Sharp estimates for the anisotropic torsional rigidity and the principal frequency

Contents lists available atScienceDirect

Journal

of

Mathematical

Analysis

and

Applications

www.elsevier.com/locate/jmaa

Sharp

estimates

for

the

anisotropic

torsional

rigidity

and the principal

frequency

Giuseppe Buttazzoa,∗, Serena GuarinoLo Biancob, Michele Marinic

a

DipartimentodiMatematica,UniversitàdiPisa,LargoB.Pontecorvo5,56126Pisa,Italy

b

DipartimentodiMatematicaeApplicazioni“RenatoCaccioppoli”,UniversitàdegliStudidiNapoli FedericoII,ViaCintia,MonteS.Angelo,80126Napoli,Italy

c_{ScuolaInternazionaleSuperiorediStudiAvanzati,viaBonomea265,34136Trieste,Italy}

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

Articlehistory:

Received25December2016 Availableonline23March2017 Submittedby G. Moscariello Keywords: Torsionalrigidity Shapeoptimization Principaleigenvalue Convexdomains

Inthispaperwegeneralizesomeclassicalestimatesinvolvingthetorsionalrigidity andtheprincipalfrequencyofaconvexdomaintoaclassoffunctionalsrelatedto someanisotropicnonlinear operators.

©2017ElsevierInc.Allrightsreserved.

1. Introduction

Let hK be the norm associated to aconvex body K (see Section 3for moredetails); given a domain

Ω⊂ RN _{withfinitemeasure,wedefinetheK-principalfrequency,λ}K

1 ,andtheK-torsionalrigidity,TK,as

λK₁ (Ω) = min u∈W1,2 0 (Ω)\{0}  Ωh_2K(∇u) dx Ωu2dx , (1.1) and TK(Ω) = max u∈W1,2 0 (Ω)\{0}   Ωu dx 2  Ωh2K(∇u) dx . (1.2) * Correspondingauthor.

E-mailaddresses:buttazzo@dm.unipi.it(G. Buttazzo),serena.guarinolobianco@unina.it(S. GuarinoLoBianco),

mmarini@sissa.it(M. Marini).

http://dx.doi.org/10.1016/j.jmaa.2017.03.055

It is convenient to introducethe functionHK = hK2 /2; when HK is sufficiently smooth,we canwrite the

Euler–Lagrangeequations fortheminimizersoftheproblems(1.1)and(1.2)togetaPDEinterpretationof theabovequantities.Indeed,theK-principalfrequencyisrelated totheeigenvalueproblem

−ΔKu = λK1u in Ω, u = 0 on ∂Ω, (1.3)

while theK-torsionalrigidityistheL1_{norm ofthesolutionu of:}

−ΔKu = 1 in Ω, u = 0 on ∂Ω. (1.4)

Here ΔK denotestheFinsler–Laplace operator givenby

ΔKu = div(DHK(∇u)). (1.5)

IntheEuclideancase,occurringwhenK istheunitaryballB,(andhK(x)=|x|)theoperatorgivenin(1.5)

coincides withtheLaplacianand λ1 andT aretheusualfirstDirichleteigenvalueandtorsionalrigidity.

As in the linear case, the quantities defined in (1.1) and (1.2) are monotone, in opposite sense, with respect tothesetinclusion,i.e. if Ω1⊂ Ω2 then

λK₁(Ω1)≥ λK1 (Ω2) and TK(Ω1)≤ TK(Ω2). (1.6)

Moreover, sinceHK isahomogeneousfunctionofdegree2,thefollowingscalingshold true:

λK₁(tΩ) = t−2λK₁ (Ω) and TK(tΩ) = tN +2TK(Ω), t > 0. (1.7)

Shapeoptimizationproblems involvingλ1 andT ,or evenmoregeneralspectral functionalsof theform

F(Ω)= Φ(λ1(Ω),T (Ω)),are widelystudiedintheliterature(seefor instance[2–6,11,12])and, asitis well

known,itispossibletogetbothlowerandupperboundsfortheprincipalfrequencyandthetorsionalrigidity intermsofquantitiesassociated tothegeometry ofthedomainΩ,suchas,forinstancetheperimeterand the volume (justthinkto theFaber–Krahn inequality and the Saint-Venant theorem, see for instance the recentbook[11]).

Asitshouldnotbeunexpected,ifweimposesomefurtherconstraintsintheclassofadmissibledomains, we can get strongerestimates. The class of convex domains, for instance, hasbeen consideredby several authors: on one hand the a priori assumption of the convexity of the domain naturally arises in many situations;ontheother,theclassofconvexsetshasstrongcompactnesspropertieswhichensuretheexistence of extremaldomainsforagreat numberofgeometricinequalities.

In this paper we are interested in estimates of the principal frequency and the torsional rigidity of a convex domainintermsoftheinradius,RΩ,i.e. theradius ofthebiggestball containedinΩ.

Animmediateconsequenceof(1.6)and(1.7)isthat,fortheEuclidean case

λ1(Ω)≤ λ1(BRΩ) = λ1(B1)R

−2

Ω . (1.8)

A classicalresultbyJ.Hersch(see[13])shows thatforanyconvex domainΩ⊂ R2 itholds

π2

4R

−2

Ω ≤ λ1(Ω), (1.9)

andtheinequalityissharp:ifweallowunboundeddomains,equalitycaseoccurswhenΩ isastrip,otherwise itisreachedonlyasymptotically,byasequenceofrectangleswithsidesa b.Hersch’stechniquehasbeen extended to convex domains of RN _{by M.H. Protter in [16] who proved the validity of (1.9) in every}

Concerning the torsional rigidity, in [15] Polya and Szego proved that, for any domain Ω ⊂ R2_{, the}

followinginequalityholdstrue

T (Ω) |Ω| ≥ 1 8R 2 Ω, (1.10)

where|Ω|, denotestheLebesguemeasure ofΩ.Again,this inequalityissharp,becominganequalityifΩ is aball.

Anupperboundfor thetorsionalrigiditywasobtainedbyE. Makaiin[14],whoprovedthat,forevery convexdomain Ω⊂ R2,itholdstruethat

T (Ω) |Ω| ≤ 1 3R 2 Ω. (1.11)

Inequality(1.11)isalsosharp,andthebestconstantisachieved,again,ifweconsiderasequenceofrectangles withsidesa b.

Theaimofthispaperistoextendtotheanisotropiccaseandtoageneraldimensiontheestimates(1.8),

(1.9),(1.10)and (1.11).Byvirtueof(1.6),and (1.7)itseemsreasonable tofindestimatesoftheform

c(N, K)(RK_Ω)−2≤ λK₁ (Ω)≤ C(N, K)(R_ΩK)−2 and c(N, K)(RK_Ω)2≤ T K_(Ω) |Ω| ≤ C(N, K)(RKΩ)2, where RK

Ω istheanisotropicinradius,i.e. thelargestnumbert suchthatx+ tK ⊆ Ω,forsomex∈ Ω. We

show that, surprisingly, for the best constants in the aboveformulas, there is no dependence on K, this meansthattheboundswhichareinforceintheEuclideancase,holdtrueforeverychoiceofthenonlinear operatorΔK.

Inthefollowingsection,weillustratemorepreciselyalltheresultsthatweprovethroughoutthispaper; herewe limitourselvesto stress that,besides theirown interest, theproofs ofsuch resultsmayprovide a moregeometrical insightfor the specialcaseK = B as well,and mayhelpto shed lighton whatare the mostrelevantassumptions thatwe needto impose onanonlinear operator, inorderto expectthose kind ofestimates.

2. Mainresults

In Section4, we extend to theanisotropic caseinequalities (1.10) and (1.11). More precisely, in Theo-rem 4.3,weshowthat,foranyconvexdomainΩ⊂ RN_{,andany2-homogeneousC}2

+-regular1function,HK,

thefollowingapriori boundsontheK-torsionalrigidityholdstrue 1 N (N + 2)(R K Ω)2≤ TK(Ω) |Ω| ≤ 1 3(R K Ω)2. (2.1)

As for the corresponding linear case, equality can be achieved in the first inequality of (2.1) when Ω coincideswithK up totranslationsanddilations,while(asymptotic)equalityinthesecond inequalitycan

be obtainedbyconsideringasequenceof rectangleswithsides a b,see Proposition 4.4(notice that1/3 is thebest constantineverydimension).

As far as we know, inthe case K = B, even if the proof is significantly easier, there is not a specific reference forestimatesoftheform (2.1)ingeneraldimensionavailableintheliterature.

The proofs of both Theorems 4.2 and 4.3 are based on the choice of a suitable one dimensional test function.Since wedefinedthetorsionalasasupremum,itis easytogetanestimate frombelowusingthe variational formulation, while to get abound from above it is important to use the PDE interpretation explained in(1.4).

InSection5,weextendtheresultbyHerschandProtterbyprovingtheformula

π2

4 (R

K

Ω)−2≤ λK1 (Ω)≤ λ1(B)(RKΩ)−2. (2.2)

As inthe linearcase thefirst inequalityholds trueas an equality ifwe consider astrip (or asequence of “thin”rectangles, ifwedonotallowunboundeddomains).Thecaseofequalityinthesecond inequality occurs when wechooseΩ= x+ rK,forsomex∈ RN and r > 0;as inthelinearcase, thelatterestimate followsbyvirtueofthemonotonicityandthescalinglaws.WeproveinProposition 5.1thatλK

1 (K)= λ1(B).

In Section 6, we remove any regularity assumption on the convex body K. This result easily follows by approximating K with smooth convex bodies Kn and then by passing to the limit the corresponding

inequalitiesforthesequenceKn.

We concludethissection byremarkingthat,inarecentpaper [9]authors giveasharpestimate forthe anisotropic principalfrequencyand torsionalrigidity2 _{intermsofthe anisotropicperimeterand Lebesgue}

measure.We pointoutthatthinrectanglesare limitsets fortheanisotropictorsionalrigidityas well;but curiously, whilesuch asequencemaximizes theratio betweentorsionalrigidityand volumeamong allthe setsofgivenanisotropicinradius,itminimizesthesameratioamongallsetsofgivenanisotropicperimeter. 3. Preliminaries

Inthissection,werecallsomebasicnotionsconcerningconvexbodiesandanisotropicdifferential opera-tors.

Associated withaconvexbodyK⊂ RN,there isthesupportfunction,hK :RN → R definedas

hK(x) = max



x· y : y ∈ K.

Thesupportfunctionisaconvexandpositive1-homogeneousfunction,whenK issymmetricwith respect to theorigin,hK isactuallyanorm, whoseunitball,K∗={x : hK(x)≤ 1},iscalledpolar body ofK.

We denotebydK theanisotropicdistance inducedbyK. Inparticular, givenaset A and apoint x we

denote distK(x, A) = inf  hK∗(a− x) : a ∈ A  . (3.1)

It isusefultodefine theanisotropic inradius intermsofsuchadistanceas

RK_Ω = supdistK(x, ∂Ω) : x∈ Ω



.

We now recall someduality relations betweenK and K∗ that weuse inthe following sections and we referto [18, Chapter 1]foramorecomprehensiveaccount onthis subject.

2

Theyconsideraslightlymoregeneralclassofoperatorssincetheyallow,inournotations,functionsHKwhicharehomogeneous

K∗∗ = K, namely ∗ is an involution in the set of convex bodies; this means thatin all the following relations the role played by K and K∗ can be interchanged. In particular hK∗ is a norm as well, K =



x : hK∗(x)≤ 1



is thecorresponding unitballand (RN,hK∗),as aBanachspace, isthedual spaceof,

(RN_,h K).

H_K∗ = h2

K∗/2 is theLegendre–Fenchel transformation of HK = h2K/2; this entails that, whenboth the

functionsabovearedifferentiable,thegradientmappingsDHK andDHK∗ areonetheinverseoftheother,

namely

DHK(DHK∗(x)) = x for every x∈ RN. (3.2)

A completedescription ofthe differentialtheory forconvex bodiesand supportfunctionscanbe found in[18],herewelimitourselvestorecallthattheobstructiontotheregularityofhK isthepossiblepresence

of “flat” parts in the boundary of K; inparticular, if K is strictly convex, then K∗ (and thence hK) is

differentiableand

DhK(x) = y, (3.3)

where y is the only point in ∂K such that theouter normal unitto theboundary of K at y is x/|x|, in otherwords,DhK is the0-homogeneous extensionoftheinversefunctionof theGaussmap.Conversely,if

K hasdifferentiableboundary,thenhK∗ isdifferentiableandK∗ isstrictlyconvex.Moreover, forx∈ ∂K

DhK∗(x)/|DhK∗(x)| = νK(x), (3.4)

whereνK(x) is theouterunitnormalto ∂K atx.

Astraightforwardconsequenceof(3.3)isthefollowing importantformula:

hK∗(DhK(x)) = 1 for every x= 0. (3.5)

AsimilarcriterionholdstrueforhigherorderdifferentiabilityofHK:indeed,whenK isC+2-regular,i.e.

theboundaryofK canbewrittenasthegraphofatwicedifferentiablefunctionwithpositiveHessian,then both HK and HK∗ areC2-regular,and theirHessianmatrices arepositive.Byvirtueof theconsiderations

abovethedifferentialoperatorΔK,actingonC2functionsas

ΔKu(x) = div(DHK(∇u(x))),

is uniformly elliptic, and classicalexistence and regularity results for the solutionsof problems (1.3)and

(1.4)canbe applied. 4. K-torsionalrigidity

Throughout thissectionwe assumethatK isaC₊2-regular symmetricconvexbody,so thatitssupport function,hK,isasmoothnormonRN.Wearegoingtoshowthatballs(withrespecttothemetricinduced

byhK)areextremalbodiesfor(2.1);inthefollowinglemmaweexplicitlycomputetheK-torsionalrigidity

forthosesets. Asweshallsee,this quantitydoesnotdependonthechoiceofthenorm. Lemma4.1. LetTK beasin (1.2),then

TK_(rK)

|rK| = r2

Proof. We startbyconstructingthesolutionoftheproblem(1.4).From (3.2)we deduce ΔKHK∗(x) = div [DHK(DHK∗(x))] = div x = N.

Then thefunctionu= (r2_{− h}2

K∗)/2N isasolutionof −ΔKu = 1 in rK, u = 0 on ∂(rK). Wehave  rK h2_K∗dx = r  0  {hK∗=t} h2 K∗ |DhK∗| dHN−1dt = r  0 t2  ∂(tK) 1 |DhK∗| dHN−1dt = r  0 tN +1  ∂K 1 |DhK∗| dHN−1dt = r N +2 N + 2  ∂K 1 |DhK∗| dHN−1 ,

where weusedtheco-areaandthechangeofvariableformulas.Letusevaluatethelastintegral.Thanksto

(3.4)and (3.5),and sincethesupportfunctionis1-homogeneous,weget  ∂K 1 |DhK∗| dHN−1=  ∂K hk  _Dh K∗ |DhK∗|  dHN−1 =  ∂K hK(νK(x)) dHN−1 =  ∂K x· νK(x) dHN−1 =  K div x dx = N|K|. Therefore weobtain TK(rK) =  rK r2− hK∗ 2N dx = 1 2N  r2|rK| − r N +2 N + 2N|K|  = r 2_|rK| N (N + 2) as required. 2

In the following proposition we prove a lower bound for the K-torsional rigidity for strictly convex domains. Insuchacase,thefunction

u(x) =1−



hΩ∗(x)

2

2N (4.1)

used in the above lemma to compute the K-torsional rigidity of balls is no longer the solution of the anisotropic torsion problem,butit still vanisheson theboundaryof the domain,and itcanbe used as a test function.Noticethat,sinceTK _{maximizesthequotient}

⎛ ⎝ Ω u dx ⎞ ⎠ 2⎛ ⎝ Ω h2_K(∇u) dx ⎞ ⎠ −1 ,

theneverytestfunctionprovidesalowerboundforthetorsion.Ourchoiceofthetestfunctionu ismotivated bythefactthatitprovides theoptimallower bound,as statedinTheorem 4.2below.Theassumption on thestrictconvexityofΩ isonlytechnicalandisremovedin

Theorem4.2. LetΩ⊂ RN _{be aconvexboundeddomain,then}

TK_(Ω) |Ω| ≥ 1 N (N + 2)(R K Ω)2. (4.2)

Moreover(4.2)issharp, andequalityoccurs onlyif Ω= x+ rK,forsome x∈ RN andr≥ 0.

Proof. We firstproveinequality(4.2)undertheassumption thatΩ isstrictly convex. Upto atranslation ofthedomain,wecanalwaysassumethatRK_ΩK⊆ Ω,sothathΩ≥ RKΩhK.Letu bethefunctionin(4.1);

u isanadmissiblefunctionfortheproblem(1.2),hence

TK(Ω) = max u∈W1,2 0 (Ω)\{0}   Ωu dx 2  Ωh2K(∇u) dx ≥   Ωu dx 2  Ωh2K(∇u) dx .

SinceΩ isastrictlyconvexdomain,thefunctionhΩ∗isdifferentiableand,byrepeatingsimilarcomputations

asthose inLemma 4.1,wehave  Ω h2_Ω∗dx = 1  0  {h2 Ω∗=t} t |Dh2 Ω∗| dHN−1dt = 1  0 t 2  {h2 Ω∗=t} hΩ(DhΩ∗) hΩ∗|DhΩ∗| dHN−1dt = 1  0 tN/2 2  ∂Ω hΩ(νΩ(x)) dHN−1dt = 1 N + 2  ∂Ω x· νΩ(x) dHN−1 = 1 N + 2  Ω div x dx = N|Ω| N + 2. Hence ⎛ ⎝ Ω u dx ⎞ ⎠ 2 = |Ω| 2 N2_{(N + 2)}2 . (4.3)

Wenowcomputethedenominator.Again,theco-areaandthechangeofvariableformulasgive  Ω h2_K(∇u) dx = 1 4N2  Ω h2_K(Dh2_Ω∗) dx

= 1 4N2 1  0  {h2 Ω∗=t} h2_K(Dh2_Ω∗) |Dh2 Ω∗| dHN−1dt (4.4) = 1 4N2 1  0 t(N−1)/2  ∂Ω h2 K(Dh2Ω∗) |Dh2 Ω∗| dHN−1dt.

Byusingthehomogeneityofthesupportfunctionsandbyrecallingequations(3.5)and(3.4),wehave

h2 K(Dh2Ω∗(x)) |Dh2 Ω∗(x)| = |Dh 2 Ω∗(x)| hΩ(DhΩ∗(x))· h2 K(Dh2Ω∗(x)) |Dh2 Ω∗(x)|2 = 2hΩ∗(x) hΩ(νΩ(x))· h 2 K(νΩ(x)) = 2t1/2h2_K(νΩ(x)) hΩ(νΩ(x)) . (4.5)

Plugging(4.5)into(4.4)weget  Ω h2_K(∇u) dx = 1 4N2 1  0 tN/2  ∂Ω 2hΩ(νΩ(x)) h2_K(νΩ(x)) h2 Ω(νΩ(x)) dHN−1(x) dt ≤ 1 4N2 1  0 tN/2  ∂Ω 2hΩ(νΩ(x)) (RK Ω)2 dHN−1(x) dt = 1 N (N + 2) |Ω| (RK Ω)2 , (4.6)

where weusedthefactthathΩ≥ RΩKhK.Combining (4.3)and(4.6)wefind:

TK_(Ω) |Ω| ≥ 1 N (N + 2)(R K Ω)2, as required.

We are now left to show the validity of (4.2) without the assumption on the strict convexity of the domain Ω.

We recallthatstrictly convexbodies areadensesubset oftheset of convexbodieswith respect tothe topologyinduced bytheHausdorffdistance.Inparticular (seefor instance[17])thereexists asequenceof convex bodies Ωn ⊂ Ω such that hΩnS

N−1 _{converges to h}

ΩSN−1 inthe C0-norm. Such a convergence

ensuresthatasn→ ∞

|Ωn| → |Ω| and RKΩn→ R

K

Ω. (4.7)

From (1.6),itfollowsthat

T (Ω)≥ T (Ωn),

and byapplying(4.2)toeachΩn equation(4.2),wefind

TK(Ω)≥ |Ωn|

N (N + 2)(R

K Ωn)

2_,

which, combinedwith(4.7),givesthedesiredresult.

Notice that,byvirtueof Lemma 4.1, inequality(4.2)issharp; moreover,ifit holdstrueas an identity, then (4.6)as wellmustbe anequality,inparticular

 ∂Ω hΩ(νΩ(x)) h2 K(νΩ(x)) h2 Ω(νΩ(x)) dHN−1(x) =  ∂Ω hΩ(νΩ(x)) (RK Ω)2 dHN−1(x).

Since the integrand function inthe left-hand side is pointwise lower than the onein the right-handside, and since they are continuous functions, they must coincide everywhere. Namely hΩ = RKΩhK, that is

Ω= RK ΩK. 2

Inthefollowing theorem weextendMakai’sresult(1.11) toageneraldimensionand to theanisotropic case. The proof given in[14] is mainlybased on a cleveruse of the Schwarz inequality, on an inequality involvingrealfunctionsandtheirderivatives,andontheinequality

|v|2_{≥ v}2

i. (4.8)

Thesecondingredientisappliedtoafunctionwhichisaparametrizationoftheboundaryofthedomain (werecallthatinMakai’ssettingtheboundaryisasetof“dimensionone”),while(4.8)isatrivialinequality fortheEuclideannorm,butisfalseforagenericnorm.Noticethatthegeometricfeatureofthespherethat ensures the validity of (4.8) is the fact that each radius connecting the center to a boundary point is perpendiculartothetangentspaceatthatpoint.

Inourproofwehavetochangestrategy:asinMakais’sproofwedivideΩ intosuitablesubdomains;then weconstruct, foreachsubdomain,afamilyofonedimensionalobstaclefunctions,andfinally weusesome lineartransformationstogetananalogueof(4.8).

Theorem4.3. LetΩ⊂ RN _{be aboundedconvexbody,then}

TK_(Ω) |Ω| ≤ 1 3  RK_Ω 2. (4.9)

Proof. Weprovetheclaim whenΩ is apolytope, andthen thevalidityof(4.9) followsbyapproximation. Letusdenote byF1,. . . ,Fn thefacetsofΩ,anddivideΩ intosubdomainsΩj definedas

Ωj=



x∈ Ω : dK(x, Fj)≤ dK(x, Fl) for every l= j



,

wheredK(x,Fj) isthedistancefunctiondefinedin(3.1).Foreveryj = 1,. . . ,n and ε> 0 wedefine

uε_j(x) =−dK(x, Fj)

2

2 (1 + ε) + R

K

ΩdK(x, Fj)(1 + 2ε).

Noticethat,ifx∈ Ωi∩ Ωj,thenuεi(x)= uεj(x).

Wedenotebyνj theouterunitnormalto Fj.SincedK(x,Fj)= dist(x,Fj)h−1K (νj),thefunctionsuεj are

smoothintheinteriorofΩj;moreover,since∇dist(x,Fj)=−νj,wehave

∇uε j(x) =  dist(x, Fj) h2 K(νj) (1 + ε)− R K Ω hK(νj) (1 + 2ε)  νj. Thus ΔK(uεj) = div  DHK(νj)  dist(x, Fj) h2 K(νj) (1 + ε)− R K Ω hK(νj) (1 + 2ε)  = 1 + ε h2 K(νj) DHK(νj)· (−νj) =−1 − ε.

Letusconsiderthefunctionuε_{: Ω→ R definedbyu}ε_ Ωj= u

ε

jΩj.Wenowprovethat,ifv isthesolutionof

theanisotropictorsionproblem onΩ,thenv(x)≤ uε(x) inΩ.Suppose,bycontradiction,thatthere exists a pointx∈ Ω such thatuε(x)< v(x); sincewe know thatuε= v on ∂Ω, thenthe functionuε− v hasa local minimum,sayx0insideΩ.

We show that x0 cannot be neither an interior point nora boundarypoint of each Ωj. If x0 ∈ int Ωj,

then ∇uε_(x

0)=∇v(x0),so

D2HK(∇uε(x0)) = D2HK(∇v(x0)). (4.10)

Moreover,since∇uε_(x

0)= 0 inΩj,thenalso∇v(x0)= 0.Thusv solves,inaneighborhoodofx0anelliptic

equation withcoefficientsinC0,α_{,and thenceisC}2,α_{-regular(seeforinstance[10, Chapter 6]).Then}

D2(uε− v)(x0)≥ 0. (4.11)

Inparticular,using(4.10)and(4.11),andsincethetraceoftheproductoftwopositive-definitematricesis positive,wehavethat

−ε = ΔKuε(x0)− ΔKv(x0) = trD2HK(∇uε(x0))D2uε(x0)  − trD2HK(∇v(x0))D2v(x0)  = trD2HK(∇uε(x0))  D2uε(x0)− D2v(x0)  = trD2HK(∇uε(x0))D2(uε− v)(x0)  ≥ 0, whichisacontradiction.

Wearelefttoshowthecontradictioninthecasewhenx0∈ Ωi∩Ωj,i= j.Letw(x)= v(x)−v(x0)+uε(x0);

then w(x0) = uε(x0) and, thanks to our assumption on x0, there exists a positive number r, such that

uε(x)≥ w(x),foreveryx∈ (x0+ rK)⊂ Ω.

Since, uε

j isaconcave functionofthedistance,wehavethat

uε_j(x)≤ uε_j(x0) + νj· (x − x0)  dist(x, Fj) h2 K(νj) (1 + ε)− R K Ω hK(νj) (1 + 2ε)  (4.12)

for everyx∈ (x0+ rK).Withoutloss ofgeneralitywecanchooser < RKΩε/(1+ ε);thischoiceofr allows

us toconcludethat

dK(x, Fj)≤ r + dK(x0, Fj),

and thusthefactor

dist(x, Fj) h2 K(νj) (1 + ε)− R K Ω hK(νj) (1 + 2ε),

appearingintheright-handsideof (4.12),nevervanishes,foreveryx∈ (x0+ rK).Indeed

dist(x, Fj) h2 K(νj) (1 + ε)− R K Ω hK(νj) (1 + 2ε)≤r(1 + ε) hK(νj) +dK(x0, Fj)(1 + ε) hK(νj) − RK Ω hK(νj) (1 + 2ε) <dK(x0, Fj)(1 + ε) hK(νj) − RK Ω hK(νj) (1 + ε)≤ 0.

Moreover, since r < RK

Ωε/(1+ ε),for any other l such thatx0 ∈ Ωl, inside x0+ rK, uεl is anincreasing

functionoftheanisotropicdistancedK,then itfollowsfromthedefinitionofthesetsΩl that

uε_l(x)≤ uε_j(x) for every x∈ (x0+ rK)∩ Ωl,

thatentailsthat

uε(x)≤ uε_j(x) for every x∈ (x0+ rK). (4.13)

Plugging(4.13)into(4.12) andrecallingthatuε_(x

0)= uεj(x0),weobtain uε(x)≤ uε(x0) + νj· (x − x0)  dist(x, Fj) h2 K(νj) (1 + ε)− R K Ω hK(νj) (1 + 2ε)  (4.14) foreveryx∈ (x0+ rK).

Arguinganalogouslyforthefunctionuε

i wefind uε(x)≤ uε(x0) + νi· (x − x0)  dist(x, Fi) h2 K(νi) (1 + ε)− R K Ω hK(νi) (1 + 2ε)  (4.15) foreveryx∈ (x0+ rK).Byimposingin(4.14)and(4.15) theconditionsv≤ uε andv = uεatx0,wefind

v(x)− v(x0)≤ νj· (x − x0)  dist(x, Fj) h2 K(νj) (1 + ε)− R K Ω hK(νj) (1 + 2ε)  and v(x)− v(x0)≤ νi· (x − x0)  dist(x, Fi) h2 K(νi) (1 + ε)− R K Ω hK(νi) (1 + 2ε)  .

Since v isdifferentiable at x0, bymultiplying both sides by the factor|x− x0|−1 and taking thelimit as

x→ x0weobtainthat∇u(x0) mustbeproportional bothtoνi andνj, thatisacontradiction.

Wecannow usethefunctionuj as abarrierfunction toget anestimateof thetorsionrigidityofΩ. In

order to compute _Ω

ju

ε

j(x)dx,it isimportant to make surethateach subdomain Ωj can be writtenas a

graphoverthefacetFj.ThisisalwaysthecasewhenK istheEuclideanball,sinceitsradiusisorthogonal

tothetangentspace,whilethisis nottrueforageneralconvexbodyK.

Togetthesamecondition,we consideralineartransformation L (seeFig. 1),suchthatLν_j⊥ = Id and

LDhK(νj)= hK(νj)νj.Noticethat,sinceDhK(νj)· νj = hK(νj)> 0,L iswell definedand

det L = 1. (4.16) MoreoverhK(νj)= hLK(νj),indeed hK(νj) = sup x∈Kx· νj = supy∈LKL −1_{y· ν} j = sup y∈LK  L−1y· νj+ (y· νj)L−1νj· νj , (4.17)

wherey = y+(y·νj)νj,andy·νj= 0.SinceLνj⊥= Id,thenL−1y·νj,andsinceDhK(νj)= hK(νj)L−1νj,

thenL−1νj· νj = 1 sothat,byrecalling(4.17)

hK(νj) = sup

Fig. 1. The linear transformation L.

WeshowthatourchoiceofthelinearapplicationL allowsusto describethesubdomainLΩj as

LΩj=



y + rνj : y∈ LFj, r∈ [0, ρj(y)]



.

ForthisweneedonlytoprovethatLΩjiscontainedinthecylinderCj = LFj× Rνj,sinceLΩj isaconvex

set.Westartnoticingthat,sincedistK(x,Fj)= distLK(Lx,LFj),

LΩj=



x∈ LΩ : dLK(x, LFj)≤ dLK(x, LFl), for every l= j



.

Suppose,bycontradiction,thatthereexistsapointx∈ LΩj\ Cj.Letr bethelargest numbersuchthat

x + rLK ⊆ LΩ.Itfollowsfrom thedefinitionofLΩj,thatthere existsapointy∈ x + rLK ∩ LFj.

Sincex + rLK ⊆ LΩ,thenormalconeofLΩ iscontainedinthenormalconeofx + rLK atthepointy,

and thenνj istheouter normalunitto x + rLK aty.Thereforex = y + rνj ∈ Cj.

Now wecompute  Ωj uε_j(x) dx =  LΩj uε_j(L−1y) dy =  LΩj  −dist2(x, Fj) 2h2 K(νj) (1 + ε) + R K Ωdist(x, Fj) hK(νj) (1 + 2ε)  dx.

We used (4.16) for the first equality, we use thefact that distK(x,Fj) = distLK(Lx,LFj) and hK(νj)=

hLK(νj) forthesecond one.Finally

 Ωj uε_j(x) dx =  LΩj  −dist2(x, Fj) 2h2 K(νj) +R K Ωdist(x, Fj) hK(νj)  dx + O(ε) =  LFj ρj(y) 0  −dist2(y + rνj, Fj) 2h2 K(νj) +R K Ωdist(y + rνj, Fj) hK(νj)  dr dy + O(ε) =  LFj ρj(y) 0  − r2 2h2 K(νj) + R K Ωr hK(νj)  dr dy + O(ε) =  LFi  − ρ3j(y) 6h2 K(νj) +R K Ωρ2j(y) 2hK(νj)  dy + O(ε).

Sinceρj(y)≤ RKΩhK(νj),foreveryy∈ LFj,then − ρ2j 6h2 K(νj) + RK_Ω ρj(y) 2hK(νj) ≤ 1 3  RK_Ω 2. (4.18) Using(4.18) wefind  Ωj uε_j(x) dx≤  LFj ρj(y)  RK Ω 2 3 dy + O(ε) =1 3  RK_Ω 2|LΩj| + O(ε). Since|LΩj|=|Ωj|, weget TK(Ω)≤  Ω uεdx = j  Ωj uε_jdx ≤ 1 3  RK_Ω 2 j |Ωj| + O(ε) = 1 3  RK_Ω 2|Ω| + O(ε) asrequired. 2

Remark.The mostimportant inequality playing arole inthe proof of thetheorem above is (4.18). This inequalitymustbestrictforsomeboundarypoint,unfortunatelythisisnotenoughtoconcludethatequality in(4.9) isnever attained, becauseour proof isbased onan approximationprocedure. Inorder to get the strictsignin(4.9)wemusttotakeintoaccount andestimatestheremindertermsin(4.18).

− ρ 2 j 6h2 K(νj) + RK_Ω ρj(y) 2hK(νj) =1 3  RK_Ω 2−1 6  RK_Ω − ρj hK(νj) 2 + ρjR K Ω 6hK(νj)− (RK Ω)2 6 ≤1 3  R_ΩK 2+ ρjR K Ω 6hK(νj)− (RK_Ω)2 6 .

Arguingasintheproofabovewe canconcludethat  Ωj v≤1 3  R_ΩK 2|Ω| + 1 6  Fj  ρ2 jRKΩ hK(νj)− ρj (RK_Ω)2  dy = 1 3  RK_Ω 2|Ωj| + 1 6  Ωj  2RK_ΩdistK(x, Fj)− (RΩK)2 dx. Summingoverj wefind TK(Ω)≤ 1 3  RK_Ω 2|Ωj| + RK Ω 6  Ω  dK(x, ∂Ω)− RKΩ dx. (4.19)

Taking into account Equation(4.19) in order to show that theminimum in (4.9)is never attained we havejustto showthat,foreveryconvexbodyΩ

2 

Ω

Fig. 2. The set Ωε_{= C}ε_{∪ D}ε_.

LetusdenotebyΩt={x∈ Ω : dK(x,∂Ω)≥ t}.Since|∇dK(x)|= hK(νt(x))−1,whereνt(x) istheouter

normalto ∂Ωt atx,thankstheco-areaformulawecanwrite

 Ω dK(x, ∂Ω) = RKΩ  0  ∂Ωt dk(x)hK(νt(x)) dHN−1(x)dt = RKΩ  0 tPK(Ωt) dt,

where PK denotestheanisotropicperimeter.

Ontheotherhand

RKΩ  0 dK(x, ∂Ω) dx = RKΩ  0 |Ωt| dt.

AsitiswellknowntPK(Ωt)+|Ωt|<|Ωt+ tK|;thisfactcanbeseenasaconsequenceofthecharacterization

of mixedvolumes(see,forinstance[18, Theorem 5.1.7]).SinceΩt+ tK⊆ Ω,wehave

2  Ω dK(x, ∂Ω) < RK Ω  0 |Ωt+ tK| dt ≤ RK Ω  0 |Ω| dt = RK Ω|Ω|.

Nonetheless, thefollowingexampletellsusthat(4.9)issharp. Proposition 4.4. LetΩε _{therectangle [−ε,ε]× [−a}

2,a2]× . . . × [−aN,aN].Then lim ε→0+ TK_(Ωε₎  RK Ωε 2 |Ωε_| = 1 3.

Proof. LetΩε= Cε∪ Dε,whereCε= [−ε,ε]× [−a2+ ε,a2− ε]× . . . × [−aN+ ε,aN− ε],andDε= Ωε\ Cε,

as inFig. 2.Settingx= (x1,z) withz ∈ RN−1 anda= (a2,. . . ,aN),we considerthe functionuε defined

by ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ uε_(x 1, z) = ε2_{− x}2 1 2h2 K(e1) in Cε uε_(x 1, z) = min  |a − z|, | − a − z| ε2− x21 2εh2 K(e1) in Dε_.

WecanestimatetheK-torsionalrigidity

TK(Ωε)≥  Ωεuε 2  Ωεh2_K(∇uε) =  Cεuε+  Dεuε 2  Cεh2_K(∇uε) +  Dεh2_K(∇uε) .

Wenowcompute  Cε uεdx =  Cε ε2_{− x}2 1 2h2 K(e1) dx = |C ε_|ε2 3h2 K(e1) and  Cε h2_K(∇uε) dx =  Cε x2 1 h2 K(e1) dx = |C ε_|ε2 3h2 K(e1) .

Wenoticethatboth_Dεuεdx and



DεHK(∇uε)dx arenegligible,sincetheygoto zeroasεN +2;moreover

|Dε_{| is negligibleas well,sinceitgoestozeroas ε}N_{.Byrecallingthat}

 RKΩε 2 = ε 2 h2 K(e1) , wehavethat lim ε→0 TK_(Ωε₎  RK Ωε 2 |Ωε| ≥ limε→0  Cεuεdx 2  RK Ωε 2 |Ωε| Cεh2_K(∇uε) dx = 1 3 which,togetherwith Theorem 4.3,concludestheproof. 2

5. Eigenvalue

Inthis section we extendto the anisotropiccase inequalities(1.8) and(1.9). As inthe linearcase, the upper bound for the principal frequency relies on the monotonicity and scaling laws. As we mentioned, allthe bestconstantinthose inequalitiesdo notdependon K;inthe followingproposition weshow that

λK

1(K)= j02,andthisgivesthesecondinequalityin(2.2).

AsisSection4,wealwaysassumethatK isaC2

+ regularsymmetricconvexbody.

Proposition5.1. LetΩ⊂ RN be aconvexbody,then

λK₁(Ω)≤ λ1(B)(RΩK)−2.

Proof. From(1.6)and(1.7)wehavethat

λK₁(Ω)≤ λK₁ (K)(RK_Ω)−2.

NowhavetoshowthatλK

1(K)λ1(B).Sincethere existsapositivefunctiong solving

−ΔKg = λK1g in K, g = 0 on ∂K,

wecanassumethatg isradial,namely g(x)= fH∗(x) (see[1]).Adirectcomputationshowsthat

f(t) + N f(t) =−λK₁ f (t) and f (1) = 0. Indeed −λK 1 g = ΔKg = div  DHK(∇g) = divDHK  f(H_K∗)DH_K∗ = divf(H_K∗)x = f(H_K∗)DH_K∗(x)· x + Nf(H_K∗).

Settingg(x)= f (|x|2_{/2) iteasyto checkthat}

Δg =−λK₁ g in B, g = 0 on ∂B.

Since g > 0,theng isthefirstDirichleteigenfunctionoftheballB andthenceλK₁ (K)= λ1(B). 2

Now we provethree technical lemmas thatare usefulinthe proof ofthe lower bound forthe principal frequency. Lemmas 5.2 and 5.3 are generalizations of a lemma by D. Gale and referred to as a private communicationbyM.H.Protterin[16].

Lemma5.2.LetΩ beaconvexdomain,letK⊆ Ω beaconvexbodywithdifferentiableboundary,andsuppose that foreveryτ > 1 andx∈ RN,x+ τ K  Ω.Thenthereexisti∈ N andx1,. . . ,xi ∈ ∂Ω∩ ∂K suchthat,

denoted by νj theouterunit normalto∂K atxj,forevery ν∈ SN−1,there exists,j,such thatν· νj≥ 0.

Proof. Let ν ∈ SN−1; for ε > 0, we consider the set Kε _{= K + εν. Notice that K}ε _{cannot be a proper}

subset of Ω,since, otherwise,also asmalldilation ofKε _{shouldbe,against ourassumptions. Then, there}

exists με_{∈ S}N−1 _suchthat

hΩ(με)≤ hKε(με) = h_K(με) + εμε· ν.

Notice thatμε isthe outerunitnormal to∂Kε at somepointxε∈ Kε\ Ω.Since K ⊆ Ω,thenhK ≤ hΩ,

and thenμε· ν ≥ 0.

Uptoextractasubsequencewecanalwaysassumethat,asε goesto0,με_{convergestosomeunitvector}

μ and xε _{convergestosomepointx∈ ∂K ∩ ∂Ω withthepropertythatμ· ν ≥ 0.}

We haveproved thatCx=



ν ∈ SN−1 : ν· νx≥ 0



, x∈ ∂K ∩ ∂Ω isacoveringof theunitsphere;we are lefttoshowthatwecanextract fromCxafinitecoveringofthesphere.

WeproceedbyinductiononthedimensionN .IfN = 1 thereisnothingtoprove,sincethesphereS0isa finiteset.LetnowN > 1 andletOx=



ν ∈ SN−1 : ν·νx> 0



.IfSN−1 ⊆ ∪xOx,then,bycompactness,we

canextractafinitecoveringofthesphereSN−1.Otherwise,thereexistsaunitvectorν,suchthatν·νx≤ 0,

foreveryx∈ ∂K ∩ ∂Ω.

Let A=x∈ ∂K ∩ ∂Ω : νx· ν = 0



; weclaimthat{Cx}x∈A is acoveringofSN−1∩ ν⊥ =SN−2.Let

η ∈ SN−1_{, such that η· ν = 0 and let, for ε > 0, η}ε _{= sin(ε)η + cos(ε)ν. Let x}ε _{∈ ∂K ∩ ∂Ω,such that}

νxε· ηε≥ 0.Sinceν_xε· ν ≤ 0,wemusthaveν_xε· η ≥ 0.

Upto extracting asubsequence xε _{convergesto somepointx. Clearlyν}

x· ν ≥ 0 and, since∂K∩ ∂Ω is

closed, alsoνx· ν ≤ 0 andthusx∈ A.Thefactthatνx· η ≥ 0 proves ourclaim.

WecanfinallyapplyourinductionhypothesistofindafinitesubsetA⊆ A suchthatSN−2⊂ ∩_x∈AC_x,

and itisstraightforwardto checkthatalsoSN−2_⊂

x∈ACx. 2

Lemma 5.3. LetΩ and K as above, thenthere exists apolyhedral convex domain T such that Ω⊆ T and,

forevery τ > 1 andx∈ RN,x+ τ K  Ω.

Proof. Let x1,. . . ,xi and ν1,. . . ,νi be as inLemma 5.2. Weset T ={x : (x− xi)· νi ≤ 0}. SinceT is

theintersectionofsupportinghalf-spacesofΩ,thenΩ⊆ T .

Supposenow, thatthere existsx,τ > 1 suchthatx + τ K⊆ T .Letj be suchthatνj· x ≥ 0.Thepoint

x_j = τ xj+ x∈ x + τK verifies

(x_j− xj)· νj≥ (τ − 1)xj· νj> 0,

Lemma5.4. Forevery invertible linearmappingL:RN_{→ R}N _wehave

λK₁ (Ω) = λLK₁ (LΩ).

Proof. Bythedefinitionofsupportfunction,wehave

hL−1K(x) = sup

y∈L−1_Ky, x = supz∈KL

−1_{z, x = sup} z∈Kz, L

−t_{x = h}

K(L−tx),

whereL−tdenotes theinverseofthetransposeofL.

Denotingbyv(x)= u(L−1x),againbyasimplecomputationwehave

∇v(x) = L−t_∇u(L−1_x)

and

hLK(∇v(x)) = hK(Lt∇v(x)) = hK(∇u(L−1x)).

Then,using thecomputationaboveandthechangeofvariable formula,wehave  LΩ_h2LK(∇v) dx LΩv2dx =  LΩ_ h2K(∇u ◦ L−1) dx LΩ(u◦ L−1)2dx =  Ωh_2K(∇u) dx Ωu2dx ,

provingthelemma. 2

Wearenow readyto provethevalidityof(2.2).Theproof iscarriedoutinthesamespiritoftheproof oftheresultachievedbyHerschin[13],eveniftheintroductionoftheanisotropyobligesustoavoidtouse Hersch’sestimatesand toadapttheproof.

Theorem5.5. LetΩ⊂ RN _{be aconvexbody,then}

λK₁(Ω)≥ π

2

4(R

K Ω)−2.

Proof. Uptoatranslationandadilationofthedomain,wecanassumethatΩ satisfiestheassumptionsof

Lemma 5.2(noticethatinthiscase,RK

Ω = 1)byrepeatingtheconstructionexplainedinLemmas 5.2 and 5.3

wefindaconvexpolyhedraldomain T⊇ Ω withthesameanisotropicunitaryinradius.Forj = 1,. . . ,i we

call Tj the pyramidaldomain obtainedbytaking theconvexhullof theoriginandthe j-thfacet Fj ofT ,

andweset Ωj= Ω∩ Tj.

Letnowu bethefirsteigenfunction,wecompute

λK₁ (Ω) =  Ωh 2 K(∇u) dx  Ωu2dx =  j  Ωjh 2 K(∇u) dx  j  Ωju 2_dx ≥ min j  Ωjh 2 K(∇u) dx  Ωju 2_dx = min_j  Tjh 2 K(∇u) dx  Tju 2_dx .

Asweshallsee, toconcludeitisenoughtogetanestimatefrombelowofthequantity  Tjh 2 K(∇u) dx  Tju 2_dx .

To thisaim,as wedidinTheorem 4.9,weconsider alinearmapL such thatLν_j⊥= Id andLDhK(νj)=

hK(νj)νj.ArguingasinLemma 5.4,itiseasytoshowthat

 Tjh 2 K(∇u) dx  Tju 2_dx =  LTjh 2 LK(∇f) dx  LTjf 2_dx ,

where f (x)= u(L−1x).MoreoveritisclearthatLTj isagraphoverFj.

Forany functionf definedonTj wehave

h2_LK(∇f) ≥ h2_LK(νj)(∇νjf )

2_{. (5.1)}

Indeed,sinceDhK(νj)∈ K,thenhK(νj)νj∈ LK andthence,byrecallingthathK(νj)= hLK(νj),wehave

hLK(∇f) ≥ hK(νj)νj· ∇f = hLK(νj)∇νjf.

Werecallthat,foranytest functionv(t): [0,]→ R,suchthatv(0)= 0 it holdstruethat  0v(t) 2_dt  0v(t)2dt ≥ π2 42 . (5.2)

Thanksto (5.1)and(5.2), wecanconcludebycomputing  Tjh 2 K(∇u) dx  Tju 2_dx =  LFj (y) 0 h 2 K(∇f) dy dt  LFj (y) 0 f2dy dt ≥  LFj (y) 0 h 2 K(νj)∇νjf dy dt  LFj (y) 0 f2dy dt = h2K(νj) ⎡ ⎢ ⎣  LFj (y) 0 ∇νjf dt (y) 0 f2dt (y) 0 f 2_dt 1 dy ⎤ ⎥ ⎦ 1 LFj (y) 0 f2dy dt ≥ h2 K(νj) ⎡ ⎢ ⎣  LFj π2(y) 0 u 2_dt 4(y)2 ⎤ ⎥ ⎦ 1 LFj (y) 0 u2dy dt ≥ h2 K(νj) π2 42 max ,

where max,byconstruction,isRKΩhK(νj). 2

6. Regularity

IntheprevioussectionswelimitourselvestothecaseofnormsassociatedtoaC2

+-regularconvexbodyK.

InSection3,wementionedthatthisassumptionisnecessarytomaketheFinslerLaplaceoperatoruniformly elliptic, nonetheless thedefinitionsoftheK-torsionalrigidityandtheK-principal frequencyvia(1.2)and

(1.1)makesense evenforlessregularnormshK.Moreover,severalinterestingexamplesofapplicationsfor

those inequalities in the case of non-Euclidean norms arise when we consider the norms associated to a square, ormoreingeneral,p-normoftheform

vp =



|vi|p

1/p

.

Notice that,for 1≤ p< 2 and p=∞, thefunction · 2

p isnot C2-regular, while for2< p <∞ itis

smooth butits Hessianmayvanish.Thus allthese cases arenotcovered bythe theorems thatweproved so far.

In this section we remove any regularity assumption on the convex body K. This result, contained in

Theorem 6.2,reliesonaresultbythefirstauthorandDalMaso(see[7])concerningtheΓ-convergence(we referto[8]foranexhaustivereviewonthetopic) ofsomeintegralfunctionalsoftheform

Fh(u) =  Ω fh(x, u, Du) dx , toafunctional F (u) =  Ω f (x, u, Du) dx

bymeansofthepointwiseconvergenceoftheintegrandfunctionsfh toafunctionf .

Forthereaderconvenienceletusstatetheabovementionedresult.

Theorem 6.1 ([7]).Assume that all fh = fh(x,u,p) areconvex in p and that the sequence (f )h converges

pointwise to a function f . Assume that there exist two increasing continuous functions μ,ν : R+ → R+, with μ(0)= ν(0)= 0 and μ concave, suchthat

|fh(x, u, p)− fh(y, v, p)| ≤ ν(|x − y|)(1 + fh(x, u, p)) + μ(|u − v|),

foreach x,y∈ Ω,u,v∈ R,p∈ Rn_.Then

Γ- lim Fh(u) = F (u).

Our strategyis thefollowing: weapproximate aconvex bodyK with a sequenceof C₊2-regularconvex bodiesKn,andwepasstotheΓ-limitthedesiredinequalities.

Theorem6.2. LetK andΩ be convexbodies,then (2.1)and(2.2)holdtrue.

Proof. Welimitourselvestoprove(2.2),beingtheproofof(2.1)completelyanalogous.LetKnbeasequence

ofC2

+-regularconvexbodiesconvergingtoK withrespect totheHausdorffmetric(theexistenceofsucha

sequenceisanoldtheorem byMinkowski,butwereferto [17]forashorterproof),andlet

Fn(u) =



Ω

h2_Kn(∇u) dx ,

u∈ X, where X is the functional space madeof all functionsinW₀1,2(Ω),with L2 _{norm equalto 1. The}

minimizersofthefunctionalsFn,sayun,satisfy

π2 4 (R Kn Ω )−2 ≤ Fn(un)≤ j 2 0(R Kn Ω )−2. (6.1) NoticethatRKn

Ω → RKΩ.Moreover,sincehKn→ hK andsincehK(v)≤ cv,forsomepositiveconstantc,

then thesecond inequalityin(6.1)givesthe equi-boundednessof theminimizersun inW1,2, thatensures

theexistence,uptoextractingasubsequence,ofaweaklimit,u.

Sincethefunctionals Fn satisfytheassumptions ofTheorem 6.1,wehavethatF (u)=



Ωh 2

K(∇u)dx is

theΓ-limitofthefunctionalsFn and,sinceminimizersconverge tominimizers,weobtain

Since both the right-handside and the left-handside of (6.1)are converging,it is possible to passto the limitthose inequalitiesto findthedesiredresult. 2

Acknowledgments

Theworkofthefirstand secondauthors ispartoftheproject2015PA5MP7“Calcolodelle Variazioni”

fundedbytheItalianMinistryofResearchandUniversity.ThethirdauthorissupportedbytheMIUR SIR-grant“GeometricVariationalProblems” (RBSI14RVEZ).TheauthorsaremembersoftheGruppoNazionale perl’AnalisiMatematica,laProbabilitàeleloroApplicazioni(GNAMPA)oftheIstitutoNazionalediAlta Matematica (INdAM).The authors wish to thank the anonymous referee for several suggestions and re-marks.

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