Variational approximation for a functional governing point-like singularities

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point-like singularities

Daniele Graziani, Gilles Aubert

To cite this version:

Daniele Graziani, Gilles Aubert. Variational approximation for a functional governing point-like

sin-gularities. [Research Report] RR-7095, INRIA. 2009, pp.35. �inria-00403232v3�

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

1

Variational approximation of a functional governing

point-like singularities.

Daniele Graziani, Gilles Aubert

N° 7095

2009

2

Unité de recherche INRIA Sophia Antipolis

point-like singularities. 4 Daniele Graziani

∗

, GillesAubert

†

5

ThèmeCOG Systèmes ognitifs 6

ProjetARIANA 7

Rapportdere her he n° 70952009xxxpages 8

Abstra t: The aim of this paper is to provide arigorous variationalformulationfor the 9

dete tion of points in

2

-d images. To this purpose we introdu e a new fun tional of the 10

al ulusofvariationwhoseminimizersgivethepointswewanttodete t. Thenwebuildan 11

approximatingsequen eoffun tionals,forwhi hweprovethe

Γ

- onvergen e,withrespe t 12

toasuitable onvergen e,totheinitialone. 13

Key-words: pointsdete tion,divergen e-measureelds,

p

- apa ity,

Γ

- onvergen e. 14

∗

ARIANAProje t-team, CNRS/INRIA/UNSA,2004Routedeslu ioles-BP93,06902Sophia-Antipolis Cedex,Fran e

†

LABORATOIREJ.A.DIEUDONNÉUniversitédeNi eSOPHIAANTIPOLIS,par valrose06108Ni e CEDEX2,FRANCE.

Résumé: Nousproposons unenouvelleméthode variationellepourisolerdespointsdans 17

uneimage

2

-D.Dans ebutnousintroduisonsuneenergiedontlespointsdeminimumsont 18

donnésparl'ensembledespointsqueonveutdéte ter. En suiteonappro he ette energie 19

parunesuitedefon tionellesplusrégulières,pourlaquelleonmontrela

Γ

- onvergen evers 20

lafon tionelle initiale. 21

Mots- lés: déte tiondepoints, hampsave divergen emesure,

p

- apa ité,

Γ

- onvergen e. 22

Contents 23 1 Introdu tion ii 24 2 Preliminaries v 25 2.1 Notation. . . v 26

2.2 Distributionaldivergen eand lassi alspa es . . . v 27

2.3

p

- apa ity . . . vi 28

3 The Variational Model vii

29

3.1 Thevariationalframework . . . ix 30

3.2 TheFun tional . . . xi 31

4

Γ

- onvergen e: The intermediate approximation xii 32

4.1 Compa tness . . . xiii 33

4.2 Lowerbound . . . xv 34

4.3 Upperbound . . . xvi 35

4.4 Variationalproperty . . . xvii 36

5 Approximation by smoothfun tion xix 37

5.1 Compa tness . . . xx 38

5.2 Lowerbound . . . xxii 39

5.3 Upperbound . . . xxiii 40

5.4 Variationalproperty . . . xxvi 41

6 De Giorgi'sConje ture xxvii

42

Referen es xxix

43

1 Introdu tion 44

The issue of dete ting ne stru tures, like points or urves in two or three dimensional 45

biologi alimages,isa ru ialtaskin imagepro essing. Inparti ularapointmayrepresent 46

aviralparti lewhosevisibilityis ompromisedbythepresen eof otherstru tureslike ell 47

membranesorsomenoise. Thereforeoneof themain goalsis dete tingthespotsthat the 48

biologistswish to ount. This operationis made harderby thepresen e of othersingular 49

stru tures. 50

In some biologi al images the image intensity is a fun tion that takes the value

1

on 51

pointsorother stru tureslikesets with Hausdordimension

0 ≤ α < 1

, andit is lose to 52

0

outside. In image pro essing these on entrationsets are alled dis ontinuities without 53

jump,meaningthatthereisnojumpa rossthesetandthereforethegradientoftheimage 54

is

0

. 55

In the literature there are few variational methods dealing with this problem. In this 56

dire tion one interesting approa h has been proposed in [3℄. In that paper the authors 57

onsider this kind of pathology as a

k

- odimension obje t, meaning that they should be 58

regardedasa singularityof a map

U : R

k+m

_{→ R}

k

, with

k ≥ 2

and

m ≥ 0

(see[6℄ for a 59

ompletesurveyonthissubje t). Inparti ularthedete tingpoint ase orrespondstothe 60

ase

k = 2

and

m = 0

. 61

Thispointofviewmakespossibleavariationalapproa hbasedonthetheoryof Ginzburg-62

Landausystems. Intheirworktheisolatedpointsin

2

-Dimagesare regardedasthe topo-63

logi alsingularitiesofamap

U : R

2

_{→ S}

1

,where

S

1

istheunitsphereof

R

2

. Startingfrom 64

theinitialimage

I : Ω ⊂ R

2

_{→ R}

,thisstrategy makes ru ialthe onstru tionofaninitial 65

ve toreld

U

0

: R

2

_{→ S}

1

with atopologi al singularityof degree

1

. Howto build su h a 66

ve toreldin arigorousway,is asubje tofa urrentinvestigation. 67

Ourrstpurposehereisndingamostnaturalvariationalframeworkinwhi harigorous denitionofdis ontinuitywithoutjump anbegiven. Inourmodeltheimage

I

isaRadon measure. It is ru ial for dete ting points that this Radon measure be able of harging points. Thepreliminary stepis nding aspa e whose elementsare ableof produ ing this kind of measures. This spa e is given by

DM

p

_(Ω)

: the spa e of

L

p

-ve tor elds whose distributionaldivergen e is aRadonmeasure,with

1 < p < 2

. Therestri tionon

p

is due to thefa t that when

p ≥ 2

thedistributional divergen e

DivU

of

U

annot beameasure on entratedonpoints(seeSe tion

3.1

below). Thenwehaveto onstru t,fromtheoriginal image

I

,adatave tor

U

0

∈ DM

p

_(Ω)

. Clearlythereare,atleastinprin iple,manywaysto dothis. Theoneweproposehereseemsto bethemostnatural. We onsiderthe lassi al ellipti problemwithmeasuredata

I

:

(

−∆u

0

= I

on

Ω

u

0

= 0

on

∂Ω.

Thenbysetting

U

0

= ∇u

0

wehave

U

0

∈ DM

p

_(Ω)

with

DivU

0

= I

. Howeverthesupportof 68

themeasure

DivU

0

istoolargeand ould ontainsseveralstru tureslike urvesorfra tals, 69

whilethesingularities,weareinterestedin,are ontainedintheatomi partofthemeasure 70

DivU

0

and therefore we have to isolate it. To do this the notion of

p

- apa ity of a set 71

plays a keyrole. Indeed when

p < 2

the

p

- apa ity of a point in

Ω

is zero and one an 72

say, in this sense, that it is a dis ontinuity with no jump. Besides everyRadon measure 73

anbede omposed(see[14℄)intwomutuallysingularmeasures: therstoneisabsolutely 74

ontinuouswithrespe ttothe

p

- apa ityandthese ondoneissingularwithrespe ttothe 75

p

- apa ity,meaningthatitisameasure on entratedonsetswith

0 p

- apa ity. 76

Asitisknownin dimension

2

,setswith

0 p

- apa ity,andhen edis ontinuitieswithout 77

jump, an beisolated points, ountable set of pointsorfra tals with Hausdordimension 78

0 ≤ α < 1

(seeSubse tion

2.3

forthedenitonof

p

- apa ityandrelatedproperties). 79

Ourgoalhereiskeepingnothingelsebut pointsintheimage. Thea hievementofsu h 80

apurposemakesne essarytheminimizationofasuitableenergythatmust removeallthe 81

dis ontinuitieswhi harenotdis ontinuitieswithoutjump,andremoveallthedis ontinuities 82

withoutjump whi harenotisolatedpoint. 83

Fromonehandwehavetofor ethe on entrationsetofthedivergen emeasureof

U

to 84

ontainonlythepointswewantto at h, andontheotherhand wehaveto regularizethe 85

initial data

U

0

outside thepoints of singularities. To this end weintrodu e the auxiliary 86

spa e

SDM

p

_(Ω)

of ve torelds belonging to

DM

p

_(Ω)

whose divergen e measure hasno 87

absolutely ontinuous part with respe t to the

p

- apa ity. Then, by taking into a ount 88

that theinitialve toreld is agradientofaSobolev fun tion,ourgoalisto minimizethe 89 followingenergy: 90

F(u) =

Z

Ω

|∆u|

2

_{dx + λ}

Z

Ω

|∇u − U

0

|

p

dx + µH

0

(supp(div

s

∇u)

0

),

where

u ∈ W

1,p

0

(Ω)

with

∇u ∈ SDM

p

_(Ω)

,

1 < p < 2

and

λ, µ

arepositive weights. The 91

gradient of a minimizer of the energy

F

is the ve toreld we are looking for, that is a 92

ve toreldwhosedivergen emeasure anbede omposedinanabsolutely ontinuous(with 93

respe ttotheLebesgue'smeasure)termplusanatomi measure on entratedonthepoints 94

wewantto isolatein theimage. 95

Evenifapointwise hara terizationofdis ontinuitywithoutjumpisnotavailable,thanks 96

toourdenitionthesingularsetofpoints anbelinkedtotheve toreld

∇u

,inthespirit 97

of the lassi al SBV formulation of the Mumford-Shah's fun tional (we refer to [1℄ for a 98

ompletesurveyontheMumfordShah'sfun tional). 99

Forfuture omputationalpurposes, thenexttaskistoprovideanapproximationinthe sense of

Γ

- onvergen e introdu ed in [16, 17℄. Our approa h is lose in the spirit to the oneusedto approximatetheMumfordShahfun tionalbyafamilyofdepending urvature fun tionalsas in[9℄. Indeed,asin theirwork(seealso[8℄),werepla etheatomi measure

H

0

bytheterm

G

ε

(D) =

1

4π

Z

∂D

1

ε

+ εκ

2

_dH

1

_;

where

D

is a properregular set ontainingthe atomi set

P

,

κ

is the urvature of its 100

boundary,and the onstant

1

4π

is anormalizationfa tor. Roughlyspeakingtheminimaof 101

thesefun tionalsarea hievedontheunionofballsofsmallradius,so thatwhen

ε → 0

the 102

sequen e

G

ε

shrinkstotheatomi measure

H

0

_{(P )}

. 103

Thisleadstoanintermediateapproximationgivenby

F

ε

(u, D) =

Z

Ω

(1 − χ

D

)|∆u|

2

dx +

Z

Ω

|∇u − U

0

|

p

dx

+

1

4π

Z

∂D

1

ε

+ εκ

2

_dH

1

_.

(1) This strategy permits to work with the perimeter measure

H

1

_⌊∂D

, that an be approxi-mated,a ordingtotheModi a-Mortola'sapproa h(see[21,22℄),bythemeasure:

µ

ε

(w, ∇w)dx = ε|∇w|

2

+

W (w)

ε

dx,

where

W (w) = w

2

_{(1 − w)}

2

isadoublewellfun tion. 104

BesidesbyusingSard'sTheoremand oareaformula(seealso[4℄forasimilarapproa h) one anformally repla etheintegralon

∂D

by anintegral omputed overthelevelsetsof

w

, whose urvature

κ

be omes

div

∇w

|∇w|

and theintegralis omputed overthelevelsets of

w

. Sothatone anformallywrite the ompleteapproximatingsequen e:

F

ε

(u, w)

=

Z

Ω

w

2

|∆u|

2

dx + µ

1

8πC

Z

Ω\{∇w=0}

(

1

β

ε

+ β

ε

div(

∇w

|∇w|

)

2

(ε|∇w|

2

+

1

ε

W (w))dx

+ λ

Z

Ω

|∇u − U

0

|

p

dx +

1

γ

ε

Z

Ω

(1 − w)

2

dx,

where,asusual,

C =

R

1

0

pW (t)dt

,

β

ε

and

γ

ε

areinnitesimalas

ε → 0

. Thelastintegralis 105

apenalizationtermthat for es

w

totendto

1

as

ε → 0

. 106

Thegoalofthe se ondpartofthis workis thento showthat thefamilyofenergies

F

ε

107

Γ

- onvergestothefun tional

F

whentheparametersarerelatedin asuitableway. 108

As in [9℄ wedealwithasuitable onvergen e offun tions involvingtheHausdor on-109

vergen eofasub-levelsets. Thisstrategyrequiresa arefulstatementofthe

Γ

- onvergen e 110

denitions andresults,in order tohavethat sequen es asymptoti allyminimizing

F

ε

on-111

vergestoaminimumof

F

. 112

Despite this approa h is inspired bysome ideas ontainedin [8, 9℄, we point out that 113

inour asetheregularizationterminvolvesase ondorderdierentialoperator,duetothe 114

fa tthat ourgoalistodete tpointsandnotsegment urves. Thisdeepdieren erequires 115

anontrivialadaptationoftheargumentsusedinthosepapers. 116

Thepaperisorganizedasfollows. Se tion

2

is devotedtonotations,preliminary deni-117

tionsandresults. In Se tion

3

weillustrate thenewvariationalmodel andwepresentthe 118

fun tionalwedealwith. Se tion

4

and

5

aredevotedtothe

Γ

- onvergen eresult. Finallyin 119

thelastSe tionwe on ludethepaperby omparingthisapproa hwiththe elebrated on-120

je turebyDeGiorgi, on erningtheapproximationofthe urvaturedependingfun tionals. 121

Wedo notgivehereexperimental resultillustratingourapproa h. Wereferthereader 122

forthatto[19℄. 123

2 Preliminaries 124

2.1 Notation 125

Inall the paper

Ω ⊂ R

2

is anopenbounded set with lips hitzboundary. The Eu lidean 126

normwill bedenoted by

| · |

, whilethesymbol

k · k

indi ates thenorm ofsomefun tional 127

spa es. Thebra kets

h, i

denotes thedualityprodu tin somedistributional spa es.

L

d

or 128

dx

isthe

d

-dimensionalLebesguemeasureand

H

k

is the

k

-dimensionalHausdor measure. 129

B

ρ

(x

0

)

isthe ball entered at

x

0

with radius

ρ

. Wesay thata set

D ⊂ Ω

isaregular set 130

ifit anbewritten as

{F < 0}

with

F ∈ C

∞

0

(Ω)

. In thefollowingwewill denoteby

R(Ω)

131

thefamily ofall regularsets in

Ω

. Finally wewill use thesymbol

⇀

fordenoting aweak 132

onvergen e. 133

2.2 Distributional divergen e and lassi al spa es 134

InthisSubse tionwere allthedenitionofthedistributionalspa e

L

p,q

_{(div; Ω)}

and

DM

p

_(Ω)

, 135

1 ≤ p, q ≤ +∞

,(see[2,12℄). 136

Denition 2.1. We say that

U ∈ L

p,q

_{(div; Ω)}

if

U ∈ L

p

_{(Ω; R}

2

₎

and if its distributional 137

divergen e

DivU = divU ∈ L

q

_(Ω)

. If

p = q

the spa e

L

p,q

_{(div; Ω)}

will be denoted by 138

L

p

_{(div; Ω)}

. 139

Wesay that afun tion

u ∈ W

1,p

_(Ω)

belongsto

W

1,p,q

_{(div; Ω)}

if

∇u ∈ L

p,q

_{(div; Ω)}

. We 140

saythat afun tion

u ∈ W

1,p

0

(Ω)

belongsto

W

1,p,q

0

(div; Ω)

if

∇u ∈ L

p,q

_{(div; Ω)}

. 141 Denition2.2. For

U ∈ L

p

_{(Ω; R}

2

₎

,

1 ≤ p ≤ +∞

,set

|DivU |(Ω) := sup{hU, ∇ϕi : ϕ ∈ C

0

∞

(Ω), |ϕ| ≤ 1}.

Wesaythat

U

isan

L

p

-divergen e measureeld, i.e.

U ∈ DM

p

_(Ω)

,if

kU k

DM

p

_(Ω)

:= kU k

_L

p

_(Ω;R

2

₎

+ |DivU |(Ω) < +∞.

Letusre allthefollowing lassi alresult(see[13℄Proposition

3.1

). 142 Theorem2.1. Let

{U

k

}

k

⊂ DM

p

_(Ω)

besu hthat 143

U

k

⇀ U

in

L

p

_{(Ω; R}

2

_),

as

k → +∞

for

1 ≤ p < +∞.

(2) Then

kU k

L

p

_(Ω;R

2

₎

≤ lim inf

k→+∞

kU

k

k

L

p

(Ω;R

2

)

,

|DivU |(Ω) ≤ lim inf

k→+∞

|DivU

k

|(Ω).

2.3

p

- apa ity 144

The

p

- apa itywill be ru ial to nd a onvenient fun tional framework to deal with. If

K ⊂ R

2

isa ompa tset and

χ

K

denotesits hara teristi fun tion,wedene:

Cap

p

(K, Ω) = inf{

Z

Ω

|∇f |

p

_{dx, f ∈ C}

∞

0

(Ω), f ≥ χ

k

}.

If

U ⊂ Ω

isanopenset ,the

p

- apa ityisgivenby

Cap

p

(U, Ω) = sup

K⊂U

Cap

p

(K, Ω).

Finallyif

A ⊂ Ω

isaBorelset

Cap

p

(A, Ω) =

inf

A⊂U⊂Ω

Cap

p

(U, Ω).

Were allthefollowingresult(seeforinstan e[20℄,Theorem2.27)thatexplainsthe relation-145

shipbetween

p

- apa ityandHausdormeasures. Su haresultis ru ial tohavegeometri 146

informationsonnull

p

- apa itysets. 147

Theorem2.2. Assume

1 < p < 2

. If

H

2−p

_{(A) < ∞}

then

Cap

p

(A, Ω) = 0

. 148

Anotherusefultoolto managesetsof

p

- apa ity

0

is providedby thefollowing hara -149

terization. 150

Theorem2.3. Let

E

bea ompa t subsetof

Ω

. Then

Cap

p

(E, Ω) = 0

if andonlyif there 151 existsasequen e

{φ

k

}

k

⊂ C

∞

0

(Ω)

, onvergingto

0

stronglyin

W

1,p

0

(Ω)

,su hthat

0 ≤ φ

k

≤ 1

152

and

φ

k

= 1

on

E

for every

k

. 153

Forageneralsurveywereferthereaderto[18,20, 25℄. 154

3 The Variational Model 155

Inthisse tionwesetthefun tionalframeworkandthefun tionaltobeminimized. 156

Roughlyspeaking in biologi alimages theimage is afun tion that ouldbeveryhigh onpointsor other stru tureslikesets withHausdor dimension

0 ≤ α < 1

, andit is lose to

0

outside. From amathemati alpointof viewitseemsto bemu hmoreappropriateto thinkoftheimageasaRadonmeasure,thatis

I = µ ∈ (C

0

(Ω))

∗

. Thenextstepisndinga spa ewhoseelementsareableofprodu ingthiskindofdis ontinuities: thespa e

DM

p

_(Ω)

, with

1 < p < 2

. Therestri tionon

p

is dueto thefa t that when

p ≥ 2

thedistributional divergen eof

U

annotbeameasure on entratedon points. Set

p ≥ 2

,a ording tothe denition,wehave

hDivU, ϕi = −

Z

Ω

U · ∇ϕdx

forall

ϕ ∈ C

∞

0

(Ω).

Sin e

p ≥ 2

thisdistributionis well-dened foranytest

ϕ ∈ W

1,p

′

0

(Ω)

,where

p

′

_{≤ 2}

isthe 157

dualexponentof

p

. Inparti ular

DivU

belongstothedualspa e

W

−1,p

′

(Ω)

oftheSobolev 158

spa e

W

1,p

0

(Ω)

. Thenin this ase,thedistributional divergen e of

U

annotbeanatomi 159

measure,sin e

δ

0

∈ W

/

−1,p

′

(Ω)

. Tosee this, one an onsider as

Ω

the disk

B

1

(0)

andthe 160

fun tion

ϕ(x) = log(log(1 + |x|)) − log(log(2))

˜

. Thisfun tion isin thespa e

W

1,p

′

0

(Ω)

for 161

every

p

′

_{≤ 2}

and therefore itis an admissible test fun tion,howeverit easy to he k that 162

hδ

0

, ϕi = +∞

. 163

When

1 < p < 2

wehavethat

DivU ∈ W

−1,p

′

(Ω)

, butin this asesin e

p < 2

,wehave 164

p

′

> 2

and hen e thefun tion

ϕ

˜

is nolonger an admissible test fun tion. One an he k 165

thatthe distribution

DivU

isanelementof

(C

0

(Ω))

∗

ableof hargingthepoints. Takefor 166

instan ethemap

U (x, y) = (

x

x

2

_+y

2

,

y

x

2

_+y

2

)

. 167

Thenextstepistotransformtheinitialimage

I

asthedivergen emeasureofasuitable 168

ve toreld. We onsidertheellipti problemwithmeasuredata

I

: 169

(

−∆u = I

on

Ω

u = 0

on

∂Ω.

(3)

Classi al results (see [24℄) ensures the existen e of a unique weak solution

u ∈ W

1,p

0

(Ω)

with

p < 2

. Then iteasyto see that thedistributional divergen eof

∇u

is givenby

I

. In parti ular bysetting

U = ∇u

, wehave

U ∈ DM

p

_(Ω)

. A ordingto theRadon-Nikodym de ompositionofthemeasure

DivU

wehave

DivU = divU + div

s

U,

where

divU ∈ L

1

_(Ω)

and

div

s

_U

isasingularmeasure withrespe tto

L

2

. Forourpurpose 170

the support of the singular measure

div

s

_U

is too large. In parti ular the measure

div

s

_U

171

ould harge sets with Hausdor dimension

0 ≤ α < 2

. So that in order to isolate the 172

singularities we are interested in, we need a further de omposition of the measure

DivU

. 173

This anbedonebyusingthe apa itaryde ompositionoftheRadonmeasure

div

s

_U

. Itis 174

known(see[14℄)thatgivenaRadonmeasure

µ

thefollowingde ompositionholds 175

µ = µ

a

+ µ

0

,

(4)

where the measure

µ

a

is absolutely ontinuous with respe t to the

p

- apa ity and

µ

0

is 176

singularwithrespe ttothe

p

- apa ity,thatis on entratedonsetswith

0 p

- apa ity. Besides 177

it isalso known (see [14℄)that everymeasure whi h is absolutely ontinuouswith respe t 178

to the

p

- apa ity an be hara terized asan element of

L

1

_{+ W}

−1,p

′

, leadingto the ner 179 de omposition: 180

µ = f − DivG + µ

0

,

(5) where

G ∈ L

p

′

(Ω; R

2

)

with

1

p

+

1

p

′

= 1

and

f ∈ L

1

_(Ω)

. 181

Byapplyingthisde ompositontothemeasure

div

s

_U

weobtainthefollowing de ompo-182

sitionofthemeasure

DivU

183

DivU = divU + f − DivG + (div

s

U )

0

,

(6)

with

G ∈ L

p

′

(Ω; R

2

₎

,

f ∈ L

1

_(Ω)

,

divU ∈ L

1

_(Ω)

,and

(div

s

_{U )}

0

isameasure on entratedon 184

aset with

0 p

- apa ity. 185

A ordingtothis de ompositionandtaking intoa ountTheorem2.3wegivethe de-186

nitionofdis ontinuitywithoutandwith jump. 187

Denition3.1. Wesay that apoint

x ∈ Ω ⊂ R

2

is apointof dis ontinuity without jump 188

of

U

if

x ∈ supp(div

s

_{U )}

0

. 189

Remark 3.1. The other singularities, where thereis a jump, are ontained in the se ond term of de omposition (6 ). Indeed the spa e

W

−1,p

′

(Ω)

ontains Hausdor measures re-stri tedtosub-manifoldsof dimension greaterthanor equalto

1

. (Werefer to[25℄Se tion

4.7

for adetaileddis ussion onthe spa e

W

−1,p

′

(Ω)

), likefor instan eHausdormeasures on entrated on regular losed urves, whi h are lassi al examples of dis ontinuities with jump. Morepre isely a ontourofaregularset

D

isthe jumpsetofthe hara teristi fun -tionof

D

andits

p

- apa ityisstri tlypositive. Thisisof ourseinagreementwithTheorem 2.3 . Indeedif therewere asequen e

{φ

k

}

k

⊂ C

∞

0

(Ω)

, onverging to

0

stronglyin

W

1,p

0

(Ω)

, su h that

0 ≤ φ

k

≤ 1

and

φ

k

= 1

on

∂D

for every

k

, it would be possible to dene the sequen e

˜

φ

k

=

(

φ

k

on

D

1

on

Ω \ D,

whi h onverges, in the

W

1,p

-norm, tothe

BV

-fun tion

1 − χ

D

, whi h annot be approxi-190

matedbyregularfun tionsinthe

W

1,p

-norm. 191

Denition3.2. Wesaythatapoint

x ∈ Ω ⊂ R

2

isapointofdis ontinuitywith jumpof

U

192

if

x ∈ supp(f − DivG)

. 193

3.1 The variational framework 194

Weshallbuildanenergywhoseminimizerswillbeve toreldswhosedivergen emeasure's 195

singularpartwillbegivenbynothingelsebut points. 196

Ea hminimizermustbean

L

p

(with

p < 2

)ve toreldwiththefollowingproperties: 197

1. Itmustbe loseto theinitialdata

U

0

whi his,in general,an

L

p

ve toreld

U

0

with 198

1 < p < 2

. 199

2. The absolutely ontinuous partwith respe tto the Lebesgue measure of

DivU

isan 200

L

2

fun tion. 201

3. Thesupportofthemeasure

(dive

s

_{U )}

0

mustbegivenbysetofpoints

P

U

with

H

0

_(P

U

) <

202

+∞.

203

A ordingto these onsiderationsitisnaturaltointrodu ethespa e 204

SDM

p

(Ω) := {U ∈ DM

p

(Ω),

f − DivG = 0},

(7)

sothat,asa onsequen e,de omposition(6)yieldsforany

U ∈ SDM

p

_(Ω)

205

DivU = divU + (div

s

U )

0

.

(8)

For our purposes the following result on erning the features of elements of the spa e 206

SDM

p

(Ω)

willplaya ru ialrole. 207

Proposition 3.1. Let

u ∈ W

1,p,2

0

(div; Ω \ P )

,with

1 < p < 2

. Let

P ⊂ Ω

be asetof nite 208

numberof points. Then

∇u ∈ SDM

p

_(Ω),

with

(div

s

_∇u)

0

= P

. 209

Proof. Weset

P = {x

1

, ..., x

n

}

. Let

ρ(h) → 0

as

h → +∞

besu hthat

B

ρ

h

(x

i

)∩B

ρ

h

(x

j

) =

210

∅

for

h

large enoughand

i 6= j

. We set

Ω

h

=

S

n

i=1

B

ρ

h

(x

i

)

and wedene the following 211 sequen e

{U

h

} ⊂ L

p

_{(Ω; R}

2

₎

. 212

(

U

h

= ∇u

on

Ω \ Ω

h

,

0

on

Ω

h

.

(9) Sin e

∆u ∈ L

2

_{(Ω \ P )}

,bystandardellipti regularitywededu ethat

u ∈ W

2,p

loc

(Ω \ P )

. In 213

parti ulartheexteriortra e

γ

ext

0

(u) ∈ W

3

2

,p

(∂Ω

h

)

. Thereforeweinferthat

u ∈ W

2,p

_(Ω\Ω

h

)

. 214

For every

i = 1, ..n

and

h

small enoughwe an nd an open set

A

i

su h that

B

ρ

h

(x

i

) ⊂

215

A

i

⊂ Ω \

S

j6=i

B

ρ

h

(x

j

)

and

A

i

doesnotdependon

h

. Let

θ

i

bea utofun tionasso iated 216 to

A

i

su hthat 217























θ

i

= 1

on

B

ρ

h

(x

i

)

forany

i = 1, ..., n,

0 ≤ θ

i

≤ 1

forany

i = 1, ..., n,

θ

i

= 0

on

Ω \ A

i

forany

i = 1, ..., n,

k∇θ

h

k

∞

≤

_d(∂A

_i

_,∂B

M

i

ρh

(x

i

))

forany

i = 1, ..., n.

(10)

Then,if

ϕ ∈ C

1

0

(Ω)

with

|ϕ| ≤ 1

,byapplyingGauss-Green'sformulaweobtain:

Z

Ω

U

h

· ∇ϕdx =

Z

Ω\Ω

h

∇u · ∇ϕdx = −

Z

Ω\Ω

h

∆uϕdx +

Z

∂(Ω\Ω

h

)

∇u · νϕdH

1

=

−

Z

Ω\Ω

h

∆uϕdx +

n

X

i=1

Z

∂(Ω\B

_ρh

(x

i

))

∇u · ν(ϕ − θ

i

ϕ(x

i

))dH

1

+

n

X

i=1

ϕ(x

i

)

Z

∂(Ω\B

ρh

(x

i

))

θ

i

∇u · νdH

1

=

−

Z

Ω\Ω

h

∆uϕdx +

n

X

i=1

Z

∂Ω

∇u · ν(ϕ − θ

i

ϕ(x

i

))

+

n

X

i=1

Z

∂B

_ρh

(x

i

)

∇u · ν(ϕ − ϕ(x

i

))dH

1

+

n

X

i=1

n

ϕ(x

i

)

Z

A

i

\B

ρh

(x

i

)

∆uθ

i

dx +

Z

A

i

\B

ρh

(x

i

)

∇u∇θ

i

dx

o

.

(11)

whereinthelastequalitywehaveappliedagaintheGauss-Green'sformulaandthedenition 218

of

θ

i

. 219

Now for every

i

we have that

{∂B

ρ

h

(x

i

)}

onverges in the Hausdor metri to the singleton

{x

i

}

. Then, sin e the support of the fun tion

ψ = ϕ − ϕ(x

i

)

is ontained in

Ω \ {x

i

}

,wehavethat

suppψ ∩ ∂{B

h

(x

i

)} = ∅

for

h

largeenough,bystandardpropertiesof theHausdor onvergen e. Thereforethethird termin (11)is equalto

0

. Moreoverfor

h

largeenoughwe anndaproperopenregularset

A

,thatdoesnotdependon

h

,su hthat

u ∈ W

2,p

_{(Ω \ A)}

. Hen eweinfer

∂u

∂ν

∈ W

1

2

,p

(∂Ω)

. Therefore,from(11)itfollowsthat

|DivU

h

|(Ω) ≤

sup

0≤ϕ≤1

Z

Ω

|∇u · ∇ϕ|dx ≤ (n + 1)C

1

(Ω)k∆uk

L

2

_{(Ω\P )}

+ 2nk

∂u

∂ν

k

W

1

2

,p

(∂Ω)

+

k∇uk

L

p

_(Ω;R

2

₎

n

X

i=1

M

i

d(∂A

i

, ∂B

ρ

h

(x

i

))

:= C(n, Ω),

for

h

largeenough. Sin e

U

h

⇀ ∇u

in

L

p

_{(Ω; R}

2

₎

, byTheorem2.1

|Div∇u|(Ω) ≤ lim inf

h→∞

|Div∇u

h

| ≤ C.

Therefore

∇u ∈ DM

p

_(Ω)

. Finally we know that

u ∈ W

1,p,2

_{(div; Ω \ P )}

and thus the 220

support ofthe measure

div

s

_∇u

isgiven bytheset

P

. Sin e

Cap

p

(P, Ω) = 0

, a ordingto 221

de omposition(6)themeasure

f − DivG

vanishesonsetswith

0 p

- apa ity,andwededu e 222

f − DivG = 0

,thatis

∇u ∈ SDM

p

_(Ω)

,with

(div

s

_∇u)

0

= P.



223

3.2 The Fun tional 224

A ordingtoourpurposethenaturalenergytodealwithisthefollowing

F : SDM

p

_{(Ω) →}

[0, ∞]

,

1 < p < 2

,givenby

F (U ) =

Z

Ω

|divU |

2

_{dx + λ}

Z

Ω

|U − U

0

|

p

dx + µH

0

(supp(div

s

U )

0

).

Fromnowonweassumewithoutloosinggeneralitythattheweights

λ

and

µ

areequalto

1

. 225

Wenotethat,if

DivU

0

6= 0

inthesenseofdistributions,then

inf F (U ) > 0

on

SDM

p

_(Ω)

. 226

Indeed if we had

inf

SDM

p

_(Ω)

F (U ) = 0

then, it would be possible exhibiting a minimizing 227

sequen e

{U

n

}

,su hthat

F (U

n

) → 0

. Thiswouldimply

U

n

→ U

0

in

L

p

and

DivU

n

→ 0

in 228

D

′

_(Ω)

. Ontheotherhand,the

L

p

-distan ebetween

U

n

and

U

0

anbearbitrarysmallonly 229

if

DivU

0

= 0

aswell,be ausethe onstraint

DivU = 0

isstable under

L

p

- onvergen e. 230

4

Γ

- onvergen e: The intermediate approximation 231

Byanalogywith the onstru tion of

U

0

we restri tourselvesto ve torelds

U

whi h are 232

thegradientofafun tion

u ∈ W

1,p

0

(Ω)

. 233

Thus thefun tional

F

is niteon the lass of fun tions whose support ofthe measure 234

(div

s

∇u)

0

is givenbyaniteset. Consequentlyitis onvenienttointrodu ethefollowing 235 spa es: 236

∆M

p

(Ω) := {u ∈ W

0

1,p

(Ω), ∇u ∈ SDM

p

(Ω)},

(12) and 237

∆AM

p,2

(Ω) = {u ∈ ∆M

p

_{(Ω) : ∆u ∈ L}

2

_{(Ω), supp(div}

s

_∇u)

0

= P

∇u

with

H

0

_(P

∇u

) < +∞}.

(13) Sothatthetarget-limitenergy

F : ∆AM

p,2

_{(Ω) → (0, ∞)}

isgivenby 238

F(u) =

Z

Ω

|∆u|

2

dx +

Z

Ω

|∇u − U

0

|

p

dx + H

0

(P

∇u

).

(14)

Inthespirit[9℄weintrodu eanintermediatevariationalapproximationofthefun tional

F

. Wedene asequen e offun tionalswhere the ountingmeasure

H

0

_(P

∇u

)

isrepla edby a fun tionaldened onregularsetsD andwhi h involvesthe urvatureoftheboundary

∂D

. Theapproximatingsequen eisgivenby:

F

ε

(u, D) =

Z

Ω

(1 − χ

D

)|∆u|

2

dx +

Z

Ω

|∇u − U

0

|

p

dx

+

1

4π

Z

∂D

1

ε

+ εκ

2

_dH

1

_.

Where

u ∈ W

1,p,2

0

(div; Ω)

,

D

isaregularset, and

κ

denotesthe urvatureofitsboundary. 239

Inordertoguaranteethatthemeasureofthesets

D

issmallwedeneanewfun tional 240

stilldenotedby

F

ε

(u, D)

givenby 241

F

ε

(u, D) =

Z

Ω

(1−χ

D

)|∆u|

2

dx+

Z

Ω

|∇u−U

0

|

p

dx+

1

4π

Z

∂D

1

ε

+εκ

2

_dH

1

₊

1

ε

L

2

_(D)

on

Y (Ω),

(15) where

Y (Ω) = {(u, D) u ∈ W

1,p,2

0

(div; Ω), D ∈ R(Ω)}

. Weendowthe set

Y (Ω)

with the 242

following onvergen e. 243

Denition 4.1. Let

h ∈ N

go to

+∞

. We say that a sequen e

{(u

h

, D

h

)}

h

⊂ Y (Ω)

H-244

onvergesto

u ∈ ∆AM

p,2

_(Ω)

ifthe following onditionshold 245

1.

L

2

_(D

h

) → 0

; 246

2.

{∂D

h

}

h

→ P ⊂ Ω

inthe Hausdormetri ,where

P

isaniteset ofpoints; 247

3.

u

h

→ u

in

L

p

_(Ω)

and

P

∇u

⊆ P

. 248

Asin [9℄weadoptethefollowingadho denitionof

Γ

- onvergen e. 249

Denition4.2. Let

h ∈ N

goto

+∞

. Wesaythat

F

ε

Γ

- onvergesto

F

ifforeverysequen e 250

ofpositive numbers

{ε

h

} → 0

andfor every

u ∈ ∆AM

p,2

_(Ω)

wehave: 251

1. for everysequen e

{(u

h

, D

h

)}

h

⊂ Y (Ω) H

- onvergingto

u ∈ ∆AM

p,2

(Ω)

lim inf

h→+∞

F

ε

h

(u

h

, D

h

) ≥ F(u);

2. thereexistsasequen e

{(u

h

, D

h

)}

h

⊂ Y (Ω)

H- onvergingto

u

su hthat

lim sup

h→+∞

F

ε

h

(u

h

, D

h

) ≤ F (u).

We point out that with this approa h, the fundamental theorem of the

Γ

- onvergen e 252

annotbeapplieddire tly,sin ewedonotdealwithametri spa e(fora ompletesurvey 253

on

Γ

- onvergen e wereferto [7, 10℄). Howeveritis still possibleto provethat asequen e 254

{(u

h

, D

h

)}

h

asymptoti allyminimizing

F

ε

(u, D)

admits a subsequen e H- onverging to a 255

minimizerof

F(u)

. Indeedwewill showat the end ofthe Se tion (seeTheorem 4.4) that 256

the onvergen eoftheminimumproblems anstillobtainedasa onsequen eof ompa tness 257

oftheminimizingsequen eof

F

ε

,

Γ − lim inf

inequality(

1

)and

Γ − lim sup

inequality(

2

). 258

4.1 Compa tness 259

Westateandprovethefollowing ompa tnessresult. 260

Theorem4.1. Let

h ∈ N

go to

+∞

and

ε

h

→ 0

su hthat 261

F

ε

h

(u

h

, D

h

) ≤ M,

(16)

thenthereexistasubsequen e

{(u

h

k

, D

h

k

)}

k

⊂ Y (Ω)

,afun tion

u ∈ ∆AM

p,2

_(Ω)

andaset 262

P ⊂ Ω

ofnite numberofpoints, su hthat

{(u

h

k

, D

h

k

)}

k

H- onvergesto

u

. 263

Proof. Weadaptan argumentof[9℄. From (16) wehaveimmediately

{D

h

} ⊂ R(Ω)

with

L

2

_(D

h

) → 0

. Then we an parametrize every

C

h

= ∂D

h

by a nite and disjoint union of Jordan urves. Let us set for every

h

,

C

h

=

S

m(h)

i=1

γ

i

. Then we havea ordingto the

2

-dimensional versionofGauss-Bonnet'sTheoremandYoung'sinequality

M ≥

1

4π

Z

∂D

h

(

1

ε

h

+ ε

h

κ

h

2

)dH

1

≥

1

4π

Z

∂D

h

2κ

h

dH

1

=

1

4π

Z

S

h

C

h

2κ

h

dH

1

= m(h).

Notethat thenumber

m(h) ≤ M

,with

M ≥ 0

, isindependentof

h

. Then itispossibleto extra t asubsequen e

C

h

k

withthe numberof urvesin

C

h

k

equalto some

n

for every

k

. Thenweset

C

h

k

= {γ

1

h

k

, ..., γ

n

h

k

}

forany

k

. From (16)we alsohavefor any

γ ∈ C

h

k

that

H

1

_{(γ) ≤ 4πM ε}

h

k

and onsequently

max{H

1

_{(γ) : γ ∈ C}

h

k

} → 0

. Thenthere existsanite setofpoint

P = {x

1

, ..., x

n

} ⊂ Ω

su hthatforanyradius

ρ

thereisanindex

k

ρ

with

γ

i

h

k

⊂ B

ρ

(x

i

)

forall

k > k

ρ

and

i ∈ {1, ..., n},

sothatifweset

∂D

h

k

=

S

n

i=1

γ

h

i

k

⊂

S

n

i=1

B

ρ

(x

i

)

,thentheHausdordistan e

d

H

(∂D

h

k

, P ) →

264

0

sin e

L

2

_(D

h

k

) → 0

andtherefore

ρ → 0

aswell. 265

Nowweprovethe ompa tnesspropertyfor

u

h

. Firstofallfromtheestimate 266

k∇u

h

k

p

_L

p

_(Ω)

≤ 2

p

(k∇u

h

− U

0

k

p

_L

p

_(Ω)

+ kU

0

k

p

_L

p

_(Ω)

),

(17)

and(16),wemayextra tasubsequen e

{u

h

k

} ⊂ W

1,p

0

(Ω)

weakly onvergentto

u ∈ W

1,p

0

(Ω)

. 267

Let

Ω

j

beasequen eofopensets

Ω

j

⊂⊂ Ω\P

invading

Ω\P

. We laimthatitispossible toextra t asequen eof

D

h

k

su h that

Ω

j

∩ ∂D

h

k

= ∅

. Indeedsin ethedistan ebetween

Ω

j

and

P

ispositiveforany

j

thereexists

η

j

su hthat

Ω

j

∩ (

S

n

i

B

η

j

(x

i

_{)) = ∅}

. Ontheother hand weknowthat forevery

ρ

we annd

k

ρ

su h that

∂D

h

k

=

S

n

i=1

γ

h

i

k

⊂

S

n

i=1

B

ρ

(x

i

)

. Theninparti ularif

ρ = η

j

thereexists

k

j

su h thatforall

k ≥ k

j

Ω

j

∩ ∂D

h

k

= ∅.

Thereforeforany

x ∈ Ω

j

thereexists

δ > 0

su hthateither

B

δ

(x) ⊂ D

h

k

or

B

δ

(x) ⊂ Ω\D

h

k

. 268

Finallybytakinginto a ountthat

L

2

_(D

h

k

) → 0

we on lude

Ω

j

∩ ∂D

h

k

= ∅

for

k ≥ k

j

. 269

Thenforevery

k ≥ k

j

wehavethat

u

h

k

∈ W

1,p,2

_{(div; Ω}

j

)

andby(16)weget 270

Z

Ω

j

|∆u

h

|

2

dx ≤

Z

Ω\D

hk

|∆u

h

k

|

2

_{dx ≤ M.}

(18)

Thenwe anextra tafurthersubsequen estilldenotedby

{u

h

k

} ⊂ W

1,p,2

_{(div; Ω}

j

)

su h that











u

h

k

→ u

in

L

p

_(Ω

j

; R

2

)

anda.e.

∇u

h

k

⇀ ∇u

in

L

p

_(Ω

j

; R

2

)

∆u

h

k

⇀ ∆u

in

L

2

_(Ω

j

).

Letnow

x ∈ Ω

′

⊂⊂ Ω \ P

. Thenthere existsasequen e

x

j

→ x

with

j ∈ N

. Byapplying 271

thediagonalargumenttothesequen e

u

h

kl

(x

j

)

weobtainasubsequen e

u

l

= u

h

kl

(x

l

)

su h 272

that

∆u

l

onvergesweaklyin

L

2

_(Ω

′

₎

to

∆u

forany

Ω

′

_{⊂⊂ Ω}

. Thenbythesemi ontinuity 273 ofthe

L

2

-norm wehave 274

sup

j

Z

Ω

j

|∆u|

2

_{dx ≤ sup}

j

lim inf

l→+∞

Z

Ω

j

|∆u

l

|

2

dx ≤ M.

Ifweset

P = P \∂Ω

˜

,thenwededu e

u ∈ W

1,p,2

0

(div; Ω\ ˜

P )

andtherefore

∇u ∈ SDM

p

_(Ω)

275

with

P

∇u

⊆ P

,byProposition3.1. Sowe on ludethat

u ∈ ∆AM

p,2

_(Ω).



276

4.2 Lower bound 277

Weprovidethelowerbound (

1

)inDenition

4.2

. 278

Theorem4.2. Let

h ∈ N

goto

+∞

. Let

{ε

h

}

h

beasequen eofpositivenumbers onverging tozero. Foreverysequen e

{(u

h

, D

h

)}

h

⊂ Y (Ω)

,H- onvergingto

u ∈ ∆AM

p,2

(Ω)

,wehave

lim inf

h→∞

F

ε

h

(u

h

, D

h

) ≥ F(u).

Proof. Uptoasubsequen ewemayassumethatthe

lim inf

isaa tuallyalimit. Asinthe proofofTheorem4.1,bysettingforevery

h

,

C

h

=

S

m(h)

i=1

γ

i

,weget

M ≥

1

4π

Z

∂D

h

(

1

ε

h

+ ε

h

k

2

)dH

1

= m(h).

Uptosubsequen eswehave

m(h) = n

forsomenaturalnumber

n

. Hen ethereexists aset 279

P

1

of

n

pointssu hthat

∂D

h

onvergesin theHausdor metri to

P

1

. Ontheother hand 280

wehavethat

∂D

h

onvergesintheHausdor metri to

P

with

P

∇u

⊆ P

. Then,sin ethe 281

limitis unique,wehave

P = P

1

. 282

Let now

{Ω

j

}

j

be a sequen e of open sets

Ω

j

⊂⊂ Ω \ P

1

invading

Ω \ P

1

. As in the proof of Theorem 4.1 we may assume up to a subsequen e, that

∆u

h

⇀ ∆u

in

L

2

_(Ω

j

)

. Furthermore,sin einthis aseallthesequen e

D

h

onvergestotheset

P

1

wehave,bythe sameargument usedin theproof ofTheorem 4.1,

Ω

j

⊂ Ω \ D

h

forh large andfor any

j

. Consequently

lim inf

h→+∞

Z

Ω\D

h

|∆u

h

|

2

dx ≥ lim inf

h→+∞

Z

Ω

j

|∆u

h

|

2

dx ≥

Z

Ω

j

|∆u|

2

_dx.

Ontheotherhand,arguingasinTheorem4.1,weinferthatthelimit

u

ofthesubsequen e 283

u

h

belongs to

∆AM

p,2

(Ω)

, with

∆u ∈ L

2

_{(Ω \ P}

1

)

and

P

∇u

⊆ P

1

. Sothat bymonotone 284 onvergen e 285

lim inf

h→+∞

Z

Ω\D

h

|∆u

h

|

2

dx ≥

Z

Ω\P

1

|∆u|

2

dx =

Z

Ω

|∆u|

2

dx.

(19)

AsintheproofofTheorem4.1,inequality(17)holds. Thenweeasilyget 286

lim

h→∞

Z

Ω

|∇u

h

− U

0

|

p

dx ≥

Z

Ω

|∇u − U

0

|

p

dx.

(20) Finallywehave 287

1

4π

Z

∂D

h

(

1

ε

h

+ ε

h

k

2

)dH

1

≥ n = H

0

(P

1

) ≥ H

0

(P

∇u

).

(21)

Eventuallyby (19),(20) (21) and bythesuperlinearitypropertyof the

lim inf

operator we 288

a hievetheresult.



289

4.3 Upper bound 290

In[9℄forthe onstru tionoftheoptimalsequen eitis ru ialtousearesultdueto Cham-291

bolle and Doveri (see [11℄). This result states that it is possible to approximate, in the 292

H

1

-norm, afun tion

u ∈ W

1,2

_{(Ω \ C)}

(where

C

is a losed set), by means of asequen e 293

of fun tions

u

h

∈ W

1,2

_{(Ω \ C}

h

)

with

C

h

onvergentto

C

in theHausdor metri . In our 294

asethis argumentdoes notapply due topresen e of ase ond orderdierentialoperator. 295

Neverthelesssin eweworkonlywithsetofpointsitispossibletobuildanoptimalsequen e 296

inamoredire tway. 297

Theorem4.3. Let

h ∈ N

go to

+∞

. Let

ε

h

be asequen eof positive onverging to

0

. For 298

every

u ∈ ∆AM

p,2

_(Ω)

thereexistsasequen e

{(u

h

, D

h

)}

h

⊂ Y (Ω)

H- onvergingto

u

su h 299 that 300

lim sup

h→+∞

F

ε

h

(u

h

, D

h

) ≤ F(u).

(22)

Proof. Westartbythe onstru tionofthesequen e

D

h

. Let

n

bethenumberofpoints

x

i

301

in

P

∇u

. Then wetake

D

h

=

S

n

i=1

B

ε

h

(x

i

)

. Sothat

L

2

_(D

h

) → 0

,

1

ε

h

L

2

_(D

h

) → 0

and

∂D

h

302

onvergeswith respe t tothe Hausdor distan eto

P

∇u

. Moreoverfor

h

largeenoughwe 303

mayassume

B

ε

h

(x

i

) ∩ B

ε

h

(x

j

) = ∅

for

i 6= j

. Nowwebuild

u

h

. Let

{ρ

h

} ⊂ R

besu h that 304

ρ

h

≥ 0

and

ρ

h

→ 0

when

h → ∞

. Let

θ

h

∈ C

∞

_(Ω)

withthefollowingproperty: 305























θ

h

= 1

on

B

ρh

2

(x

i

)

forany

i = 1, ..., n

0 ≤ θ

h

≤ 1

on

B

ρ

h

(x

i

) \ B

ρh

₂

(x

i

)

forany

i = 1, ..., n

θ = 0

on

Ω \ B

ρ

h

(x

i

)

forany

i = 1, ..., n

k∇θ

h

k

∞

≤

_ρ

1

_h

.

(23)

Weset

u

h

= (1 − θ

h

)u

. Itisnotdi ultto he kthat

{(u

h

, D

h

)}

h

⊂ Y (Ω)

andH- onverges to

u

. We laimthatthepair

(u

h

, D

h

)

realizestheinequality(22)forasuitable hoi eofthe sequen e

ρ

h

. Bymakingthe omputationwehave

∇u

h

= (1 − θ

h

)∇u − u∇θ

h

.

Then

Z

Ω

|∇u

h

− U

0

|

p

dx =

Z

Ω

|∇u − U

0

− θ

h

∇u − u∇θ

h

|

p

dx,

sothat 306

lim sup

h→+∞

Z

Ω

|∇u

h

−U

0

|

p

dx ≤ lim sup

h→+∞



(

Z

Ω

|∇u−U

0

|

p

dx)

1

p

+(

Z

Ω

|θ

h

∇u|

p

dx)

1

p

+(

Z

Ω

|∇θ

h

u|

p

dx)

1

p



p

.

(24) Sin e

|∇u|

p

_{∈ L}

1

_(Ω)

,wehavebyapplyingthedominated onvergen etheorem

R

Ω

|θ

h

∇u|

p

dx →

307

0

. Let us fo us on theterm

R

Ω

|∇θ

h

u|

p

. Bythe Sobolev embedding we have

u ∈ L

p

∗

(Ω)

308

with

p

∗

₌

2p

2−p

andhen e

|u|

p

_{∈ L}

p

∗

p

(Ω)

, with

p

∗

p

=

2

2−p

. 309

By (23), using Holder's inequality with dual exponents

2

2−p

and

2

p

, and taking into a ountthat

p < 2

Z

Ω

|∇θ

h

u|

p

dx

≤

n

X

i=1

Z

B

ρh

(x

i

)\B

ρh

2

(x

i

)

|∇θ

h

u|

p

dx =

n

X

i=1



Z

B

ρh

(x

i

)

|∇θ

h

u|

p

dx −

Z

B

ρh

2

(x

i

)

|∇θ

h

u|

p

dx



≤

n

X

i=1

(

Z

B

ρh

(x

i

)

|∇θ

h

|

2

dx)

p

2

kuk

L

2

2−p

_(Ω)

≤

n

X

i=1

kuk

_L

p

∗

(Ω)

(

π

2

_ρ

2

h

ρ

p

_h

) → 0.

(25)

From(24)itfollowsthat

lim sup

h→+∞

Z

Ω

|∇u

h

− U

0

|

p

dx ≤

lim

h→+∞



(

Z

Ω

|∇u − U

0

|

p

dx)

1

p

| + (

Z

Ω

|θ

h

∇u|

p

dx)

1

p

+ (

Z

Ω

|∇θ

h

u|

p

dx)

1

p



p

=



(

Z

Ω

|∇u − U

0

|

p

dx)

1

p



p

=

Z

Ω

|∇u − U

0

|

p

dx.

(26)

Nowwe ompute

∆u

h

. Theidentity

div(f A) = f divA + ∇f · A

yields

∆u

h

= (1 − θ

h

)∆u − 2∇θ

h

∇u − ∆θ

h

u.

Thenby hoosing

ρ

h

smallenoughwehavefrom(23) 310

lim sup

h→+∞

Z

Ω\D

h

|∆u

h

|

2

dx ≤ lim

h→+∞

Z

Ω\D

h

|∆u|

2

dx →

Z

Ω

|∆u|

2

dx.

(27)

Finallysin efor

h

largewehave

B

ε

h

(x

i

) ∩ B

ε

h

(x

j

) = ∅

for

i 6= j

weget 311

lim

h

1

4π

Z

∂D

h

(

1

ε

h

+ ε

h

k

2

)dH

1

= lim

h

n

X

i=1

1

4π

Z

∂B

_εh

(ε

h

1

ε

h

k

2

)dH

1

= n = H

0

(P

∇u

).

(28)

By re allingthat the

lim sup

is sublinear operation and by (26),(27),(28), wea hievethe 312

result.



313

4.4 Variational property 314

We on lude this se tionby properly statingand provingthe parti ular versionof funda-315

mentalTheorem,whi his,inthis ase,adire t onsequen eofTheorems4.1,4.2,4.3. The 316

proof anbea hievedbya lassi alargument(see[7℄, Se tion

1.5

). Howeverweprefer to 317

give the proof in order to make lear that the lassi al variational setting is not dire tly 318

available,andthereforethevariationalpropertyhastobeproven. 319

Theorem4.4. Let

h ∈ N

go to

+∞

. Let

F

ε

and

F

begiven respe tively by(15) and(14). If

{ε

h

}

is a sequen e of positive numbers onverging to zeroand

{(u

h

, D

h

)} ⊂ Y (Ω)

su h that

lim

h→+∞

(F

ε

h

(u

h

, D

h

) − inf

Y

(Ω)

F

ε

h

(u, D)) = 0,

then thereexists asubsequen e

{(u

h

k

, D

h

k

)} ⊂ Y (Ω)

and aminimizer

u

of

F(u)

with

u ∈

320

∆AM

p,2

(Ω)

,su hthat

{(u

h

k

, D

h

k

)}

H- onverges to

u

. 321

Proof. WeknowfromTheorems4.2and4.3that

F

ε

Γ

- onvergesto

F

. Let

u ∈ ∆AM

p,2

_(Ω)

besu hthat

F(u) ≤

inf

∆AM

p,2

_(Ω)

F(u) + δ.

FromTheorem4.3there existsasequen e

{(§ ˜

u

h

, ˜

D

h

)} ⊂ Y (Ω)

,su hthat

inf

∆AM

p,2

_(Ω)

F + δ ≥ F(u) ≥ lim sup

h→+∞

F

ε

h

( ˜

u

h

, ˜

D

h

).

Thensin e

δ

isarbitraryitfollowsthat 322

lim sup

h→+∞

inf

Y

(Ω)

F

ε

h

≤ lim sup

_h→+∞

F

ε

h

( ˜

u

h

, ˜

D

h

) ≤

∆AM

inf

p,2

_(Ω)

F.

(29)

Letnow

{(u

h

, D

h

)} ⊂ Y (Ω)

besu h that

lim

h→+∞

(F

ε

h

(u

h

, D

h

) − inf

Y

(Ω)

F

ε

h

(u, D)) = 0

. ThenfromTheorem4.1,uptosubsequen es,thesequen e

{(u

h

, D

h

)}

h

H- onvergestosome

u ∈ ∆AM

p,2

(Ω)

. ThenbyTheorem4.2 andtakingintoa ount(29)wededu e

inf

∆AM

p,2

_(Ω)

F ≤ F(u) ≤ lim inf

_h→+∞

_Y

inf

_(Ω)

F

ε

h

≤ lim sup

h→+∞

inf

Y

(Ω)

F

ε

h

≤

∆AM

inf

p,2

_(Ω)

F.

Thenweeasilygetthethesis.



323

5 Approximation by smooth fun tion 324

ByfollowingtheBraides-Mar h'sapproa hin[9℄weapproximatethemeasure

H

1

_⌊∂D

bythe 325

Modi a-Mortola'senergydensitygivenby

(ε|∇w|

2

₊

1

ε

W (w))dx

where

W (w) = w

2

_{(1 − w)}

2

326 and

w ∈ C

∞

_(Ω)

. Thenextstepistorepla etheregularset

D

withthelevelset of

w

. Let 327

us set

Z = {∇w(x) = 0}

. BySard's Lemma we havethat

L

1

_{(w(Z)) = 0}

. In parti ular, 328

if

w

takes values into the interval

[0, 1]

, we infer that for almost every

t ∈ (0, 1)

the set 329

Z ∩ w

−1

_(t)

is empty. Consequently foralmost every

t ∈ (0, 1)

the

t

-level set

{w < t}

is a 330

regularsetwithboundary

{w = t}

. Now,sin ewewanttorepla etheset

D

,weneedthat 331

{w < t} ⊂⊂ Ω

. Thenwerequire

1 − w ∈ C

∞

0

(Ω; [0, 1])

. Furthermore foralmostevery

t

,we 332

have

k({w = t}) = div(

∇w

|∇w|

).

Fromallofthisweareled todenethefollowingspa e: 333

S(Ω) = {(u, w); u ∈ W

₀

1,p,2

(div; Ω); 1 − w ∈ C

0

∞

(Ω; [0, 1])}

(30)

and having in mind the oarea formula, the following sequen e of fun tionals dened on

S(Ω)

G

ε

(u, w) =

Z

Ω

w

2

|∆u|

2

dx +

1

8πC

Z

Ω\{∇w=0}

(

1

β

ε

+ β

ε

div(

∇w

|∇w|

)

2

(ε|∇w|

2

+

1

ε

W (w))dx

+

Z

Ω

|∇u − U

0

|

p

dx +

1

γ

ε

Z

Ω

(1 − w)

2

dx,

(31) with

C =

R

1

0

pW (t)dt

. Thelast termfor es

w

ε

be

1

almosteverywhereinthelimit. From 334

nowontheparameters

ε

,

β

ε

,

γ

ε

willberelatedasfollows 335

lim

ε→0

+

β

ε

γ

ε

= 0,

(32) 336

lim

ε→0

+

ε| log(ε)|

β

ε

= 0.

(33)

The onvergen e that playsthe role of the H- onvergen e is the following. With a slight 337

abuseofnotationthis onvergen ewillbestill denotedbyH. 338

Denition5.1. Let

h ∈ N

goto

+∞

and

{(u

h

, w

h

)}

h

beasequen e

S(Ω)

. Set

D

t

h

= {w

h

<

339

t}

. We say that

{(u

h

, w

h

)}

h

H- onverges to

u ∈ ∆AM

p,2

_(Ω)

, if for every

t ∈ (0, 1)

the 340

sequen e

{(u

h

, D

t

h

)}

h

in

Y (Ω)

H- onvergesto

u

. 341

AsinthepreviousSe tion,weadoptetheadho denitionof

Γ

- onvergen ewithrespe t 342

tothe onvergen eabove. 343

Denition 5.2. Let

h ∈ N

go to

+∞

. We say that

G

ε

Γ

- onverges to

F

if, for every 344

sequen e ofpositive numbers

ε

h

→ 0

andfor every

u ∈ ∆AM

p,2

_(Ω)

,we have: 345