexpressionof

thefun tional (11);so,thereis no ondition orrespondingto ( 1). Infa t,in the ase N =1,givena

alibration (S;S 0 ),theve toreld '=(' x ;' z ):R!R n R dened as ' x (x;z):= z S(x;z); ' z (x;z):= [div x S℄(x;z) S 0 (x)

isa alibrationinthesenseof Alberti,Bou hitte,DalMaso. Indeed, ' turns outtobedivergen e-free,

andtheremaining onditionsofthes alarformulationfollowfrom onditions(a1),(a2),(b1), and(b2)

stated above. Conversely, givenany divergen e-free ve toreld ' = (' x

;' z

), we an alwayswrite ' x

as the derivativewith respe t to z 2 R of asuitablefun tion S : R !R n

, and using the relation

 z ' z = div x ' x

(whi hfollowsfrom( 1)),we andedu ethatthereexistsafun tion S

0 ofthevariable x su h that ' z (x;z)= [div x S℄(x;z) S 0

(x). If we rewritenow onditions(a1),(a2), (b1), and (b2)

ofthes alarformulationbyusing theseexpressions of ' x

and ' z

, weobtainthat thepair (S;S

0 ) isa

alibration.

The formulation in termsof (S;S

0

) is related to lassi aleld theory for multiple integralsof the

form F 0 (u)= Z f(x;u;ru)dx:

Inthis ontext a suÆ ient ondition for the minimality of a andidate u 2C 1 (;R N ) is obtainedby omparing F 0

with the integral of anull-lagrangian of divergen e type, whi h is onstru ted starting

fromasuitablydened slopeeld P, alledWeyleld,andafun tion S 2C 2 (R N ;R n ),theeikonal

map asso iated with P ( f., e.g.,[18℄). InSe tion 5.2 weprovethat, under suitable assumptionson f

and ,wheneveraWeyleldexistsforafun tion u2C 1

(;R N

) (sothat uisaDiri hletminimizerfor

F

0

),thenthereexistsa alibrationfor u withrespe ttothefun tional F (whi hisgivenbytheeikonal

map S andby S

0

0),so u isalsoaDiri hletminimizerfor F among SBV fun tions.

Someexamplesandappli ationsarepresentedinSe tion 5.3. InExamples5.14,5.16,5.17,and5.18

wedeal withminimizers ofthe Mumford-Shahfun tional, andwegeneralize someresultsprovedin [2℄

forthes alar ase. Apurelyve torialexampleisgivenbyExample5.15,wherewestudytheminimality

of ontinuous solutionsofthe Eulerequations forafun tional arisingin fra tureme hani s,whi h an

bedenedonlyonmapsfrom R n

into R n

.

Finally,wepointoutthat,asmentionedin[2℄,one ouldtrytogeneralizethe alibrationtheoryfrom

thes alar asetotheve torialonebyrepla ingdivergen e-freeve toreldsby losedn-formson R N

,

a tingonthegraphsof thefun tions v, viewedas suitablydened surfa esin R N

. This ouldlead

totheideathat our hoi eofwritingthe alibrationin termsofthepair (S;S

0

) issomehowrestri tive

when N >1. This is notthe aseat all, sin ethe existen eof a alibrationexpressed viadierential

formsimpliestheexisten eofa alibrationexpressedintermsofapair (S;S

0

),as showninSe tion5.4.

TheresultsofChapter2areobtainedin ollaborationwithGianniDalMasoandMassimilianoMorini,

andarepublished in[11℄,whiletheresultsofChapter3area hievedin ollaborationwithMassimiliano

Moriniandpublishedin[27℄. The ontentofChapter4willappearin[25℄,whilethe ontentofChapter5

Preliminary results

Inthis hapterwe olle tsomepreliminaryresultswhi hwill beusefulin thesequel. InSe tion 1.1we

re all somebasi results from the theoryof fun tions with bounded variation. InSe tions 1.2 and 1.3

wedeal withne essaryand suÆ ient onditionsfortheminimality ofthehomogeneousMumford-Shah

fun tionalons alarmaps: in Se tion1.2wewritetheEuler-Lagrangeequations,whileinSe tion 1.3we

presentthetheoryof alibrations.

Letusxsomenotation. Given x;y2R n

,wedenotetheirs alarprodu tby hx;yi,andtheeu lidean

norm of x by jxj. Weset S n 1 :=fx2R n : jxj=1g. Givena set B R n

, wedenotethe Lebesgue

measureof B by L n

(B) andthe (n 1)-dimensionalHausdormeasureof B by H n 1

(B). If a;b2R,

themaximumand theminimumof fa;bg aredenotedby a_b and a^b,respe tively.

1.1 Fun tions of bounded variation

Let beabounded open subset of R n , let u2 L 1 lo (;R N ), and let x 0

2. Wesay that u has an

approximatelimitat x 0 2 ifthere exists z2R N su hthat lim r!0 + 1 L n (B r (x 0 )) Z Br(x0) ju(x) zjdx=0; (1.1) where B r (x 0

) istheball ofradius r entredat x

0

. Theset S

u

of pointswhere thispropertydoesnot

holdis alledtheapproximatedis ontinuitysetof u. Foranyx

0 2nS

u

theve tor z (whi hisuniquely

determinedby(1.1))is alled theapproximatelimitof u at x

0

anddenoted by u(x~

0 ).

Wesaythat afun tion u:!R N

hasboundedvariation in , andwewrite u2BV(;R N ),if u belongsto L 1 (;R N

) anditsdistributionalderivative Du isaniteRadon R nN

-valuedmeasurein .

If hasLips hitzboundary,we an speakaboutthetra eof u on ,whi hbelongsto L 1

(;H n 1

)

andwillbestilldenotedby u.

If u2BV(;R N ), then S u is ountably (H n 1

;n 1)-re tiable, that is, it an be overed,upto

an H n 1

-negligibleset, by ountablymany C 1

-hypersurfa es. Moreover,for H n 1 -a.e. x 0 2S u there existsatriplet (u + (x 0 );u (x 0 ); u (x 0 ))2R N R N S n 1 su hthat u + (x 0 )6=u (x 0 ),  u (x 0 ) isnormal to S u

in anapproximatesense,and

lim r!0 + 1 L n (B  r (x 0 )) Z B  r (x0) ju(x) u  (x 0 )jdx=0; (1.2) where B  r (x 0 ) istheinterse tionof B r (x 0

) withthehalf-plane fx2R n :hx x 0 ; u (x 0 )i0g. The triplet (u + (x 0 );u (x 0 ); u (x 0

)) isuniquelydetermined upto apermutation of (u + (x 0 );u (x 0 )) and a

hangeofsignof 

u (x

0

). Condition(1.2)saysthat 

u (x

0

) pointsfromthesideof S

u

orrespondingto

u (x

0

) totheside orrespondingto u + (x 0 ). Foreveryu2BV(;R N

),byapplyingtheRadon-Ni od ymtheoremwe ande omposethemeasure

Du as D a u+D s u,where D a

u istheabsolutely ontinuouspartwith respe ttotheLebesguemeasure

L n

and D s

u isthesingularpart. Thedensityof D a

u withrespe tto L n

isdenotedby ru andagrees

with the approximate gradient of u. The measure D s

u an be in turn written as D j u+D u, where D j u istherestri tionof D s u to S u

andis alled thejump part,while D

u istherestri tionto nS

u

andis alledtheCantorpart. Thedensityof D j

u withrespe ttothemeasure H n 1 bS u isgivenbythe tensorprodu t (u + u ) u

. Wealso allthesum D a

u+D

u thediuse partofthederivativeof u

anddenoteitby ~

Du.

We say that afun tion u: ! R N

is a spe ial fun tion of bounded variation, and we write u 2

SBV(;R N ), if u2BV(;R N ) and D u=0.

Finally,forevery u2BV(;R N

) wedene asgraph of u theset

graphu:=f(x;u(x))~ : x2nS

u g:

Inthes alar ase N =1,forevery u2BV() we all 1

u

the hara teristi fun tionofthesubgraph

of u in R, namelythefun tiondenedby 1

u

(x;z):=1 for zu(x) and 1

u

(x;z)=0 for z>u(x).

We dene as omplete graph of u (and we denote it by

u

) the measure theoreti boundary of the

subgraphof u,thatisthesingularsetof 1

u

. Wenotethat,assuming u and S

u

suÆ ientlyregular,the

omplete graph

u

onsistsof the unionof the graphof u and ofall segmentsjoining (x;u (x)) and

(x;u +

(x)) with x rangingin S

u .

Formoredetailswereferto thebook[6℄byL.Ambrosio,N.Fus o,andD. Pallara,wherea

self- on-tainedpresentationof BV and SBV spa es anbefound.

1.2 The Euler-Lagrange equations for the Mumford-Shah

fun tional

LetdenoteaboundedopensubsetofR 2

withLips hitzboundary,andletus onsiderthehomogeneous

Mumford-Shahfun tional MS(u)= Z jruj 2 dx+H 1 (S u ) (1.3) for u2SBV().

Inthesequelwewillrefertothefollowingdenitionofminimizers.

Denition1.1 An absoluteminimizerof(1.3)in isafun tion u2SBV() su hthat

Z jruj 2 dx+H 1 (S u ) Z jrvj 2 dx+H 1 (S v ) (1.4)

for every v2SBV(), while a Diri hlet minimizerin is afun tion u2SBV() su h that(1.4) is

satisedforevery v2SBV() withthe sametra eon  as u.

Letusfo usourattentiononne essaryoptimality onditionsneararegularportionof S

u

. Let u be

aDiri hletminimizerof MS andlet U  beanopenset su hthat S

u

\U isagraph,thatis

forsomeopen set DR and :D !R. Set U +

:=f(t;s)2U : s> (t)g and U :=f(t;s)2U :

s< (t)g. Let '2 C 1

(U) be afun tion vanishing in aneighbourhood of U +

nS

u

; by omparing u

withthefun tion v:=u+"',fromtheminimalityof u weobtainthat

Z

U +

hru;r'idx=0:

Thismeansthat u isaweaksolutionofthefollowingproblem:

(
u=0 in U + ,   u=0 onU + \S u . (1.5)

Asimilarproblemissolvedby u in U .

TheEulerequation(1.5)hasbeenobtainedby onsideringonlyvariationsof uandkeepingS

u xed.

By onsideringalsovariationsof S

u

weexpe ttoderiveatransmission onditionfor u along S

u ,whi h

takesinto a ounttheintera tionbetweenthebulkandthesurfa epartofthefun tional. Assumethat

u belongsto W 2;2

(U +

[U ) and suppose that S

u

\U isthegraphof a C 2

-fun tion (that is, isof

lass C 2

). Thenit anbeprovedthat

div r p 1+jr j 2 ! =j(ru) + j 2 j(ru) j 2 onS u \U, (1.6)

wheretheleft-handsideisthe urvatureof S

u

,whileattheright-handside (ru) 

denotethetra esof

ru on S

u

\U from U 

,respe tively.

Wenote that, if is known to be onlyof lass C 1;

, equation(1.6) a tually still holds in a weak

sense. Thenusing (1.6) it ispossibleto provethat, as soon as we knowthat S

u \U is of lass C 1; , then S u

\U turnsouttobeinfa t of lass C 1

.

Thefollowing onje tureisstillanopenproblem.

Conje ture (De Giorgi). If u is aDiri hlet minimizer of MS, then S

u

is analyti near itsregular

points.

We on ludethisse tion bysomeremarkson theregularityofthe dis ontinuityset ofaminimizer,

whi hrepresentsavery hallengingmathemati alproblem. In[31℄D.MumfordandJ.Shah onje tured

that, if u isaDiri hletminimizerof MS,then S

u

islo allytheunionofnitelymany C 1;1

embedded

ar s;moreover,theyshowedthat, ifthe onje tureistrue,thenonly twokindsofsingularity an o ur

inside : either aline terminates at somepoint, theso- alled \ ra k-tip",or three lines meet forming

equalanglesof 2=3,theso- alled\triple jun tion".

In[6,Theorem8.1℄thefollowingregularityresultisproved.

Theorem 1.2 If u2 SBV() is a minimizer of MS, there exists an H 1

-negligible set   S

u \

relatively losedin su hthat \S

u

n isa urveof lass C 1;1

.

This result is still farfrom Mumford-Shah onje ture, sin e we areonly ableto saythat  is H 1

-negligible,andnotthat ithaslo allynite H 0

measure.

1.3 The alibration method for the Mumford-Shah fun tional

Inthisse tionwepresentthe alibrationmethod forthehomogeneousMumford-Shahfun tionalintwo

Werstintrodu eamoregeneralnotionofminimalitywhi hwillbeusefulinthesequel. Let bea

xedboundedopensubsetof R 2

withLips hitzboundary,and 



itsinnerunitnormal. Let A denote

anopensubsetof R withLips hitzboundary,whose losure anbewritten as

A=f(x;y;z)2R: 

1

(x;y)z

2 (x;y)g;

wherethetwofun tions 

1 ; 2 :![ 1;+1℄ satisfy  1 < 2 .

Denition1.3 Wesaythatafun tion u2SBV() isanabsolute A-minimizerof MS ifthe omplete

graph of u is ontainedin A and MS(u)MS(v) for every v2SBV() su hthat

v

A, while u

isaDiri hlet A-minimizerif weaddthe requirementthatthe ompetingfun tions v have thesametra e

on  as u.

Foreveryve toreld ':A!R 2

R wedene themaps ' xy :A!R 2 and ' z :A!R by '(x;y;z)=(' xy (x;y;z);' z (x;y;z)):

Weshall onsider the olle tion F of all pie ewise C 1

-ve torelds ' : A ! R 2

R with the following

property: thereexistanitefamily(A

i )

i2I

ofpairwisedisjointopensubsetsofAwithLips hitzboundary

whose losures over A,andafamily ('

i ) i2I ofve toreldsin C 1 (A i ;R 2

R) su hthat ' agreesatany

pointwithoneofthe '

i .

Let u 2SBV() be su h that

u

A. A alibration for u on A (with respe t to the fun tional

MS)isabounded ve toreld '2F satisfyingthefollowingproperties:

(a1) ' z (x;y;z) 1 4 j' xy (x;y;z)j 2 for L 2

-a.e. x2 andeveryz2[

1 ; 2 ℄; (a2) ' xy

(x;y;u(x;y))=2ru(x;y) and ' z

(x;y;u(x;y))=jru(x;y)j 2 for L 2 -a.e. x2; (b1)     Z t 2 t1 ' xy (x;y;z)dz     1for H 1

-a.e. (x;y)2,andevery t

1 ;t 2 in [ 1 ; 2 ℄; (b2) Z u + (x;y) u (x;y) ' xy (x;y;z)dz= u (x;y) for H 1 -a.e. (x;y)2S u ;

( 1) ' isdivergen e-freeinthesenseofdistributions in A.

Ifalsothefollowing onditionissatised:

( 2) h' xy ;  i=0 H 2 -a.e. on A\(R);

then ' is alledanabsolute alibration for u on A.

Wenote that, in orderto prove ondition( 1), itisenoughto showthat div'

i

=0 in A

i

for every

i2I,and thefollowingtransmission onditionissatised:

h' i ; Ai i=h' j ; Aj i H 2 -a.e. on A i \A j , where  Ai and  Aj

denotetheunit normalve torto A

i

and A

j

,respe tively.

We annowstatethefundamentaltheoremofthe alibrationmethod,whi hisprovedin [1℄and[2℄.

Theorem 1.4 Let u2 SBV() be su h that

u

A. If there exists a alibration for u on A (with

respe t to MS), then u is a Diri hlet A-minimizer of the homogeneous Mumford-Shah fun tional. If

thereexistsanabsolute alibration for u on A, then u isan absolute A-minimizer.

Lemma1.5 Let U be an open subset of R 2

and I, J be two real intervals. Let u : UJ ! I be a

fun tion of lass C 1

su hthat

 u(
;
;s) isharmoni for every s2J;

 there existsa C 1

-fun tion t:UI !J su hthat u(x;y;t(x;y;z))=z.

Then, ifwedene in UI theve toreld

'(x;y;z):=(2ru(x;y;t(x;y;z));jru(x;y;t(x;y;z))j 2

);

where ru(x;y;t(x;y;z)) denotes the gradient of u with respe tto the variables (x;y) omputedat the

point (x;y;t(x;y;z)), ' isdivergen e-freein UI.

Proof.{ Letus omputethedivergen eof ':

div '(x;y;z)=2
u(x;y;t(x;y;z))+2h s ru(x;y;t(x;y;z));rt(x;y;z)i +2 z t(x;y;z)hru(x;y;t(x;y;z)); s ru(x;y;t(x;y;z))i; (1.7)

where
u(x;y;t(x;y;z)) denotestheLapla ianof u withrespe tto (x;y) omputedat (x;y;t(x;y;z)),

and rt(x;y;z) denotesthegradientof t withrespe t to (x;y). Bydierentiatingtheidentityveried

bythefun tion t rstwithrespe t to z andwithrespe tto (x;y), wederivethat

 s u(x;y;t(x;y;z)) z t(x;y;z)=1; ru(x;y;t(x;y;z))+ s u(x;y;t(x;y;z))rt(x;y;z)=0:

Usingtheseidentities andsubstitutingin(1.7),wenallyobtain

div '(x;y;z)=2
u(x;y;t(x;y;z))=0;

sin ebyassumption u isharmoni withrespe tto (x;y). 2

Letus onsider nowageneralfun tionaloftheform

F(u):= Z f(x;u;ru)dx+ Z Su (x;u ;u + ; u )H n 1 ;

where isaboundedopensubsetof R n

withLips hitzboundary,theunknown ubelongsto SBV(),

and f, areBorelfun tions.

Let f 

and 



f denote the onvex onjugateand the subdierentialof f withrespe tto thelast

variable. Were allthatthesubdierentialofafun tion g:R n

![0;+1℄ atthepoint 2R n

isdened

asthesetofve tors 2R n

su hthat g()+h; ig() forevery 2R n

.

Asbefore,letA beanopensubsetof R withLips hitzboundarywhose losure anbewrittenas

A=f(x;z)2R:  1 (x)z 2 (x)g; where  1 ; 2 :![ 1;+1℄ satisfy  1 < 2 .

Theregularityassumptionson ' anbeweakenedbyrequiringthat ' is approximatelyregular,i.e.

itisboundedandforeveryLips hitzhypersurfa e M in R n+1 thereholds aplim (x;z)!(x0;z0) h'(x;z); M (x 0 ;z 0 )i=h'(x 0 ;z 0 ); M (x 0 ;z 0 )i for H n -a.e. (x;z)2M\A, where  M (x 0 ;y 0

) is the unit normal to M at (x

0 ;y

0

). It is easy to see that, if ' 2 F, then ' is

approximatelyregular.

Let u2SBV() besu h that

u

A. A alibration for u on A withrespe t tothe fun tional F

(a1) ' z (x;z)f  (x;z;' x (x;z)) for L n

-a.e. x2 andevery z2[

1 ; 2 ℄; (a2) ' x (x;u(x))2 

f(x;u(x);ru(x)) and ' z (x;u(x))=f  (x;u(x);' x (x;u(x))) for L n -a.e. x2; (b1) Z t2 t1 h' x (x;z);idz  (x;t 1 ;t 2 ;) for H n 1 -a.e. x 2, every  2S n 1 , and every t 1 <t 2 in [ 1 ; 2 ℄; (b2) Z u + (x) u (x) h' x (x;z); u (x)idz= (x;u (x);u + (x); u (x)) for H n 1 -a.e. x2S u ;

( 1) ' isdivergen e-freeinthesenseofdistributions in A.

Ifalsothefollowing onditionissatised:

( 2) h' x ;  i=0 H n -a.e. on A\(R);

then ' is alledanabsolute alibration.

Thefollowingtheoremisprovedin[2℄.

Theorem 1.6 Let u2 SBV() be su h that

u

 A. If there exists a alibration for u on A with

respe tto F, then u isaDiri hlet A-minimizer of F,that is F(u)F(v) for every v2SBV() with

the same tra e on  as u and su hthat

v

A. If there exists an absolute alibration for u on A

withrespe tto F, then u isanabsolute A-minimizerof F, thatis F(u)F(v) for every v2SBV()

su hthat

v A.

Calibrations for minimizers with a

re tilinear dis ontinuity set

Inthis hapterweshowtherstexamplesof alibrationsfordis ontinuousfun tions,whi harenotlo ally

onstant. In parti ular, we onsider solutions w of theEuler-Lagrangeequations forthe homogeneous

Mumford-Shahfun tional MS(w)= Z jrw(x;y)j 2 dxdy+H 1 (S w ); (2.1)

andweassumethatthedis ontinuityset S

w

isastraightlinesegment onne tingtwoboundarypoints

ofthedomain. Weprovethat,undertheadditionalassumptionsthatthetangentialderivatives

 w and

 2

w of w do notvanish onbothsidesof S

w

, theEuler onditionsare alsosuÆ ient forthe Diri hlet

minimalityinsmalldomains.

Let bea ir lein R 2

with entreonthe x-axis,andset

0 :=f(x;y)2:y6=0g; S:=f(x;y)2:y=0g: If w2C 1 ( 0 ) with R 0 jrwj 2

dxdy <+1, then itis easyto see that w satises theEuler onditions

fortheMumford-Shahfun tional(seeSe tion1.2)ifandonlyif w hasoneofthefollowingforms:

w(x;y)= ( u(x;y) ify>0; u(x;y)+ 1 ify<0; (2.2) or w(x;y)= ( u(x;y)+ 2 ify>0; u(x;y) ify<0; (2.3) where u2C 1

() isharmoni withnormalderivativevanishingon S and

1 ,

2

arereal onstants. For

ourpurposes,itisenoughto onsiderthe ase

1

=0 in(2.2)and

2

=1 in (2.3).

In both ases we will onstru t an expli it alibration for w in the ylinder UR, where U is a

suitableneighbourhood of (x

0 ;y

0

). Sin e this onstru tionis elementary when (x

0 ;y 0 )2= S w (see [2℄),

we onsideronlythe ase (x

0 ;y 0 )2S w .

InSe tion2.1 we onsider thespe ial aseofthefun tion

w(x;y):= (

x ify>0;

x ify<0;

andgivein full details theexpression ofthe alibrationfor w (see Theorem2.1); thenin Theorem 2.3

weadaptthesame onstru tiontothefun tion

w(x;y):= (

x+1 ify >0;

x ify <0:

(2.5)

InSe tion 2.2we onsiderthegeneral ases(2.2)and (2.3): theformer ase(2.2)isstudiedin Theorem

2.4 by a suitable hange of variables and by adding two new parameters to the onstru tion used in

Theorem2.1;theminor hangesfor(2.2)are onsideredin Theorem2.5.

2.1 A model ase

Inthisse tionwedealwiththeminimalityofthefun tions(2.4)and(2.5). Theaimofthestudyofthese

simpler ases(butwewillseethattheyinvolvethemain diÆ ulties)isto larifytheideasofthegeneral

onstru tion.

Theorem 2.1 Let w:R 2

!R be the fun tiondenedby

w(x;y):= (

x ify>0;

x ify<0:

Thenevery point (x

0 ;y

0

)6=(0;0) hasanopenneighbourhood U su hthat w isaDiri hletminimizerin

U of the Mumford-Shah fun tional(2.1).

Proof. { The resultfollows from Example 4.10 of [2℄ if y

0

6= 0. We onsider now the ase y

0 = 0,

assumingforsimpli ity that x

0

>0. Wewill onstru talo al alibrationof w near (x

0 ;0). Letus x ">0 su hthat 0<"< x 0 10 ; 0<"< 1 32 : (2.6)

For 0<Æ<" we onsidertheopenre tangle

U :=f(x;y)2R 2

: jx x

0

j<";jyj<Æg

andthefollowingsubsetsof UR (seeFig.2.1):

A 1 := f(x;y;z)2UR: x (y)<z<x+(y)g; A 2 := f(x;y;z)2UR: b+()y <z<b+()y+hg; A 3 := f(x;y;z)2UR: h<z<hg; A 4 := f(x;y;z)2UR: b+()y h<z< b+()yg; A 5 := f(x;y;z)2UR: x ( y)<z< x+( y)g; where (y):= p 4" 2 (" y) 2 ; h:= x 0 3" 4 ; ():=  4 1  ; b:=2h+()Æ; := 1 4" 2h :

Wewill assumethat

Æ< x

0 3"

z

y

y=δ

y=−δ

A

₂

A

₃

A

₄

A

₅

z=-x

z=x

A

₁

Figure2.1: Se tionof thesets A

1 ;:::;A

5

sothatthesets A

1 ;:::;A

5

arepairwise disjoint.

Forevery (x;y;z)2UR, letusdenetheve tor '(x;y;z)=(' x ;' y ;' z )(x;y;z)2R 3 asfollows: 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : 2(" y) p (" y) 2 +(z x) 2 ; 2(z x) p (" y) 2 +(z x) 2 ;1 ! if(x;y;z)2A 1 ;  0;;  2 4  if(x;y;z)2A 2 ; (f(y);0;1) if(x;y;z)2A 3 ;  0;;  2 4  if(x;y;z)2A 4 ; 2("+y) p ("+y) 2 +(z+x) 2 ; 2(z+x) p ("+y) 2 +(z+x) 2 ;1 ! if(x;y;z)2A 5 ; (0;0;1) otherwise; where f(y):= 1 h Z (y) 0 " y p t 2 +(" y) 2 dt Z ( y) 0 "+y p t 2 +("+y) 2 dt ! : Note that A 1 [A 5

is an open neighbourhood of graphw\(UR). Thepurpose of the denition

of ' in A

1 and A

5

(seeFig.2.2)is toprovideadivergen e-freeve toreldsatisfying ondition(a2)of

Se tion1.3andsu hthat

' y (x;0;z)>0 for jzj<x; ' y (x;0;z)<0 for jzj>x:

Thesepropertiesare ru ial inordertoobtain(b1)and(b2) simultaneously.

Theroleof A

2 and A

4

is togivethemain ontributionto theintegralin (b2). Toexplainthisfa t,

suppose,foramoment,that "=0;in this asewewouldhave A

1 =A 5 =;and Z x x ' y (x;0;z)dz=1;

sothatthe y- omponentofequality(b2) wouldbesatised.

Thepurposeofthedenitionof ' in A

3

isto orre tthex- omponentof ',inordertoobtain(b1).

Weshallprovethat,forasuitable hoi eof Æ,theve toreld' isa alibrationforw inthere tangle

U.

Inequality(a1)is learlysatisedin allregions: theonlynontrivial aseis A

3 , whereusing(2.6)we have jf(y)j 4((y)+( y)) x 0 3"  8 p 3" x 0 3" <2:

Onthegraphof w wehave

'(x;y;w(x;y))= (

y

x

y=

−δ

y=

δ

x=z

Figure2.2: Se tion oftheset A

1

at z= onstant.

so ondition(a2)issatised.

Notethat foragiven z2R wehave

 x ' x (x;y;z)+ y ' y (x;y;z)=0 (2.8)

forevery (x;y) su h that (x;y;z)2A

1 [A

5

. This implies ' is divergen e-freein A

1 [A

5

. Moreover

div '=0 in theothersets A

i

,andthenormal omponentof ' is ontinuousa ross A

i

: the hoi eof

() ensuresthatthispropertyholdsfor A

2

and A

4

(seeFig.2.3). Therefore ' isdivergen e-freein

thesenseofdistributionsin UR.

Wenow ompute Z x x ' y (x;y;z)dz:

Letusx y with jyj<Æ. Sin e ' y (x;y;z) depends on z x,wehave Z x x (y) ' y (x;y;z)dz= Z x+(y) x ' y (;y;x)d: (2.9)

Using(2.8)andapplyingthedivergen e theoremtothe urvilineartriangle

T =f(;)2R 2 :  >x; <y; (" ) 2 +(x ) 2 <4" 2 g

Figure2.3: Se tionoftheset A

2

y

ξ

η

ε

x

y

Figure2.4: The urvilineartriangle T.

Z x+(y) x ' y (;y;x)d= Z y " ' x (x;;x)d=2(y+"): (2.10)

From(2.9)and(2.10),weget

Z x x (y) ' y (x;y;z)dz=2(y+"): (2.11)

Similarlywe an provethat

Z x+( y) x ' y (x;y;z)dz=2( y+"): (2.12)

Usingthedenitionof ' in A

2 , A 3 , A 4 ,weobtain Z x x ' y (x;y;z)dz=1: (2.13)

Ontheotherhand,bythedenitionof f,wehaveimmediatelythat

Z

x

' x

From theseequalities it followsin parti ular that ondition(b2) issatisedon thejump set S

w \U =

f(x;y)2U :y =0g.

Letusbeginnowtheproofof(b1). Letusx (x;y)2U. Forevery t

1 <t 2 weset I(t 1 ;t 2 ):= Z t2 t 1 (' x ;' y )(x;y;z)dz:

Itisenoughto onsiderthe ase x ( y)t

1 t

2

x (y). We anwrite

I(t 1 ;t 2 ) = I(t 1 ; x)+I( x;x)+I(x;t 2 ); I(t 1 ; x) = I(t 1 ^( x+( y)); x)+I(t 1 _( x+( y)); x+( y)); I(x;t 2 ) = I(x;t 2 _(x (y)))+I(x (y);t 2 ^(x (y))): Therefore I(t 1 ;t 2 )=I( x;x)+I(t 1 ^( x+( y)); x)+I(x;t 2 _(x (y))) +I(t 1 _( x+( y));t 2

^(x (y))) I( x+( y);x (y)): (2.15)

Let B betheballofradius 4" entredat (0; 4"). Wewantto provethat

I(x;t)2B (2.16)

for every t with x (y)  t  x+(y). Let us denote the omponents of I(x;t) by a x

and a y

.

Arguingas intheproofof(2.11),wegettheidentity

a y =2(" y) 2 p (t x) 2 +(" y) 2 0: As j' x j2,wehavealso (a x ) 2 4(t x) 2 =(2(" y) a y ) 2 4(" y) 2 :

Fromtheseestimatesitfollowsthat

(a x ) 2 +(a y +4") 2 16" 2 ;

whi h proves(2.16). Inthesamewaywe anprovethat

I(t; x)2B (2.17)

forevery t with x ( y)t x+( y):

If f(y)0,wedene

C :=([0;2hf(y)℄[0; 1

2

2"℄)[(f2hf(y)g[0;1 4"℄);

if f(y)0, wesimplyrepla e [0;2hf(y)℄ by [2hf(y);0℄. >Fromthe denition of ' in A

2 , A 3 , A 4 , it followsthat

I( x+( y);x (y))=(2hf(y);1 4") (2.18)

and

for x+( y)s

1 s

2

x (y). Let D:=C (2hf(y);1 4"), i.e.,

D=([ 2hf(y);0℄[ 1+4"; 1

2

+2"℄)[(f0g[ 1+4";0℄);

for f(y)0;theinterval [ 2hf(y);0℄ isrepla edby [0; 2hf(y)℄ when f(y)0. >From(2.15),(2.13),

(2.14),(2.16), (2.17),(2.18)and(2.19)weobtain I(t 1 ;t 2 )2(0;1)+2B+D: (2.20)

As f(0)=0, we an hoose Æ sothat(2.7)issatisedand

j2hf(y)j= x 0 3" 2 jf(y)j" (2.21)

for jyj <Æ. It is then easyto see that, by(2.6), the set (0;1)+2B+D is ontainedin the unit ball

entredat (0;0). Sothat(2.20)implies(b1). 2

Remark2.2 Theassumption (x

0 ;y

0

)6=(0;0) in Theorem2.1 annotbedropped. Indeed, thereis no

neighbourhood U of (0;0) su hthat w isaDiri hletminimizeroftheMumford-Shahfun tionalin U.

Toseethisfa t,let beafun tiondenedonthesquare Q=( 1;1)( 1;1) satisfyingtheboundary

ondition =w on Q andsu h that S =(( 1; 1=2)[(1=2;1))f0g. For every ", let

" be the fun tiondenedonQ " :="Qby "

(x;y):=" (x=";y="). Notethat

"

satisestheboundary ondition

"

=w on Q

"

. Letus omputetheMumford-Shahfun tional for

" on Q " : Z Q" jr " j 2 dxdy+H 1 (S " )=" 2 Z Q jr j 2 dxdy+": Sin e Z Q" jrwj 2 dxdy+H 1 (S w )=4" 2 +2"; wehave Z Q" jr " j 2 dxdy+H 1 (S " )< Z Q" jrwj 2 dxdy+H 1 (S w )

for " suÆ ientlysmall. 2

The onstru tionshownintheproofofTheorem2.1 anbeeasilyadaptedto denea alibrationfor

thefun tion w in (2.5).

Theorem 2.3 Let w:R 2

!R bethe fun tion denedby

w(x;y):= (

x+1 if y>0;

x if y<0:

Then everypoint (x

0 ;y

0 )2R

2

has anopen neighbourhood U su hthat w isaDiri hletminimizerin U

of theMumford-Shahfun tional (2.1).

Proof.{ TheresultfollowsbyExample4.10of[2℄if y

0

6=0. We onsidernowthe ase y

0

=0;wewill

onstru talo al alibrationof w near (x

0

;0),usingthesamete hniqueasinTheorem2.1. Wegiveonly

thenewdenitionsofthesets A

1 ;:::;A

5

Letusx ">0 su hthat 0<"< 1 24 ; 0<"< 1 32 : (2.22)

For 0<Æ<" we onsidertheopenre tangle

U :=f(x;y)2R 2

: jx x

0

j<";jyj<Æg

andthefollowingsubsetsof UR

A 1 := f(x;y;z)2UR: x+1 (y)<z<x+1+(y)g; A 2 := f(x;y;z)2UR: b+()y+3h<z<b+()y+4hg; A 3 := f(x;y;z)2UR: x 0 +3"+2h<z<x 0 +3"+3hg; A 4 := f(x;y;z)2UR: b+()y<z<b+()y+hg; A 5 := f(x;y;z)2UR: x ( y)<z<x+( y)g; where (y):= p 4" 2 (" y) 2 ; h:= 1 6" 5 ; ():=  4 1  ; b:=x 0 +3"+()Æ; := 1 4" 2h :

Wewill assumethat

Æ<

1 6"

10j()j

; (2.23)

sothatthesets A

1 ;:::;A

5

arepairwisedisjoint.

Forevery (x;y;z)2UR, letusdenetheve tor '(x;y;z)2R 3 asfollows: 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > : 2(" y) p (" y) 2 +(z x 1) 2 ; 2(z x 1) p (" y) 2 +(z x 1) 2 ;1 ! if(x;y;z)2A 1 ;  0;;  2 4  if(x;y;z)2A 2 ; (f(y);0;1) if(x;y;z)2A 3 ;  0;;  2 4  if(x;y;z)2A 4 ; 2("+y) p ("+y) 2 +(z x) 2 ; 2(z x) p ("+y) 2 +(z x) 2 ;1 ! if(x;y;z)2A 5 ; (0;0;1) otherwise; where f(y):= 2 h Z (y) 0 " y p t 2 +(" y) 2 dt+ Z ( y) 0 "+y p t 2 +("+y) 2 dt !

2.2 The general ase

Inthisse tionwedenoteby aball in R 2

entredat (0;0) andwe onsider as u in(2.2)andin (2.3)

ageneri harmoni fun tion with normalderivativevanishing on S. We addthe te hni alassumption

thattherstandse ondordertangentialderivativesof u arenotzeroon S.

Theorem 2.4 Let u: !R be aharmoni fun tion su h that 

y

u(x;0) =0 for (x;0) 2, and let

w:!R be thefun tion denedby

w(x;y):= ( u(x;y) for y>0; u(x;y) for y<0: Assumethat u 0 :=u(0;0)6=0,  x u(0;0)6=0, and  2 xx

u(0;0)6=0. Thenthere existsanopen

neighbour-hood U of (0;0) su hthat w isaDiri hletminimizerin U ofthe Mumford-Shah fun tional(2.1).

Proof.{ Wemayassumeu(0;0)>0and

x

u(0;0)>0. Weshallgivetheproofonlyfor 2

xx

u(0;0)>0,

and we shall explain at the end the modi ation needed for  2

xx

u(0;0) < 0. Let v : ! R be the

harmoni onjugateof u that vanishes on y = 0, i.e., the fun tion satisfying 

x v(x;y) =  y u(x;y),  y v(x;y)= x u(x;y),and v(x;0)=0.

ConsiderasmallneighbourhoodU of(0;0)su hthatthemap(x;y):=(u(x;y);v(x;y))isinvertible

on U and 

x

u>0 on U. We all theinversefun tion (u;v)7!((u;v);(u;v)), whi hisdened in

theneighbourhood V :=(U) of (u

0

;0). Notethat, if U is smallenough,then (u;v)=0 ifandonly

if v=0. Moreover, D =   u   v   u   v   = 1 jruj 2   x u  x v  y u  y v  ; (2.24)

where,inthelastformula,allfun tionsare omputedat (x;y)= (u;v),andso 

u  = v ,  v =  u  and  u (u;0)=0,  v

(u;0)>0. Inparti ular,  and  areharmoni ,and

 2 uu (u;0)=0;  2 vv (u;0)=0: (2.25)

On U wewillusethe oordinatesystem (u;v) givenby . By(2.24)the anoni albasisofthetangent

spa eto U atapoint (x;y) isgivenby

 u = ru jruj 2 ;  v = rv jrvj 2 : (2.26)

For every (u;v)2V, let G(u;v) bethematrixasso iatedwiththe rstfundamentalform of U in the

oordinatesystem (u;v),andlet g(u;v) beitsdeterminant. By(2.24)and(2.26),

g=(( u ) 2 +( v ) 2 ) 2 = 1 jru( )j 4 : (2.27) Weset (u;v):= 4 p g(u;v).

The alibration '(x;y;z) on UR willbewrittenas

'(x;y;z)=

1

2

(u(x;y);v(x;y))

(u(x;y);v(x;y);z): (2.28)

Wewilladoptthefollowingrepresentationfor :VR!R 3 : (u;v;z)= u (u;v;z) + v (u;v;z) + z (u;v;z)e ; (2.29)

where e

z

isthethirdve torofthe anoni albasisof R 3

,and 

u , 

v

are omputedatthepoint (u;v).

We now reformulate the onditions of Se tion 1.3 in this new oordinate system. It is known from

dierentialgeometry (see, e.g., [9, Proposition 3.5℄) that, if X =X u  u +X v  v is ave toreldon U,

thenthedivergen e of X isgivenby

divX= 1 2 ( u ( 2 X u )+ v ( 2 X v )): (2.30)

Using(2.26),(2.27),(2.28),(2.29),and(2.30)itturnsoutthat 'isa alibrationifthefollowing onditions

aresatised: (a1) ( u (u;v;z)) 2 +( v (u;v;z)) 2 4 z

(u;v;z) forevery (u;v;z)2VR;

(a2)  u

(u;v;u)=2,  v

(u;v;u)=0,and  z

(u;v;u)=1 forevery (u;v)2V;

(b1) Z t s  u (u;v;z)dz 2 + Z t s  v (u;v;z)dz 2  2

(u;v) forevery (u;v)2V, s;t2R;

(b2) Z u u  u (u;0;z)dz=0 and Z u u  v

(u;0;z)dz= (u;0) forevery (u;0)2V;

( 1)  u  u + v  v + z  z

=0 forevery (u;v;z)2VR.

Givensuitableparameters ">0, h>0, >0,that willbe hosenlater,andassuming

V =f(u;v): ju u

0

j<Æ;jvj<Æg; (2.31)

with Æ<",we onsiderthefollowingsubsetsof VR

A

1

:= f(u;v;z)2VR : u (v)<z<u+(v)g;

A

2

:= f(u;v;z)2VR : 3h+(u;v)<z<3h+(u;v)+1=g;

A

3

:= f(u;v;z)2VR : h<z<hg;

A

4

:= f(u;v;z)2VR : 3h+(u;v) 1=<z< 3h+(u;v)g;

A 5 := f(u;v;z)2VR : u ( v)<z< u+( v)g; where (v):= p 4" 2 (" v) 2 ;

and  is asuitablesmoothfun tion satisfying (u;0)=0, whi h willbedenedlater. Itiseasyto see

that,if " and h aresuÆ ientlysmall,while  issuÆ ientlylarge,thenthesets A

1 ;:::;A

5

arepairwise

disjoint, provided Æ is small enough. Moreover,sin e (u;0) = 

v

(u;0) > 0, by ontinuity we may

assumethat

(u;v)>128" and 

v

For (u;v)2V and z2R theve tor (u;v;z) introdu edin(2.28)is denedasfollows: 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > : 2(" v) p (" v) 2 +(z u) 2  u 2(z u) p (" v) 2 +(z u) 2  v +e z inA 1 ; (u;v) v p (u a) 2 +v 2  u +(u;v) u a p (u a) 2 +v 2  v +e z inA 2 ; f(v) u +e z inA 3 ; (u;v) v p (u a) 2 +v 2  u +(u;v) u a p (u a) 2 +v 2  v +e z inA 4 ; 2("+v) p ("+v) 2 +(z+u) 2  u + 2(z+u) p ("+v) 2 +(z+u) 2  v +e z inA 5 ; e z otherwise, where a<u 0 11Æ; >0 (2.33) f(v):= 1 h Z (v) 0 (" v) p t 2 +(" v) 2 dt Z ( v) 0 ("+v) p t 2 +("+v) 2 dt ! ; (u;v):= 1 2 (a+ p (u a) 2 +v 2 ;0) 2": (2.34)

We hoose  as thesolutionoftheCau hyproblem

8 > < > : (u;v)( v u +(u a) v )=( 1) p (u a) 2 +v 2 ; (u;0)=0: (2.35)

Sin e the line v =0 is not hara teristi for theequation near (u

0

;0), there exists a unique solution

2C 1

(V),provided V issmallenough.

In the oordinatesystem (u;v) thedenition of theeld  in A

1 , A

3

, and A

5

is thesame as the

denitionof ' intheproofofTheorem2.1. The ru ialdieren eisinthedenitiononthesets A

2 and

A

4

, wherenowwearefor edtointrodu etwonewparameters a and . Notethat thedenitiongiven

inTheorem 2.1 anberegardedasthelimiting aseas a tendsto +1.

Inordertosatisfy ondition(a1),itisenoughtotaketheparameter  su hthat

 2 4  2 (u;v)

Sin e jf(v)j (v)+( v) h  4" h ; (2.37)

inequality(2.36)istrueifweimpose

2"h:

Looking atthedenitionof  on A

1 and A

5

,one an he kthat ondition(a2)issatised.

Bydire t omputationsitiseasyto seethat  satises ondition( 1)on A

1 and A 5 . Similarly,the ve toreld v p (u a) 2 +v 2 ; u a p (u a) 2 +v 2 ! isdivergen e-free;sin e (u a) 2 +v 2

is onstantalongtheintegral urvesofthis eld,by onstru tion

thesamepropertyholdsfor ,so that  satises ondition( 1)in A

2 and A 4 . In A 3

, ondition( 1)istriviallysatised.

Notethatthenormal omponentof is ontinuousa rossea h A

i

: fortheregionA

3

this ontinuity

isguaranteedbyour hoi eof . Thisimpliesthat( 1)issatisedinthesenseofdistributionson VR.

Arguingasin theproofof(2.11), (2.12),(2.14)in Theorem2.1,wendthatforevery (u;v)2V

Z u+( v) u  u (u;v;z)dz+ Z h h  u (u;v;z)dz+ Z u u (v)  u (u;v;z)dz=0; Z u+( v) u  v (u;v;z)dz+ Z h h  v (u;v;z)dz+ Z u u (v)  v (u;v;z)dz=4":

Now,itiseasytoseethat

Z u u  u (u;v;z)dz= 2(u;v) v p (u a) 2 +v 2 ; (2.38) Z u u  v (u;v;z)dz=4"+2(u;v) u a p (u a) 2 +v 2 ; (2.39)

sin efor v=0 wehave

(u;0)= 1

2

(u;0) 2";

ondition(b2) issatised.

By ontinuity,if Æ issmallenough,wehave

Z u u  v (u;v;z)dz> 7 8 (u;v) (2.40)

forevery (u;v)2V.

Fromnowon,weregardthepair ( u

; v

) asave torin R 2

. Toprove ondition(b1)weset

I ";a (u;v;s;t):= Z t s ( u ; v )(u;v;z)dz

forevery (u;v)2V, andforevery s;t2R. Wewantto omparethebehaviourofthefun tions jI

";a j

2

and 2

;tothis aim,wedenethefun tion

d (u;v;s;t):=jI (u;v;s;t)j 2

2

Wehavealreadyshown( ondition (b2))that

d

";a

(u;0; u;u)=0: (2.41)

Westartbyprovingthat, if V issuÆ ientlysmall, ondition(b1) holdsforevery (u;v)2V,for s lose

to u and t lose to u. Usingthe denition of (u;v;z) on A

1 and A

5

, one an omputeexpli itly

d

";a

(u;v;s;t) for js+uj( v) and for jt uj(v). Bydire t omputations oneobtains

r v;s;t d ";a (u;0; u;u)=0 (2.42) for (u;0)2V.

Wenowwantto omputetheHessianmatrix r 2 v;s;t d ";a atthepoint (u 0 ;0; u 0 ;u 0 ). By(2.34)and

(2.27),after someeasy omputations,weget

 2 vv (u;0)= 1 2(u a)  u (u;0)= 1 2(u a)  2 uv (u;0):

Usingthisequalityandtheexpli itexpressionof d

";a near (u 0 ;0; u 0 ;u 0 ), weobtain  2 vv d ";a (u 0 ;0; u 0 ;u 0 )= 8" (u 0 a) 2 ( v (u 0 ;0) 4")+ 2 u 0 a  v (u 0 ;0) 2 uv (u 0 ;0)  2 vv ( 2 )(u 0 ;0):

Sin e  and donotdependon a and ",forevery" satisfying(2.32)we annd a so loseto u

0 that  2 vv d ";a (u 0 ;0; u 0 ;u 0 )<0: (2.43)

Moreover,weeasilyobtainthat

 2 tt d ";a (u 0 ;0; u 0 ;u 0 )= 2 ss d ";a (u 0 ;0; u 0 ;u 0 )=8 4 "  v (u 0 ;0);  2 vt d ";a (u 0 ;0; u 0 ;u 0 )= 2 vs d ";a (u 0 ;0; u 0 ;u 0 )= 4 u 0 a ( v (u 0 ;0) 4");  2 st d ";a (u 0 ;0; u 0 ;u 0 )=8:

Bytheaboveexpressions,itfollowsthat

det 0   2 vv d ";a  2 vt d ";a  2 vt d ";a  2 tt d ";a 1 A (u 0 ;0; u 0 ;u 0 )= 16 (u 0 a) 2  v (u 0 ;0)( v (u 0 ;0) 4")+ 1 (") u 0 a + 2 ("); where 1 ("), 2

(") aretwo onstantsdepending onlyon ". Then,if " satises(2.32), a an be hosen

so loseto u 0 that det 0   2 vv d ";a  2 vt d ";a  2 vt d ";a  2 tt d ";a 1 A (u 0 ;0; u 0 ;u 0 )>0: (2.44)

Atlast,thedeterminantoftheHessianmatrixof d

";a at (u 0 ;0; u 0 ;u 0 ) isgivenby detr 2 v;s;t d ";a (u 0 ;0; u 0 ;u 0 )= 32 " 2 (u 0 a) ( v (u 0 ;0)) 2  2 uv (u 0 ;0)( v (u 0 ;0) 4")+ 3 ("); where 3

(") isa onstantdependingonlyon ". Sin e,by(2.24),

 2 uv (u 0 ;0)=  2 xx u(0;0) ( u(0;0)) 3 ;

given " satisfying(2.32), we an hoose a so loseto u 0 that detr 2 v;s;t d ";a (u 0 ;0; u 0 ;u 0 )<0: (2.45)

By(2.43), (2.44), and(2.45), we an on ludethat, by asuitable hoi eofthe parameters,the Hessian

matrixof d ";a (withrespe tto v;s;t)at (u 0 ;0; u 0 ;u 0

) isnegativedenite. Thisfa t,with(2.41)and

(2.42),allowsustostatetheexisten eofa onstant  >0 su hthat

d ";a (u;v;s;t)<0 (2.46) for js+u 0 j<, jt u 0

j<, (u;v)2V, v6=0, provided V is suÆ ientlysmall. So, ondition(b1) is

satisedfor js+u

0

j< and jt u

0

j<. We anassume Æ< <(v) forevery (u;v)2V.

Fromnowon,sin eatthis pointtheparameters ", a havebeenxed, wesimplywrite I insteadof

I

";a

. Wenowstudythemoregeneral ase js+uj<( v) and jt uj<(v).

Letusset

m

1

(u;v):=maxfjI(u;v;s;t)j: js+uj( v); jt uj(v); jt u

0 jg: Bythedenitionof A 1 ;:::;A 5 ,for =(Æ)+Æ wehave( u ; v )=0 on (V[u 0 ;u 0 +℄)nA 1 and (V[ u 0 ; u 0 +℄)nA 5

. Thisimpliesthat

m

1

(u;v)=maxfjI(u;v;s;t)j: js+u

0

j;  jt u

0 jg

for (u;v)2 V. The fun tion m

1

, as supremum of afamily of ontinuous fun tions,is lower

semi on-tinuous. Moreover, m

1

isalsouppersemi ontinuous;indeed,suppose,by ontradi tion,thatthere exist

twosequen es (u

n ), (v

n

) onvergingrespe tivelyto u, v, su hthat (m

1 (u n ;v n )) onvergesto alimit l>m 1

(u;v);then, thereexist (s

n ), (t n ) su hthat js n +u n j( v n ); jt n u n j(v n ); jt n u 0 j; (2.47) and m 1 (u n ;v n ) = jI(u n ;v n ;s n ;t n

)j. Up to subsequen es, we an assume that (s

n ), (t

n

) onverge

respe tivelyto s, t su h that,by(2.47),

js+uj( v); jt uj(v); jt u

0 j;

hen e,wehavethat

m

1

(u;v)jI(u;v;s;t)j= lim

n!1 jI(u n ;v n ;s n ;t n )j=l;

whi h isimpossiblesin e l>m

1

(u;v). Therefore, m

1

is ontinuous.

Let B betheopenballofradius 4" entredat (0; 4"). Arguingasin (2.16),we an provethat

I(u;v;u;t)2B (2.48)

whenever 0<jt uj(v). Inthesamewaywe an provethat

I(u;v;s; u)2B (2.49)

for 0<js+uj( v). We an write

I(u;v;s;t)=I(u;v;s; u)+I(u;v; u;u)+I(u;v;u;t): (2.50)

So,for js+uj( v), jt uj(v),and jt u

0

j,by(2.49),(2.38),(2.39), and(2.48), weobtain

hen e,by(2.32),I(u;0;s;t)belongstotheopenballofradius (u;0) entredat(0;0),andso,m

1

(u;0)<

(u;0). By ontinuity,if V issmallenough,

m

1

(u;v)< (u;v) (2.51)

forevery (u;v)2V.

Analogously,wedene

m

2

(u;v):=maxfjI(u;v;s;t)j: js+uj( v); js+u

0

j; jt uj(v);g:

Arguingasin the aseof m

1

,we anprovethat,if V issmallenough,

m

2

(u;v)< (u;v) (2.52)

forevery (u;v)2V.

By (2.51), (2.52), and (2.46), we an on ludethat I(u;v;s;t) belongs to theball entred at (0;0)

withradius (u;v),for js+uj( v)and jt uj(v). Morepre isely,letE(u;v)betheinterse tion

ofthis ballwiththeupperhalf planebounded bythehorizontalstraightlinepassingthroughthepoint

(0; 3

4

(u;v)): by(2.50), (2.40),(2.48),(2.49), and(2.32),wededu ethat

I(u;v;s;t)2E(u;v) (2.53)

for js+uj( v) and jt uj(v).

We an now on ludethe proof of (b1). It is enoughto onsider the ase u ( v) s  t 

u+(v). We anwrite I(u;v;s;t)=I(u;v;s^( u+( v));t_(u (v))) +I(u;v;s_( u+( v));t^(u (v))) I(u;v; u+( v);u (v)): (2.54) By(2.53),itfollowsthat I(u;v;s^( u+( v));t_(u (v)))2E(u;v): (2.55) Let C 1

(u;v) betheparallelogramhavingthree onse utiveverti esatthepoints

(2hf(v);0); (0;0); (u;v) ( v;u a) p (u a) 2 +v 2 ; let C 2

(u;v) bethesegmentwithendpoints

(2hf(v);0); (2hf(v);0)+2(u;v) ( v;u a) p (u a) 2 +v 2 ;

andlet C(u;v):=C

1

(u;v)[C

2 (u;v).

Fromthedenition of ' in A

2 , A 3 , A 4 ,itfollowsthat I(u;v; u+( v);u (v))=(2hf(v);0)+2(u;v) ( v;u a) p (u a) 2 +v 2 (2.56) and I(u;v;s ;s )2C(u;v) (2.57)