An extremal eigenvalue problem arising in heat conduction Journal de Mathématiques Pures et Appliquées

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Journal de Mathématiques Pures et Appliquées

www.elsevier.com/locate/matpur

An extremal eigenvalue problem arising in heat conduction

Grégoire Nadin,Yannick Privat^∗

CNRS,UniversitéPierreetMarieCurie(Univ.Paris 6),UMR7598,LaboratoireJacques-LouisLions, F-75005,Paris,France

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received20March2015 Availableonline3March2016

MSC:

49J15 49K15 34E05

Keywords:

Extremalproblem Calculusofvariation Sturm–Liouvilleeigenvalue Lebesguedensitytheorem Lateralsurfaceconstraint

This article is devoted to the study of two extremalproblems arising naturally in heat conduction processes. We look for optimal configurations of thermal axisymmetric finsandmodelthisproblem astheissue of(i) minimizing (for the worst shape) or (ii) maximizing (for the best shape) the first eigenvalue of a selfadjointoperatorhavingacompactinverse.Weimposeapointwiselowerbound on the radius of thefin,as well as a lateral surface constraint.Using particular perturbationsandunderasmallnessassumptiononthepointwiselowerbound,one shows that theonlysolution isthecylinder inthefirstcase whereasthere isno solution in thesecond case. Wemoreover construct a maximizing sequence and provide theoptimal valueof theeigenvalue inthiscase. As a byproduct of this result,andto proposearemedyto thenon-existenceinthesecondcase,wealso investigatethewell-posednesscharacterofanotheroptimaldesignproblemsetina classenjoyinggoodcompactnessproperties.

© 2016ElsevierMassonSAS.All rights reserved.

r é s u m é

Cetarticleestdédiéàl’étudedesproblèmesextrémauxsurvenantdefaçonnaturelle danslesprocécédés deconductiondelachaleur.Onrecherchedesconfigurations optimalesd’ailettesthermiquesaxisymétriquesetonmodéliseceproblèmecomme celui(i)deminimiser(pourlapireforme),ou(ii)demaximiser(pourlameilleure forme) la première valeur propre d’un opérateur autoadjoint d’inverse compact.

On imposeuneborne inférieure ponctuelle surle rayon del’ailette, ainsiqu’une contrainte sur la surfacelatérale. En utilisant des perturbations particulières et sousuneconditiondepetitessedelacontraintedeborneinférieure,onmontreque l’uniquesolutionestdonnéeparlecylindredanslepremiercas,tandisqu’iln’ya pasdesolutiondansledeuxièmecas.Onconstruitdeplusunesuitemaximisante et on détermine la valeur propre optimale dans ce cas. Afin de proposer un remèdedansle secondcas,onétudieégalementlecaractère bienposéd’unautre

✩ TheresearchleadingtotheseresultshasreceivedfundingfromtheEuropeanResearchCouncilundertheEuropeanUnion’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreementn. 321186 – ReaDi – “Reaction–DiffusionEquations, PropagationandModelling”heldbyHenriBerestycki.ThisworkwasalsopartiallysupportedbytheFrenchNationalResearch Agency(ANR),withintheprojectNONLOCALANR-14-CE25-0013.

✩✩ ThesecondauthorwaspartiallysupportedbytheANRprojectOPTIFORM,ANR-12-BS01-0007.

* Correspondingauthor.

E-mailaddresses:gregoire.nadin@upmc.fr(G. Nadin),yannick.privat@upmc.fr(Y. Privat).

http://dx.doi.org/10.1016/j.matpur.2016.02.005

0021-7824/© 2016ElsevierMassonSAS.All rights reserved.

problèmed’optimisationdeformedansuneclassepossédantdebonnespropriétés decompacité.

© 2016ElsevierMassonSAS.All rights reserved.

1. Introduction

The current workis inspiredand motivated by[1], where theauthors consideredtheproblem of maxi- mizing,with respectto thecrosssectionalarea,therateofheat transferthroughabarof givenmass.For the sakeof clarity,we firststate theextremal problem wewill investigate inSection1.1 andwe will thus provide severalexplanationsonthephysicalframeofourstudyinSection1.2.

1.1. Settingof theextremalproblems

Let us introduce the extremal problems we will dealwith. Let a0 > 0.For the reasons evoked inSec- tion1.2,theadmissibleset willconsistofradiia(·) belongingtoW^1,∞(0,) suchthat

(H₁) Pointwiseconstraint. Thereholdsa(x)≥a₀ foreveryx∈[0,];

(H2) Lateralsurfaceconstraint.Thereholds 0

a(x)

1 +a(x)²dx≤S₀.

Letus thusintroducetheclassofadmissiblefunctionsdefinedby Sa0,,S0 =

a∈W^1,∞(0, ) satisfying (H1) and (H2) , whereS0> a0 isgiven,so thattheclassSa0,,S0 benon-empty.

According toSection1.2, thefunctional weaim atoptimizing isa→λ₁(a),where λ₁(a) stands forthe firsteigenvalueoftheinverseofacompactoperator(thatwewill denotebyLainthesequel),definedby

λ₁(a) = min

ϕ∈H¹(0,) ϕ=0

[a, ϕ] (1)

with

[a, ϕ] = α

0a(x)²ϕ(x)²dx+β 0 a(x)

1 +a(x)²ϕ(x)²dx+σa()²ϕ()²

0a(x)²ϕ(x)²dx+δϕ(0)² , (2) where α,β,δ,σ denotepositivereal numbers.

Note that, according to the Sturm–Liouville eigenfunctions theory, it is standard that λ₁(a) is simple (seee.g.[2,3])andthatitsassociatednormalizedeigenfunctiondenotedϕ_1,asolvestheordinarydifferential system

−α

a(x)²ϕ_1,a(x)

+βa(x)

1 +a(x)²ϕ1,a(x) =λ1(a)a(x)²ϕ1,a(x), x∈(0, ) γa(0)²ϕ_1,a(0) =−λ1(a)ϕ1,a(0)

ϕ_1,a() =−^σ_αϕ_1,a(), (3)

with γ=α/δ.

This quantityλ₁(a) can be viewedas theexponential cooling rate ofdecayof anaxisymmetricbar(or fin)of lengthwith radius a(·).This will beclarified inSection1.2 below.Weare thusled to investigate thetwofollowing extremalproblems:

• Minimization of λ₁(a) (worstshapeofafin).

inf{λ1(a), a∈ Sa0,,S0} . (4)

• Maximization of λ1(a) (bestshapeofafin).

sup{λ₁(a), a∈ Sa0,,S0} , (5)

Remark1.TheissueofoptimizingeigenvaluesofSturm–Liouvilleoperatorsisalongstory.Forasurveyof suchproblems,onerefersforinstance to[4].Moreover,onealsomentions[5,6]wheretheauthorsdealwith a“lateralsurface”constraintsimilartotheoneconsideredinthisarticle.Nevertheless,totheopinionofthe authors,thetechnics, basedon thestandardchangeof variable forSturm–Liouvilleequation y =x

0 dt a(t)²

withthenotationsofthepaper,cannotbeadaptedinasimplewaytosolvetheproblemsinvestigatedinthis article.Indeed,thischangeofvariablewasusedtointroduceanauxiliaryproblemforwhichoneshowedthat theoptimalvaluecoincidedwiththeoneoftheinitialoptimaldesignproblem.Italsoallowed toconstruct minimizing/maximizing sequences. Unfortunately, such technic does not provide a sharp estimate of the optimalvalue,andwehavetouseanotherapproach.

1.2. Motivations inconvection–conduction theory

Stateoftheartaboutshapeoptimizationinconvection–conductiontheory Amongmanyapplicationsofthe optimaldesignproblemwewillinvestigate,letusmentionthestrongimportanceinthecomputerindustry offindingcoolingfinsinmicroprocessorshavinggoodperformances.

Many engineering works focused on modeling the direct problem in order to assess the efficiency of differentfinshapes. Noticethatthese studiesaremainlynumericalandnomathematicalapproachisused todeterminetheoptimalprofilesoffins(seeforinstance[7–10]).

In amore generalcontext, let us mention several studies dealing essentially with numericalaspects of conduction/convectionproblemsinshapeoptimization.ThemodelusedcombinesaFluidMechanicspartial differentialequationwith aparabolic equationinvolvingatransport term(seee.g.[11,12]).

In [1], the authors investigate the problem of maximizing λ1(a) under a volume constraint, namely

0a=V₀, for agivenV₀ >0,and underthesimplifiedassumptions β = 0 and σ= +∞ (inother words, ϕ()= 0).BywritingandanalyzingtheEuler–Lagrangeequation,theyprovideanexplicitcharacterization of themaximizing shape with the helpof asymmetrization argument. Theyfind a(x)=C/cosh²(x−), where C is anormalizing constant. The main difference with the present work comes from thefact that thelateralconvectiveheat transfersarenotneglectedanymorehere,leadingto adifferent behavior ofthe maximizingsequences,as itwillbehighlightedinthesequel.Itisinterestingto noticethat,insomesense, theirstudycanbeinterpretedas alimitcaseoftheprobleminvestigatedinthepresentarticle.

In[13],asimilarproblemisinvestigated,wherethelateralconvectiontermβa(x)

1 +a(x)²isreplaced byagivenfunctionβP(x),independentofa.Theauthorsthenminimizethequantity

0 a(x)dxforagiven decayrate λ1. Theyshow that,when β = 0 (equivalently,P ≡0),theminimizingshape ais thesameas for the“dual problem” studiedin[1]. When β = 0, theyprovide an algorithm enabling to determine the solution.

Fig. 1.Scheme of the axisymmetric fin.

In[14], theauthors dealtwithasimplifiedone-dimensionalstationary modelofaxisymmetricfintaking intoaccountthelateralheattransfersofthefin.Theyanalyzedtheoptimaldesignproblemandinparticular theexistenceissuesaswell asthedeterminationofmaximizingsequences.

In thiswork, wewill consider amoreaccuratemodelof one-dimensionalthermal barinnon-stationary regime,whereconvectivephenomenafromthesideofthefinareconsidered.Accordingtoourmaintheorems (seeTheorems 2 and3inSection3),weshowinthisarticlethatthistermplaysacrucialrolefordetermining theoptimalshape ofthefin.

Modeling oftheproblem Letusconsider anaxisymmetricfinrepresentedbyadomain Ω_aoflength>0 and radiusa(x) atabscissax,asdisplayedinFig. 1, definedinaCartesiancoordinatesystemby

Ω_a={(rcosθ, rsinθ, x)|r∈[0, a(x)), θ∈S¹, x∈(0, )}, (6) wherea∈W^1,∞(0,) issuchthata(x)≥a0 foreveryx∈[0,] witha0 apositiveconstant.

Fig. 1 sums-up the situation and the notations we will use throughout this article. According to the approachandthemodeldescribedin[1,Sections 1,2and6],wemakethetwo followingassumptions:

(i) theconvectivecoefficienth, modelingtheheattransferbetweenthefinsurfaceandthefluidflow,does not depend onthe variable x and θ. This hypothesis allowsto reduce the three-dimensional problem to an axisymmetric one, which justifies thatthe temperature T along the fincan be considered as a functionoft(thetime),randxonly.

(ii) thefincanbeviewedasthermallythinalongther-axis.Asaconsequence,itsradialthermalresistanceis lowenoughincomparisonwiththeconvectiveheattransferhanditisrelevanttoclaimthat∂T /∂r0 almosteverywhereinΩ_a.ThisiswhywewillimposefromnowonthatthetemperatureT isafunction ofthevariablest(thetime)andxonly.

For instance, if the convective heat transfer coefficient h modeling the heat transfer between the fin surface and the fluid flow, and hr the convective coefficient characterizing the heat transfer over the tip are smallenough,thenthefincanbe viewedas thermallythinalong ther-axis.Therefore, theradialBiot numberBirdeterminingwhetherornotthetemperatureainsideabodywillvarysignificantlywithrespect tothevariablerwillbesmallenough(<0.1 inpractice)toconsidertheone-dimensionalconductionmodel as significant.Wereferto[7]formoredetailsonthesemodelingissues.

Theinletofthefin,aswellasthefluidsurroundingthefinareassumedtobeataconstanttemperatures, denotedrespectivelyT_d andT_∞.Consideringprocesseswherethefinaimsatcoolingathermalsystem,i.e.

wheretheheatflowsfromitsbasistowardsthefluid,wewillassumethat0< T_∞< T_d(·) almosteverywhere in(0,).Moreover,wewillconsider abase massM0 attachedattheend pointx= 0.

Let Td ∈ L²(0,). The temperature T is then solution of the following parabolic partial differential equation:

a(x)^2∂T_∂t =_ρc^k _∂x^∂

a(x)^2∂T_∂x

−_ρc^ha(x)

1 +a(x)²(T−T_∞)t >0, x∈(0, ), cM₀^∂T_∂t(0, t) =ka(0)^2∂T_∂x(0, t) t >0,

∂T

∂x(, t) =−^h_k^r(T(, t)−T_∞) t >0,

T(x,0) =T_d(x) x∈(0, ), (7)

wherekdenotesthethermalconductivityofthefin,ρitsdensity,citsspecificheatcapacity.Wewillassume inthe sequelthatthe realnumbers k, M0, ρ,c, hand hr arepositive. Somephysical explanations about thederivationofthetemperaturemodelmaybefoundin[7,15].

Fromnow on,wewillratherusethenotations α= k

ρc, β= h

ρc, γ= k cM0

and σ= h_r

ρc, (8)

forthesakeofreadability.Itcanbeprovedusingstandardsemigroupsargumentsandsincea≥a₀on[0,], thatthesolutionT ofthepartialdifferentialequation (7)belongstoL²(0,T,H¹(0,)).

Asdidtheauthorsof[1],itisconvenienttorepresentthesolutionT intermsofseriesofeigenfunctions.

Forthatpurpose,letusintroducetheoperator

La: C⁰([0, ]) −→ H¹(0, ), f −→ ϕ_a,

(9)

whereϕadenotes theuniquesolutionoftheo.d.e.

α

a(x)²ϕ_a(x)

−βa(x)

1 +a(x)²ϕ_a(x) =a(x)²f(x), x∈(0, ), γa(0)²ϕ_a(0) =f(0),

ϕ_a() =−^σ_αϕa(). (10)

According to Lax–Milgram’s theorem and since a ∈ W^1,∞(0,) and a ≥ a0 on [0,], this system has a unique solutionthat belongsto H¹(0,). Let us introducethe inner-product ·,·a in thespace C⁰([0,]) definedby

f, ga= 0

a(x)²f(x)g(x)dx+α

γf(0)g(0),

forevery(f,g)∈(C⁰([0,]))².ThecompletionofC⁰([0,]) forthetopologyinheritedfromtheinner-product

·,·aisaHilbertspace,andthedefinitionof·,·a extendsclearlytoelements ofthatspace.Wedenoteit byCa.Wealsodefinethenorm · a inducedbytheinnerproduct ·,·a.

Withaslightabuseofnotation,letusstilldenotebyLathisextension.Onehasthusthefollowingresult, whoseproof ispostponedto Appendix A.

Lemma 1.The operatorLais selfadjointandcompactinCa.

As a consequence of Lemma 1, the operator La is diagonalizable in Ca and there exist a sequence of positive real numbers (λn)n∈IN^∗ diverging to +∞ and a sequence (ϕn)n∈IN of elements of Ca such that Laϕn= _λ¹

nϕn foreveryn∈IN^∗, whichrewrites

−α

a(x)²ϕ_n(x)

+βa(x)

1 +a(x)²ϕ_n(x) =λ_n(a)a(x)²ϕ_n(x), x∈(0, ), γa(0)²ϕ_n(0) =−λ_n(a)ϕ_n(0),

ϕ_n() =−^σ_αϕn(). (11)

Moreover, accordingto theso-calledmin–max principlebyCourant–Fisher(seee.g.[2]),thereholds λ_n(a) = max

E⊂H¹(0,) s.t. dimE=n

ϕmin∈E ϕ=0

[a, ϕ],

where [a,ϕ] is definedby(2).

These considerationsallowus todecomposethesolutionT of(7)as T(t, x)−T_∞=

+∞ n=1

Td−T_∞, ϕnae^−λntϕn(x), (12)

Toward anextremal problem Accordingto(12),onehasthefollowingasymptoticforthesolutionof(7) T(t,·)−T_∞ ∼

t→+∞T_d(·)−T_∞, ϕ₁ae^−λ1(a)tϕ₁,

providedthatTd(·)−T_∞ benon-orthogonaltoϕ1fortheinner-product·,·a,whichisnonrestrictiveand will beassumedfrom now.

Sincewearelookingfortheshapeofafinoptimizingitscoolingproperties,itisthennaturaltoconsider:

• the problem offinding the bestshapeof a thermalfin, bymaximizing the firsteigenvalueλ1(a) with respecttothefunctiona,sothatthetemperatureofthematerialtocoolwill becomeclosetothefluid temperatureT_∞ asquickaspossible.

• the problem of finding the worst shape of a thermal fin, by minimizing λ1(a) with respect to the functiona.

Finally,letusbriefly commentonthechoiceof theadmissiblesetof radiia.Wewillimpose:

(i) a regularity assumption, namely a ∈ W^1,∞(0,), to guarantee that the surface element be defined almost everywhere.

(ii) a pointwise lower bound assumption that prevent the fin to collapse: there exists a0 > 0 such that a(x) ≥ a0 for every x ∈ [0,]. Moreover, to consider a class of shapes as large as possible, we will choosea0 suitablysmall(theprecisesense oftheword“small” willbe madeexplicitinthestatement ofthemain theoremsofthispaper).

(iii) a global lateralsurfaceassumption,to modelalimitation onthe manufacturingcost. Moreprecisely, weassumeanupperboundonthelateralsurfaceofΩ_a,thatisgivenby

lateral surface of Ωa= 2π

0

a(x)

1 +a(x)²dx.

Inthenextsection,wesum-up thepreviousconsiderationsandstatetheextremalproblemswewill solve.

2. SolvingofProblem(4) (lookingforthe worstshape)

2.1. Main results

Thissectionisdevotedto theinvestigationof Problem(4).

As highlighted in [14, Lemma 3.1], the class Sa0,,S0 does not share nice compactness properties. In particular, it is not closed nor bounded in W^1,∞(0,) (endowedwith the strong topology),whereas it is boundedinL^∞(0,).Thisdrives ustointroduceanewoptimaldesignprobleminasubclassofW^1,∞(0,) enjoyinggoodtopological properties.

Tothisaim, letM > a₀ andletusdefine thetruncated class Sa^M0,,S0=

a∈W^1,∞(0, ) satisfying (H₁), (H₂) anda

1 +a²≤M a.e. in (0, )

. (13) SinceSa0,,S0 isaboundedsetofL^∞(0,),itiseasytoseethatSa^M0,,S0 isboundedandclosedinW^1,∞(0,).

Inparticular,itinheritsusefulcompactnesspropertiesinaweaksensethatwillbemadepreciseinthesequel.

Inthefollowingtheorem,weinvestigatetheminimizationofa→λ1(a) overtheclassSa^M0,,S0.

Theorem1.Letα,β,δ,a0, andS0 be positivereal numberssuchthat S0> a0and σ≥0.The constant functiona(·)=a0 minimizesthefunctional λ1 over theclassSa^M0,,S0.

Asaconsequence,we inferthefollowingresultfortheoriginalproblem (4).

Corollary1.Foreverypositiverealnumbersα,β,δ,everypositivenumbersa0,andS0suchthatS0> a0, andσ≥0,theconstant functiona(·)=a₀ istheunique solutionof theextremalproblem (4).

This result is quite natural from a physical point of view: in order to cool the material as slowly as possible along a fin with prescribed lateral surface, one needs to use a very long fin, so that the spatial temperaturedecays,and thusthelateralheattransferisverysmooth.

The approachused to provethe results aboverests upon the use of aparticular perturbation that we willintroduceinSection2.2.TheproofsofTheorem 1and Corollary 1arethen gatheredinSection2.3.

Remark2.Theoptimal valueofthe functionλ₁(·) can be explicitlycomputed. Forexample, assumethat a0<

βδ σ

1/3

.Recallthat,formodelingreasons,suchasmallnessassumptionona0isofparticularinterest intheframeworkofourstudy,as underlinedinSection1.2.Hence,accordingto(1) andconsideringϕ≡1 asatestfunction,weclaimthat

a₀λ₁(a₀)−β= min

ϕ∈H¹(0,) ϕ=0

αa³₀

0ϕ(x)²dx+σa³₀ϕ()²−βδϕ(0)² a²₀

0ϕ(x)²dx+δϕ(0)² ≤σa³₀−βδ a²₀+δ <0.

Astraightforwardcomputationleadstothefollowing expressionoftheassociatedeigenfunction ϕ_1,a0(x) =A

cosh(ω₁(a₀)x)− λ₁(a₀)

γa²₀ω1(a0)sinh(ω₁(a₀)x)

withω₁(a₀)²= β−λ1(a0)a0

αa₀ ,whereAdenotesthenormalizationconstantforthenorm·a0.Theboundary conditionatx=yieldsthat,λ=λ₁(a₀) isthefirstpositiverootofthetranscendentalequation

Fig. 2.Construction ofλ1(a0).

λ

γa²₀ω =αωsinh(ω) +σcosh(ω)

αωcosh(ω) +σsinh(ω) with ω²= β−λ₁(a₀)a₀ αa0

.

Theconstructionofλ1(a0) isillustratedonFig. 2.Noticethatanapproximatedvalueoftheeigenvaluecan be easily obtained numerically,by solvingfor instance thetranscendental equation abovewith a Newton method.

2.2. Akey technicallemma

Thissectionisdevotedtothedescriptionofparticularperturbationsthatwewillusetosolveatthesame time theproblems ofminimizingand maximizingλ₁over theclass Sa^M0,,S0.

Let ε>0, S₀≥a₀, a∈ Sa^M0,,S0 suchthata(·)≡a₀, andset b =a√

1 +a². Letus introducethe two familiesofperturbationswewilluseinthesequel.Theirconstructionisbasedonthestraightforwardclaim holding,uptoanull(Lebesgue)measureset:

(0, ) ={b=M} ∪ {a=a0} ∪ {a0< b < M} since {b=a0}={a=a0}.

Perturbation of type I (worst shape) Assumethatthefunctiona isnotidenticallyequalto a₀. It follows thattheset {a₀+c< b≤M}is ofpositivemeasure forsomec>0.We willthen considerthe particular perturbation ofboftheform

b_ε=b−cχ_Vx

0(ε),

wherex0denotesaLebesguepointoftheset{a0+c< b≤M}andVx0(ε)={a0< b≤M}∩(x0−ε,x0+ε), and

0bε(x)dx≤S0.

Perturbation oftypeII(bestshape) AssumethatM islarge,morepreciselythatM > S₀.Therefore,one hasnecessarilyb≡M inthesensethatthemeasureoftheset {b=M}isstrictly lowerthan.

Assume that the set {a0 < b < M} has a positive measure. We could then take c > 0 such that {a0+c≤b≤M−c}hasapositivemeasure andconsidertheparticularperturbationsofb oftheform

bε=b+c

χ_Vx0(ε)−rεχ_Vy0(ε)

,

where c>0,x₀isaLesbeguepointof{a₀≤b≤M−c},y₀=x₀ isaLesbeguepointof{a₀+c< b≤M}, themeasurable setsVx0(ε) andVy0(ε) aredefinedby

Vx0(ε) ={a₀≤b < M−c} ∩(x₀−ε, x₀+ε), Vy0(ε) ={a₀+c < b≤M} ∩(y₀−ε, y₀+ε), with ε is small enough so thatthese two sets do not intersect, and rε := ^|V_|V^x0(ε)|

y0(ε)|. The Lebesgue density theorem yieldsthatlimε→0rε= 1.Obviously,sincea(·)≡a0,there holdsa0≤bε≤M almosteverywhere in(0,).

Using a choice of function b_ε as above, we will now construct a perturbation a_ε of a within the class Sa^M0,,S0.Thisisthecontentofthefollowing lemma:

Lemma2.Letb_ε∈L^∞(0,)beaperturbationofbeither oftypeI,orII.Then,thereexistsafamily(a_ε)_ε>0 suchthat

• aε∈ Sa^M0,,S0 forevery ε>0,

• aε

1 +a_ε²=bε almosteverywherein (0,)andforevery ε>0,

• onehasthereminderestimate:aε−aL^∞(0,)≤Cε²,wheretheconstantC onlydependsonM,c and theconstantsα,β,δ,a₀,,S₀ andσ.

The statementof this lemma is close to [14, Lemma 3.4]. Nevertheless, anotable difference lies in the factthatwehavetodealwithperturbationsofbthatarethesumsofcharacteristicfunctionsofmeasurable sets, insteadofopen sets.

Letusprovideaqualitativeinterpretationofthislemma.Theperturbationa_εcanbeseenasaninfinites- imal perturbation(in L^∞)oftheoriginal element a. Theoscillationscreatedonthegraphof a_εaremade sothatthelateralsurfaceelementbεisanapproximationoftheDiracmeasureatthefirstorder.Themain difficultyconsistsinbuilding thefunctionaε insuchaway thatitisanadmissibleelementofSa^M0,,S0. Proof. The constructions of the function aε satisfying the aforementioned assertions when bε is eitherof type I,or II are quite close. Nevertheless, since the case of aperturbation of typeII requires a little bit moretechnicity,wefocusonitinthisproof.Thecontentoftheproofcanthenbeeasilyadapted(andeven simplified)to deducetheconstructionofa_εforperturbationsoftype I.

Firststep:casewherebissmooth Letusassumethatb∈C^∞([0,]) andletusnowdescribetheconstruc- tionofa_ε.WewillconsiderhereageneralconstantM > a₀+ 2csuchthat{a₀≤b< M−c}hasapositive Lebesguemeasure.

Letusfirstset

aε(x) =a(x) if x /∈ Vx0(ε)∪ Vy0(ε).

Withoutlossofgenerality,wewillfocuswithinthisproofonthecharacterizationoftheperturbationa_ε−a onVx0(ε),thedefinitionofaε−aonVy0(ε) beingsimilar (inabsolutevalue).

Hence,letus defineaε onVx0(ε). Sinceb iscontinuous, thisset isafinite unionofopen intervalsIi, in otherwords

Vx0(ε) ={x∈(x0−ε, x0+ε) s.t.b(x)< M−c}=∪^Ni=1Ii.

ThedifficultyliesincontrollingtheL^∞distance betweenaandaε. Thealgorithmicproceduretodefine aεwrites asfollows:

i oneconsidersaregularsubdivisionofI_i intok intervalsoflengthη,withη <min{c²/2M,ε²}. ii oneverysubinterval(¯x,x¯+η) ofthissubdivision,onecreatesoneoscillationbysetting

a_ε=a_η,2 on (¯x, ξ) and a_ε=a_η,1 on (ξ,x¯+η), wherethefunctionsaη,1andaη,2satisfyinparticularaη,i

1 +a_η,i² =bεfori∈ {1,2},aη,1isdecreasing andaη,2 isincreasing.

Moreprecisely,wedefinethefunctionaη,2 asasolutionof theCauchyproblem a_η,2(x) =

bε(x)²−aη,2(x)²

aη,2(x) x∈(¯x,x¯+η) aη,2(¯x) =a(¯x),

andthefunctionaη,1as asolutionoftheCauchyproblem

a_η,1(x) =−^bε(x)_aη,1^2−a_(x)^η,1(x)2 x∈(¯x,x¯+η), aη,1(¯x+η) =a(¯x+η),

whereξ∈(¯x,x¯+η) ischosensothataη,1(ξ)=aη,2(ξ).

Itremainstoverifythatsuchaconstructionispossible.First,|(a²_η,2)|= 2a_η,2|a_η,2|≤2M sinceb_ε≤M, whichyields

|a²_η,2(x)−a²(x)| ≤2M|x−x| ≤2M η

for all x∈ [x,x+η) such that aη,2(x) is defined. Moreover, as bε = b+c on (¯x,x¯+η), this function is smoothand onehas

b²_ε(x)−a²(x) =

c+a(x)

1 +a(x)²2

−a²(x)≥c².

Hence, b²_ε(x)−a²_η,2(x)≥c²−2M η > 0 due to ourchoice forη. Applying the Cauchy–Lipschitz theorem yields that aη,2 is uniquelydefined at least on[x,x+η). The samearguments yield thataη,1 is uniquely definedon(x−η,x].

We now need to check that the graphsof aη,1 and aη,2 intersect at apoint whose abscissa belongs to [¯x,x¯+η],whichcomesto

a_η,2(¯x+η)≥a(¯x+η) and a_η,1(¯x)≥a(¯x).

accordingtotheintermediatevaluetheorem.

Letus showthataη,i≥aeverywherein(¯x,x¯+η] fori∈ {1,2}. Becauseofthesymmetricaldefinitions of aη,1 and aη,2, it suffices to prove this factfor i = 2. Assume by contradiction and by continuity of a and a_η,2 theexistenceofx_η ∈(¯x,x¯+η] andδ_η >0 suchthata(x)> a_η,2(x) on(x_η,x_η+δ_η). Then,since b_ε=b+c onthis interval,weclaimthat

a_η,2(x) =

b_ε(x)² aη,2(x)² −1>

b(x)²

a(x)² −1 =a(x), foralmost everyx∈(xη,xη+δη).Integratingbothsidesofthisinequalityleadsto

xη+δx

xη

a_η,2(x)dx >

xη+δx

xη

a(x)dx,

Fig. 3.Left: Zoom on one oscillation. Right: the perturbationaε.

foreveryδx∈(0,δη),whichrewritesaη,2(xη+δx)> a(xη+δx) sinceaη,2(xη)=a(xη).Thisisincontradiction withtheassumptionaboveandprovesthatforeveryx∈(¯x,x+¯ η],thereholdsa_η,2(x)≥a(x).Thisjustifies thedefinitionabove.Theconstructionofa_εisillustratedonFig. 3.

Moreover,thelateralsurfaceconstraintremainssatisfiedbyaεsince

0

bε(x)dx= 0

b(x)dx+c(|Vx0(ε)| −rε|Vy0(ε)|)≤S0,

andobviously bε≤b≤M on(0,).Thus,there holdsaε∈ Sa^M0,,S0.

Atthisstep,thefunctiona_εdefinedaspreviouslysatisfiesthetwofirstassertionsofthelemma.Itremains nowto estimatetheL^∞-norm ofa_ε−a. Since

1

2(aε(x)²−aε(¯x)²) = x

¯ x

aε(s)a_ε(s)ds≤ x

¯ x

bε(x)dx

and 1

2(a(x)²−a(¯x)²) = x

¯ x

a(s)a(s)ds≤ aL^∞(0,)aL^∞(0,)

foreveryx∈(¯x,x¯+η),there holds

a(x)²≤aε(x)²≤2

¯ x+η

¯ x

bε(x)dx+a(¯x)²

≤2η(b+cL^∞(0,)+ 2aL^∞(0,)aL^∞(0,)) +a(x)²,

foreveryx∈[¯x,x¯+η]. Moreover,sincea∈W^1,∞(0,) andaccordingto(H₁),onehas 0≤a_ε(x)−a(x)≤ (b+cL^∞(0,)+aL^∞(0,)aL^∞(0,))

a₀ ε²

for everyx∈[¯x,x¯+η].As aL^∞(0,)≤M/a₀ and, accordingto Lemma 2in[14], aL^∞(0,) is bounded by aconstantwhichonlydepends onS₀ and ,it followsthata_ε−aL^∞(¯x,¯x+η)≤Cε², foraconstantC onlydepending onM,candtheconstants ofourproblem.

Theexpectedconclusionthen followsinthecasewhereb issmooth.

The general case Byusingstandarddensitytheorems, thereexistsasequence(a_n)_n∈IN inC^∞([0,]) con- vergingto aintheSobolevspace W^1,1(0,). Letusintroduceb_n =a_n

1 +a_n².Weclaimthat,using the previous assumptionsonaand b, itisnotrestrictive toassume (forexamplebyconsideringconvolutions) thatan ≥a0in[0,] andthat|a_n|≤M/a0.Hence,(an)_n∈INand(bn)nareboundedinL^∞(0,) respectively bytwopositiveconstantsC >˜ 0 accordingto[14,Lemma3.1]andM˜.Therefore,stilldenotingwithaslight abuseofnotationby(an)n∈INanyextractedsubsequenceof(an)n∈IN,theArzelà–Ascolitheoremyieldsthat (an)n∈INconvergesto ainL^∞(0,).

First,(bn)n∈INconverges tobinL¹(0,).Indeed,thereholds

0

|bn(x)−b(x)|dx= 0

an(x)

1 +a_n²(x)−a(x)

1 +a(x)²dx

≤ 0

(a_n(x)²−a(x)²)(1 +a_n(x)²) bn(x) +b(x)

dx

+

0

a(x)²(a_n(x)²−a(x)²) bn(x) +b(x)

dx

≤ (1 +M/a₀)(C+aL^∞(0,))

2a0 a_n−aL^∞(0,)

+a²_L∞(0,)(M/a0+aL^∞(0,))

2a0 a_n−aL¹(0,),

sinceb_n≥a₀andb≥a₀a.e.in(0,),andtheright-handsideconvergesto0 asn→+∞.Replacing(b_n)_n∈IN byawell-chosenextractedsubsequence,wecanthusassumethat(b_n)_n∈INconvergestobalmosteverywhere in (0,) and, thus, only consider Lebesgue points x0,y0 such thatbn(x0) →b(x0) and bn(y0) →b(y0) as n→+∞. Hence,themeasure ofthesets

Vxⁿ0(ε) ={x∈(x₀−ε, x₀+ε), s.t.b_n(x)< M −c} and Vyⁿ0(ε) ={x∈(y₀−ε, y₀+ε), s.t.b_n(x)> a₀+c} are positivewhenevernislargeenough.

Let us now apply theconstruction of thefirst step to theelements a_n. It follows thatfor everyε>0, there exists(a_n,ε)_n∈INsuchthat

bn,ε:=an,ε

1 +a_n,ε² =bn+c

χ_Vxⁿ

0(ε)−|Vxⁿ0(ε)|

|Vyⁿ0(ε)|χ_Vyⁿ

0(ε)

(14)

and

0≤an,ε(x)−an(x)≤ b_n+cL^∞(0,)+a_nL^∞(0,)a_nL^∞(0,)

a0/2 ε²≤ M˜ +c+MC/a˜ ₀

a0/2 ε² (15) for everyx ∈[0,] and n ∈ IN.In particular, for a given n∈ IN,the family (a_n,ε)_ε>0 converges to a_n as ε0.

Moreover,byconstruction,(an,ε)ε>0 isauniformlyLipschitzfunctionsfamilywithrespectto n(andε) andaccordingtotheArzelà–Ascolitheorem,itconvergesuptoasubsequencetoafunctionaε∈W^1,∞(0,).

Lettingntendto+∞in(15)yieldsthat

0≤aε(x)−a(x)≤M˜ +c+MC/a˜ 0

a₀/2 ε² foreveryx∈[0,].Inparticular, itfollowsthataε≥a≥a0in[0,].

Next,since(bn)n∈IN convergestob inL¹(0,),onegets

n→+∞lim bn,ε=b+c

χ_Vx0(ε)−rεχ_Vy0(ε)

=bε in L¹(0, ), bypassingtothelimitin(14)

Moreover,usingthesamedecompositionasabove,weclaimthat

(a_n,ε)²−(a_ε)²= (b_n−b)(b_n,ε+b_ε)−(a²_n,ε−a²_ε)(1 +a_n,ε² ) a²_ε

for almost every x ∈ (0,)∩ {a_n,ε = a_ε}. Using the C⁰-boundedness of the families (a_n,ε)n∈IN,ε>0, (bn,ε)n∈IN,ε>0, thestrong C⁰ convergence of(an,ε)_n∈IN to aε andthe L¹-convergenceof (bn)_n∈IN to b, one gets that(a_n,ε)² converges to (a_ε)² inL¹(0,). Therefore, using the samereasonings as above,it follows:

that(bn,ε)n∈INconvergesto bε:=aε

1 +a_ε²inL¹(0,).

Theproofofthelemmaisthen complete. 2 2.3. Proofs ofTheorem 1 andCorollary 1

Proof of Theorem1 Wefirstinvestigatethe existenceofaminimizer withintheclassSa^M0,,S0.Inviewof that,wewillneedthefollowingstraightforwardlemma.

Lemma3.Let(u_n)_n∈IN be asequenceof L²(0,)converging tosomefunctionuweaklyin L²(0,).Assume moreover that(u_n)_n∈IN isboundedinL^∞(0,).Then,

1 +u²_n belongstoL²(0,)forevery n∈IN andthe sequence

1 +u²_n

n∈INconvergesweaklyinL²(0,)toafunctionU satisfying √

1 +u²≤U a.e.in(0,).

Proof. Even if this result is straightforward, we nevertheless provide elements of proof for the sake of completeness. Since

1 +u²_n ≤1+|un| a.e.in (0,), thefirst claim follows. By assumption,there exists u >0 suchthat|un|≤ua.e.in(0,),foreveryn∈IN.Noticemoreoverthatthefunctionalv→√

1 +v²is convex andcontinuous forthe strongL²-topologyontheset Uu={v ∈L²(0,)| v∞ ≤u}.Indeed, the convexityisobvious andthecontinuity isobtainedbyconsideringasequenceofUu denoted(vn)n∈IN that convergesstronglyinL² toafunctionv,andbywriting

0

1 +v_n²−

1 +v²2

dx= 0

(vn−v)²(vn+v)² 1 +v_n²+√

1 +v²

2dx≤u²v_n−v²L².