Asymptotic behavior of non linear eigenvalue problems involving $p-$Laplacian type operators

THIERRYCHAMPIONANDLUIGIDEPASCALE

Abstra t. Westudy the asymptoti behavior of two nonlinear eigenvalue

problemswhi hinvolve

p

-Lapla iantypeoperators. Intherstproblem we onsiderthe limitas

p → ∞

ofthe sequen esof the

k

-th eigenvaluesof the

p−

Lapla ianoperators. These ondproblemwestudyisthehomogenization ofnonlineareigenvalueproblemsforsome

p

-Lapla iantypeoperatorswith

p

xed. Our asymptoti analysisrelies on a onvergen e resultfor parti ular

riti alvaluesofa lassofRayleighquotients,statedinauniedframework,

andonthenotionof

Γ

- onvergen e.

Keywords. Non-lineareigenvalues,

p

-Lapla ian,

∞

-eigenvalueproblems, homogenization,

Γ

- onvergen e.

MSC 2000. 35P30,35J70,35J20,35B27,49R50,74Q99.

1. Introdu tion

Aneigenvalueofthe

p

-Lapla ianisarealnumber

λ

∈ R

su hthattheproblem



−div(|Du|

p−2

_{Du) = λ|u|}

p−2

_u

in

Ω,

u

= 0

on

∂Ω,

hasat least onenon trivialsolutionin

W

1,p

0

(Ω)

. Here solution isintended in the distributional sense and

Ω

is assumed to be a regular, bounded, open subset of

R

N

. Oneeasily provesthat

λ

is an eigenvalueif and only ifit is a riti alvalue of theRayleighquotient

k∇vk

p

_Lp

kvk

p

Lp

. The

k

-theigenvalueis obtainedbythe lassi al Ljusternik-S hnirelmantheory(see[13,18,19℄foradetailed des ription),anditis

denedasfollows

λ

k

_p

:= inf



sup

v∈G

k∇vk

p

_L

p

kvk

p

_L

p

: G ⊂ L

p

, genus(G) ≥ k



where

genus(G)

isthe Krasnoselskiigenusof

G

(forapre ise statement and pre-sentationwerefertose tionŸ2).

In this paper, we study the asymptoti behavior of the sequen e of the

k

-th eigenvalues asso iated with families of monotone operators of

p−

Lapla iantype. Thestudyofthistypeofproblemarisesindierentsettingsforvariousappli ations,

seeforexample[5℄,[9℄,[25℄andthereferen estherein. Anaturalwaytodealwith

this asymptoti problem in the linear ase is thestudy of the onvergen eof the

resolventoperator(see[16℄ h. 10,lemmaXI9.5,morereferen eonthelinear ase

anbefoundin[5℄). Thepresentworkismotivatedbythestudyoftwoasymptoti

behaviorproblemsinvolvingsequen esof

p−

lapla iantypeoperatorsforwhi hwe propose aunied approa hin the generalsettingof the onvergen eof parti ular

The rstproblem we onsider is the study ofthe asymptoti behavioras

p

→

∞

of the

k

-th non-linear eigenvalue of the

p

-Lapla ian operator. This problem wasin parti ular studied in [21℄ and [20℄, where the onvergen e of the rst two

eigenvalueswasexaminedin details and partialresultsand onje turesforhigher

order eigenvalueswere given. Our main ontributionto this problem is theproof

of the onvergen e for the generalizedsequen e of the

k

-th eigenvalues(suitably renormalized)for anypositiveinteger

k

and avariational hara terization of this limit(seeŸ4,Theorem4.3).

These ondproblem we onsideris theasymptoti behaviorof

k

-th eigenvalues asso iatedwithafamily

(A

ε

)

ε

ofp-Lapla iantypeoperator

A

ε

(v) = −div(a

ε

(·, ∇v(·))

, withxed

p

. Assumingthat

A

ε

derivesfroma onvexandp-homogeneousintegral fun tional

F

ε

(v) =

R

Ω

f

ε

(x, ∇v(x))dx

, the

k

-theigenvalue

λ

k

ε

of

A

ε

is givenby

λ

k

_ε

:= inf



sup

v∈G

F

ε

(v)

kvk

p

_L

p

: G ⊂ L

p

_{, genus(G) ≥ k}



,

sothat

λ

k

ε

isaparti ular riti alpointoftheRayleighquotient

F

ε

(v)

kvk

Lp

(wealsorefer

to se tion Ÿ2 fora morepre ise presentation). Themain ontributionwegive to

the problem ofthis asymptoti study is apositiveanswerto aquestionraised in

[5℄. In this last work it was proved that the limit of any sequen e of eigenvalues

isaneigenvalueofthelimitproblemandthat thesequen eofthersteigenvalues

(i.e. when

k

= 1

) onvergesto thersteigenvalueofthelimitoperator. Itfollows from the presentwork that thesequen e of the k-theigenvalues onvergesto the

k-theigenvalueforanygreater

k

(seeŸ5, Theorem5.1).

Thesetwoproblemsof asymptoti behaviorhavethesamestru ture

λ

n

:= inf



sup

v∈G

F

n

(v)

kvk

L

pn

: G ⊂ L

p

n

_{, genus(G) ≥ k}



,

wherethefamily

(F

n

)

n

onvergestosomelimitfun tional. Thete hniqueweuseto studytheseproblemsisbasedonthenotionof

Γ

- onvergen e(seethenextse tion fordetails). The

Γ

- onvergen ewasintrodu edin [14℄to dealwith onvergen eof minimizers ofsequen es of fun tionals. Here wedeal with some riti al valuesof

sequen es

(F

n

)

n

of fun tionals whi h

Γ−

onverge. This is doneby introdu ing a newsequen eoffun tionalswhoseargumentsare ompa tsetsofthe

L

p

spa es,and

byendowingwiththeHausdor metri theset of ompa tsubsets ofthesuitable

spa e. This gives rise to a unied treatment in a general framework of the two

problemsmentionedabove(seeŸ3,Theorem3.3).

Arelativeadvantageindealingwith theseproblemsusing the

Γ

- onvergen eis thatwedonotneedtohavealimitoperatorandthatweknowthatthelimit riti al

value is of saddletype. Let us nally underline that our method givesageneral

s hemetodealwith

Γ

- onvergen eand riti alpointsobtainedviaanindextheory.

2. Definitions andpreliminaryresults

2.1.

Γ

- onvergen e.

Let

X

beametri spa e,asequen eoffun tionals

F

n

: X → R

issaidto

Γ

- onverge to

F

∞

at

x

if

where

(

Γ − lim inf F

n

(x) = inf

 lim inf F

n

(x

n

) : x

n

→ x in X ,

Γ − lim sup F

n

(x) = inf

 lim sup F

n

(x

n

) : x

n

→ x in X

.

(2.2)

The

Γ−

onvergen e was introdu ed in [14℄, foran introdu tionto this theory we referto[15℄and[4℄. Thefollowingisavariationofa lassi altheoremwhi hreports

propertiesof

Γ

- onvergen ethat weshallusein thefollowing.

Theorem2.1. Assumethatthe sequen e

(F

n

)

n∈N

of fun tionalsissu hthat

Γ − lim inf

n→+∞

F

n

≥ F

∞

on X,

where

F

∞

is lower-semi ontinuous on

X

. Assume in addition that the sequen e

(F

n

)

n

isequi- oer ive on

X

,then (1)

lim inf

n→+∞



inf

x∈X

F

n

(x)



≥ inf

x∈X

F

∞

(x)

, (2) if

lim

n→+∞



inf

x∈X

F

n

(x)



= inf

x∈X

F

∞

(x)

,then onehas

F

∞

(x

∞

) = inf

x∈X

F

∞

(x)

for any luster point

x

∞

of asequen e

(x

n

)

n∈N

su hthat

∀n ∈ N

F

n

(x

n

) ≤ inf

x∈X

F

n

(x) + ε

n

with

ε

n

→ 0

as

n

→ ∞

. 2.2. Krasnoselskiigenus.

Were allheresomebasi fa tsabouttheKrasnoselskiigenuswhi hwillbeused

inwhatfollows. Wereferto[26,27℄foramore ompleteintrodu tionontheindex

theories.

Denition2.2. Let

E

bearealBana hspa eandlet

A

⊂ E

beanonempty losed symmetri set (i.e.

A

= −A

). The genus

γ(A)

of the set

A

is the integer dened as

inf{m ∈ N :

there existsa ontinuousandoddmapping

ϕ

: A → R

m

_{\ {0}},}

wherethe above inmumisassumedtobe

+∞

if theset above isempty. Remark 2.3. Thefollowingelementarypropertieshold:

(1) If

0

belongsto

A

then

γ(A) = +∞

, (2)

A

1

⊂ A

2

implies

γ(A

1

) ≤ γ(A

2

).

Wewill needthefollowing ontinuitypropertyofthegenus(seeProposition5.4

of[27℄orProposition7.5of[26℄foraproof).

Proposition 2.4. Assume that

A

⊂ E

is ompa t, then there is a symmetri neighborhood

N

of

A

in

E

su hthat

γ(N ) = γ(A)

.

2.3. Hausdor onvergen eof ompa t sets.

Let

(X, d)

beametri spa e,anddenoteby

K

thesetofthe ompa tsubsetsof

X

. Thedistan e

d

H

: K × K → R

+

isdenedas

d

H

(G, F ) := sup

x∈G

d(x, F ) + sup

y∈F

d(y, G).

Proposition2.5. If

X

is ompa tthen

(K, d

H

)

is ompa t.

Asanappli ationofProposition2.5weobtaintwoexampleswhi hwillbeuseful

inthesequelofthepaper.

Example 2.6. Let

Ω

be an open subset of

R

N

with Lips hitz boundary. By a

standardappli ation oftheSobolev ompa tembeddingtheoremsweget:

•

Let

p

∈ [1, ∞)

and

q

≥ 1

su hthat

q

∈ [N, ∞]

or

q

∗

_:=

N q

N

−q

> p

,thenthe set

{G ⊂ W

0

1,q

(Ω) | G

is losedandboundedby

C

in

W

1,q

0

(Ω)}

equippedwiththeHausdor distan eindu edbythe

L

p

normis ompa t

forany onstant

C >

0

.

•

Let

q

∈ (N, ∞]

and

C >

0

,thentheset

{G ⊂ W

0

1,q

(Ω) | G

is losedandboundedby

C

in

W

1,q

0

(Ω)}

equipped with the Hausdor distan e indu ed by the sup norm of

C

0

is ompa t.

Wewillneedthefollowing hara terizationoftheHausdor onvergen ewhi h

anbeobtaineddire tlyfromthedenition:

Proposition2.7. If

(K

n

)

n

isa sequen e in

K

,then

K

n

→ K

with respe tto

d

H

ifandonly if

(1) for ea h sequen e

(x

n

)

n

su h that

x

n

∈ K

n

for all

n

, any a umulation point

x

∈ X

for

(x

n

)

n

belongsto

K

,

(2) for ea h point

x

∈ K

we an nd asequen e

x

n

with

x

n

∈ K

n

onverging to

x

.

Thenextlemmaofelementaryproofwillbeusefultostudy thebehaviorofthe

genuswithrespe tto theHausdor onvergen eof ompa tsets.

Lemma 2.8. Let

(K

n

)

n

is a sequen e in

K

onverging to

K

with respe t to

d

H

. Then any open set

A

⊂ X

whi h ontains

K

also ontains

K

n

for

n

largeenough. 2.4. Nonlineareigenvalues of

p

-Lapla ian type operators.

Inthisse tionweintrodu ethebasi denitionfortheeigenvaluesofthe

p

-Lapla ian andsomegeneralizationto

p−

Lapla iantypeoperators.

Fromnowon,

Ω

denotesabounded onne tedopensubsetof

R

N

withLips hitz

boundary. Inthefollowing,

p

is arealnumberin

]1, +∞[

and weshalldenote by

k.k

p

theusualnormof

L

p

_(Ω)

(or

L

p

_{(Ω; R}

N

₎

whendealingwiththegradientofsome

elementof

W

1,p

0

(Ω)

).

An eigenvalue of the

p−

Lapla ianoperator

−∆

p

is areal number

λ

for whi h theproblem



−∆

p

u

:= −div(|∇u|

p−2

∇u) = λ|u|

p−2

u

in

Ω,

u

= 0

on

∂Ω,

hasanon-zerosolutionin

W

1,p

0

(Ω)

. Thisproblem(anditsgeneralizationsto mono-tone ellipti operators)hasbeenwidelystudied in theliteratureandformore

de-tailedtreatmentwereferto[2,8, 13,18,19,20,24℄. Mu hisstillunknownabout

known resultswhi hwill berelevantfor thispaper. Everyeigenvalueis a riti al

valuefortheRayleighquotient

v

7→

R

Ω

|∇v|

p

dx

R

Ω

|v|

p

dx



=

k∇vk

p

p

kvk

p

p



whi hisaGateauxdierentiablefun tional on

W

1,p

0

(Ω)

outsidetheorigin. More-over, a sequen e

(λ

k

p

)

k≥1

of eigenvalues an be obtained as follows (we refer to [18℄ and[24℄ fordetails). Denote by

Σ

k

p

(Ω)

theset of thosesubsets

G

of

W

1,p

0

(Ω)

whi haresymmetri (i.e.

G

= −G

), ontainedin theset

{v : kvk

p

= 1}

,strongly ompa tin

W

1,p

0

(Ω)

andsu hthat

γ(G) ≥ k

, andset

λ

k

_p

=

inf

G∈Σ

p

_k

(Ω)

u∈G

sup

k∇uk

p

p

.

Then ea h

λ

k

p

dened as above is an eigenvalue of the

p

-Lapla ianoperator and

λ

k

p

→ +∞

as

k

→ ∞

. Moreoverit is known that

λ

1

p

isthe smallesteigenvalueof

−∆

p

,thatitissimple(see[7℄forashortproof)andthattheoperator

−∆

p

doesn't haveanyeigenvaluebetween

λ

1

p

and

λ

2

p

.

Asa onsequen eof ourresults(seeCorollary3.6),weshallgetthat theabove

denitionofthe

k

-th eigenvaluemayberewritten

λ

k

_p

=

min

G∈G

_k

p

(Ω)

u∈G

sup

k∇uk

p

p

.

(2.3)

In the above formula

G

k

p

(Ω)

is the set of those subsets

G

of

W

1,p

0

(Ω)

whi h are symmetri , ontainedin theset

{v : kvk

p

= 1}

, losed and bounded in

W

1,p

0

(Ω)

(andthus ompa tin

L

p

_(Ω)

)and su h that

γ(G) ≥ k

, where

γ(G)

is thegenusof

G

asasubsetof

L

p

_(Ω)

.

Moregenerally, onsider a

p

-Lapla iantypeoperator

A : W

1,p

0

(Ω) → W

−1,p

(Ω)

oftheform

A(u) = −div(a(x, ∇u(x)))

wherethefun tion

a

: Ω×R

N

_{→ R}

N

satises

(h1)

a

is a Carathéodoryfun tion i.e.

a(x, ·)

is a ontinuous fun tion for a.e.

x

∈ Ω

and

a(·, ξ)

ismeasurableforevery

ξ

∈ R

n

,

(h2)

a(x, ·)

ispositivelyhomogeneousofdegree

p

− 1

fora.e.

x

, (h3)

a(x, ·)

isoddfora.e.

x

(h4)

a

is y li allymonotone,i.e.

m

X

i=1

ha(x, ξ

i

), ξ

i+1

− ξ

i

i ≤ 0

fora.e.

x

∈ Ω

,any

m

≥ 2

and

ξ

1

, . . . , ξ

m

∈ R

N

(with

ξ

m+1

= ξ

1

) (h5) thereexists

β > α >

0

su hthat thefollowinggrowth onditionshold

α|ξ|

p

≤ ha(x, ξ), ξi

and

|a(x, ξ)| ≤ β|ξ|

p−1

forall

ξ

∈ R

N

andfora.e.

x

∈ Ω

.

Thenaneigenvalue

λ

for

A

isarealnumberforwhi htheproblem



−div(a(x, ∇u(x))) = λ|u(x)|

p−2

_u(x)

fora.e.

x

in

Ω,

u

= 0

on

∂Ω

hasanon-trivialdistributionalsolutionin

W

1,p

0

(Ω)

. Under thehypothesis(h1-5), there exists an integrand

f

: Ω × R

N

_{→ R}

(a1)

f

isaCarathéodoryfun tion,

(a2)

f

(x, ·)

is onvex,dierentiablewithgradient

a(x, ·)

, (a3)

f

(x, ·)

ispositivelyhomogeneousofdegree

p

, (a4)

f

(x, ·)

iseven,

(a5) thefollowinggrowth onditionsholds

α|ξ|

p

_{≤ f (x, ξ) ≤ β|ξ|}

p

forall

ξ

∈ R

N

andfora.e.

x

∈ Ω

,

Thenforanyinteger

k

≥ 1

one andenethe

k

-theigenvalueof

A

asbeing

λ

k

:=

inf

G∈G

_k

p

(Ω)

u∈G

sup

Z

Ω

f

(x, ∇u)dx.

(2.4)

Then, part of the study of the homogenization pro ess for the eigenvalue

prob-lems asso iated to a family

A

ε

:= −div(a

ε

(·, ·))

of monotone ellipti operator of

p−

Lapla ian type redu es to the study of the limit of a family of problems like (2.4).

3. A general onvergen eresult

Inthisse tionwestateandprovethemainresultofthepaper. Inthefollowing,

(F

ε

)

ε≥0

isafamilyoffun tionalsdenedon

L

1

_(Ω)

withvaluesin

[0, +∞]

su hthat: (A1) forany

ε >

0

,

F

ε

is onvexand

1

-homogeneous;

(A2) there exists

β > α >

0

su hthat forany

ε >

0

there exists

p

ε

∈ [1, +∞]

forwhi h

(

αk∇vk

p

ε

≤ F

ε

(v) ≤ βk∇vk

p

ε

if

v

∈ W

1,p

ε

0

(Ω)

,

F

ε

(v) = +∞

otherwise

;

(A3) thefamily

(p

ε

)

ε>0

onvergestosome

p

0

∈ [1, +∞]

and thefamily

(F

ε

)

ε>0

Γ

- onvergesin

L

p

0

_(Ω)

(or

C

0

(Ω)

if

p

0

= +∞

)tosomefun tional

F

0

. Noti ethatthefun tional

F

0

alsosatises(A1) and(A2).

Remark 3.1. Inthe ase

p

ε

= +∞

,the notation

W

1,∞

0

(Ω)

standsfor

W

1,∞

_{(Ω) ∩}

C

0

(Ω)

. Noti ethatthislastspa estri tly ontainsthe losureof

C

∞

c

(Ω)

in

W

1,∞

_(Ω)

,

andisthuslargerthanthespa e usuallydenotedby

W

1,∞

0

(Ω)

: thereasonforour notationisthat

W

1,∞

_(Ω)∩C

0

(Ω)

isthenaturallimitspa efor

W

1,p

0

(Ω)

as

p

→ +∞

. Inthefollowing,thenotation

G

k

∞

(Ω)

generalizesthatgivenin Ÿ2.4.

Wenowdenea ommonframeworkforthestudy ofthenon-lineareigenvalues

of the family

(F

ε

)

ε

. Inthe rest ofthis se tion, weshall denote by

K

s

(Ω)

the set of ompa t symmetri subsets of

L

p

0

(Ω)

(or

C

0

(Ω)

if

p

0

= +∞

), and by

d

H

the Hausdordistan eindu edon

K

s

(Ω)

bytheusualnormof

L

p

0

(Ω)

.

Lemma 3.2. There exists

ε

0

>

0

su h that for any

ε < ε

0

, the set

G

k

p

ε

(Ω)

is in luded in

K

s

(Ω)

. Moreover, the genus of an element

G

∈ G

k

p

ε

(Ω)

is the sameas itsgenus asanelementof

K

s

(Ω)

.

Proof. We rst onsider the ase

p

0

∈ [1, +∞[

: as

p

ε

→ p

0

, there exists

ε

0

>

0

su hthat

p

ε

>

N

N

+p

0

p

0

for any

ε < ε

0

. Forsu h

ε

, sin eanelement

G

of

G

k

p

ε

(Ω)

is losedand bounded in

W

1,p

ε

0

(Ω)

weinferfrom theSobolev ompa tembedding theoremsthat

G

isin fa ta ompa tsubsetof

L

p

0

_(Ω)

. Inthe ase

p

0

= +∞

,itis su ienttotake

ε

0

>

0

su hthat

p

ε

≥ N + 1

forany

ε < ε

0

.

Let

ε < ε

0

, andassume that

p

ε

≥ p

0

: then theidentitymapping

i

: L

p

ε

_{(Ω) →}

L

p

0

_(Ω)

is ontinuous,and sin e

G

is ompa tin

L

p

ε

_(Ω)

, the sets

G

and

i(G)

are homeomorphi sothatthegenusof

G

asassubsetof

L

p

ε

_(Ω)

isthesameasitsgenus

asasubsetof

L

p

0

_(Ω)

. When

p

ε

≤ p

0

,thesameargumentworkswith theidentity mapping

i

: L

p

0

_{(Ω) → L}

p

ε

_(Ω)

.



For any integer

k

≥ 1

and number

ε

≥ 0

, we asso iate to

F

ε

the fun tional

J

k

ε

: K

s

(Ω) → [0, +∞]

givenby

J

_ε

k

(G) :=

(

sup

v∈G

F

ε

(v)

if

G

∈ G

k

p

ε

(Ω),

+∞

otherwise

.

Generalizingthedenition (2.3) introdu ed in Ÿ2.4, we denethe

k

-theigenvalue ofthefun tional

F

ε

as

λ

k

_ε

:=

inf

G∈G

k

pε

sup

v∈G

F

ε

(v)

andwededu efromLemma3.2 thatfor

ε

smallenoughthis anberewritten

λ

k

_ε

= inf

J

k

ε

(G) : G ∈ K

s

(Ω) .

We annowstatethemainresultofthisse tion,whi hinparti ularyieldsthat

λ

k

ε

→ λ

k

0

as

ε

→ 0

:

Theorem3.3. Let

k

beapositiveinteger,andassumethatthefamily

(F

ε

)

ε>0

satis-es(A1-3). Thenthereexists

ε

0

>

0

su hthatthefamily

(J

k

ε

)

0≤ε<ε

0

isequi oer ive on

(K

s

(Ω), d

H

)

,

Γ − lim inf

ε→0

J

k

ε

≥ J

0

k

on

K

s

(Ω)

and

lim

ε→0



inf

G∈K

s

(Ω)

J

_ε

k

(G)



=

inf

G∈K

s

(Ω)

J

₀

k

(G).

Proof. Wedividetheproofinthreesteps.

Step 1. Equi oer ivity. We rst onsider the ase

p

0

∈ [1, +∞)

, and dene an exponent

q

≥ 1

dependingonthedimension

N

and

p

0

asfollows:

q(P

0

, N) := q :=

(

1

if

N

= 1

or

N >

1

and

p

0

∈ [1,

N

N

−1

],

N p

0

N

+p

0

if

p

0

∈ (

N

N−1

,

+∞).

Let

δ

=

p

0

−q

2

and

ε

0

>

0

su h that for any

ε < ε

0

wehave

p

ε

≥ q + δ

(noti e thatfor

p

0

= 1

onehas

δ

= 0

thus

p

ε

≥ q

forany

ε >

0

). Observethatthe riti al exponent

(q + δ)

∗

isalwaysstri tlygreaterthan

p

0

. Let

ε < ε

0

and

G

ε

∈ K

s

(Ω)

besu hthat

J

k

ε

(G

ε

) ≤ C

. Bydenition of

J

k

ε

and property(A2)theestimate

kuk

W

0

1,pε

(Ω)

≤

C

α

holdsforall

u

∈ G

ε

andthisimplies

kuk

_W

1,q+δ

0

(Ω)

≤ |Ω|

1

q+δ

−

1

pε

C

α

≤ K

where

K

isa onstantindependentof

ε < ε

0

. ByProposition2.5andtherstpart ofExample2.6,thesublevels

{J

ε

≤ C}

are ontainedina ommon ompa tsubset of

(K

s

(Ω), d

H

)

for

ε < ε

0

,sothat thefamily

(J

k

ε

)

0≤ε<ε

0

isequi oer ive.

In the ase

p

0

= ∞

we follow the same s heme but we hoose

ε

0

su h that

Step 2. Weshow the

Γ

-liminf estimate. Tothis end, let

G

∈ K

s

(Ω)

and

(G

ε

)

ε>0

beafamilysu hthat

G

ε

d

H

→ G

,weshallprovethat

lim inf

n→∞

J

k

ε

(G

ε

) ≥ J

0

k

(G).

Without loss of generality, we may assume that there exists a onstant

C >

0

su h that

J

k

ε

(G

ε

) ≤ C

for any

ε >

0

. Let us rst show that

γ(G) ≥ k

. By Proposition2.4thereexistsanopensymmetri neighborhood

N

of

G

in

L

p

0

_(Ω)

(or

C

0

(Ω)

for

p

0

= +∞

)su h that

γ( ¯

N) = γ(G)

. Wetheninferfrom Lemma2.8 that

G

ε

⊂ N ⊂ ¯

N

for

ε

smallenough. Bythese ondpropertyinRemark 2.3,forsu h an

ε >

0

weget

k

≤ γ(G

ε

) ≤ γ( ¯

N) = γ(G).

Let now

u

∈ G

, by the sequential hara terization of the Hausdor onvergen e of ompa t sets (Proposition 2.7), there exists a (generalized) sequen e

u

ε

∈ G

ε

whi h onvergesto

u

in

L

p

0

_(Ω)

. Bythe

Γ−

liminfinequalityforthefun tionals

F

ε

(assumption(A3))

F

0

(u) ≤ lim inf

ε→0

F

ε

(u

ε

) ≤ lim inf

ε→0



sup

G

ε

F

ε



= lim inf

ε→0

J

k

ε

(G

ε

).

Takingthesupremumfor

u

∈ G

givesthe laim.

Step3. BythetwopreviousstepsandTheorem2.1,weinferthatitonlyremains

toprovethat

lim sup

ε→0



inf

G∈K

s

(Ω)

J

_ε

k

(G)



≤

inf

G∈K

s

(Ω)

J

0

k

(G).

Weassumethat

inf

G∈K

s

(Ω)

J

0

k

(G) < +∞

,otherwisethereisnothingtoprove. Wex

δ

∈ ]0, 1[

and rststudythe ase

p

0

∈ ]1, +∞]

. Let

G

0

∈ K

s

(Ω)

besu h that

inf

G∈K

s

(Ω)

J

0

k

(G) ≥ J

0

k

(G

0

) − δ.

Sin e

G

0

is ompa t in

L

p

0

_(Ω)

, there exists a nite family

(u

i

₎

1≤i≤m

in

G

0

su h that

G

0

⊂

m

[

i=1

B

L

p0

(Ω)

(u

i

,

δ

5

)

We infer from hypothesis (A3) that for any

i

∈ {1, . . . , m}

there exists afamily

(u

i

ε

)

ε

in

L

p

0

(Ω)

su hthat

u

i

_ε

→ u

i

in

L

p

0

_(Ω)

and

F

ε

(u

i

ε

) → F

0

(u

i

)

as

ε

→ 0.

Taking

ε

0

asinstep1,forany

ε

∈ ]0, ε

0

[

wedene

C

ε

tobethe onvex losureof thenitesymmetri set

{±u

i

ε

: i = 1, . . . , m}

. Wemayassumethat

F

ε

(u

i

ε

) < +∞

forany

i

andanysu h

ε

,sothatthenitedimensionalset

C

ε

isa ompa t onvex subsetof

W

1,p

ε

0

(Ω)

and

L

p

0

_(Ω)

. Nowlet

1 < q < p

0

besu hthat

∀i ∈ {1, . . . , m}

ku

i

k

q

≥ 1 −

δ

5

.

Wedenoteby

P

ε

theproje tiononto

C

ε

forthenormof

L

q

_(Ω)

,forwhi h

C

ε

isalso ompa t. Thenwenoti ethatforany

v

∈ G

0

thereexists

i

∈ {1, . . . , m}

su hthat

kv − u

i

_k

p

0

≤

δ

5

,therefore

kP

ε

(v)k

q

≥

ku

i

ε

k

q

− kP

ε

(u

i

) − u

i

ε

k

q

− kP

ε

(v) − P

ε

(u

i

)k

q

≥

ku

i

ε

k

q

− ku

i

− u

i

ε

k

q

−

δ

5

.

Sin e

u

i

ε

→ u

i

in

L

p

0

_(Ω)

, thus alsoin

L

q

_(Ω)

, we getthat for any

ε

smallenough onehas

P

ε

(G

0

) ⊂ C

ε

\ B

L

q

_(Ω)

(0, 1 −

δ

2

).

Alsonoti e thatthe element

P

ε

(G

0

)

of

K

s

(Ω)

satises

γ(P

ε

(G

0

)) ≥ k

. Then on-siderthefun tional

ϕ

ε

: P

ε

(G

0

) → W

1,p

ε

0

(Ω)

givenby

ϕ

ε

(v) :=

v

kvk

pε

andset

∀ε ∈ ]0, ε

0

[ ,

G

ε

:= ϕ

ε

(P

ε

(G

0

)).

Sin e

ϕ

ε

is ontinuouson

P

ε

(G

0

)

,

G

ε

belongsto

G

k

p

ε

(Ω)

. Moreoverfor

ε >

0

small enoughonehas

p

ε

> q

sothat

∀v ∈ P

ε

(G

0

)

1 −

δ

2

≤ kvk

q

≤ kvk

p

ε

|Ω|

1

q

−

1

pε

_.

Asa onsequen eonegets

J

_ε

k

(G

ε

) =

sup



F

ε



v

kvk

p

ε



: v ∈ P

ε

(G

0

)



≤

|Ω|

1

q

−

1

pε

1 −

δ

2

sup {F

ε

(v) : v ∈ P

ε

(G

0

)}

≤

2|Ω|

1

q

−

1

pε

2 − δ

sup {F

ε

(v) : v ∈ C

ε

} =

2|Ω|

1

q

−

1

pε

2 − 2δ

1≤i≤m

max

F

ε

u

i

ε

.

Asa onsequen ewehave

lim sup

ε→0



inf

G∈K

s

(Ω)

J

_ε

k

(G)



≤

lim sup

ε→0

J

ε

(G

ε

)

≤

2|Ω|

1

q

−

1

p0

2 − δ



inf

G∈K

s

(Ω)

J

k

0

(G) + δ



.

The on lusionofstep3thenfollowsbyletting

δ

goto

0

and

q

goto

p

0

.

For the ase

p

0

= 1

, onehas to slightly modify theabove argument. Wejust noti ethat

G

0

beingbounded in

W

1,1

0

(Ω)

, itis in fa t ompa tin

L

2N

2N −1

_(Ω)

. We anthenpro eedasif

p

0

=

2N

2N −1

andfollowthesamearguments. Noti ethatthis worksbe auseif

u

i

_ε

→ u

i

in

L

1

_(Ω)

and

F

ε

(u

i

ε

) → F

0

(u

i

)

as

ε

→ 0

andifthefamily

(F

ε

)

ε

satisestheuniformgrowth ondition(A2),oneinfers that

(u

i

ε

)

ε

isboundedin

W

1,1

_(Ω)

sothat

u

i

_ε

→ u

i

in

L

2N

2N −1

(Ω).



Remark 3.4.Theargumentusedinthethirdstepoftheaboveproofisanadaptation

of theproof ofLemma 4.3 in [23℄. Noti e that theapproximatingfamily

(G

ε

)

ε>0

doesnot onvergeto

G

0

in

K

s

(Ω)

, butto asetthat anbesomewhatlarger: this

is due to the onvexi ation pro edure, whi h on the one hand ensures that the

genusdoesnotde rease,butontheotherhandenlargetheapproximatingsets.

Remark 3.5. The proof of Theorem 3.3 presented above allows to handle in a

uniedwayawidevarietyofasymptoti problems,namelyallofthose overedby

thehypotheses

(A1 − 3)

,su hasthoseofthetwofollowingse tions. Weonlynoti e thatthe ase

p

0

= 1

requiresaspe i treatmentinStep3tohandlethefa t that the proje tionin the

L

1

-norm maybemultivaluedsin e this norm isnot stri tly

onvex.

We noti e that in the third step of theabove proof, for any positive

ε

the set

G

ε

onstru ted above belongs to

Σ

k

p

ε

: indeed, it is nite dimensional, and thus ompa t in

W

1,p

ε

0

(Ω)

. As a onsequen e,if Theorem 3.3is applied to a onstant family

F

ε

:= F

0

forany

ε >

0

, oneobtainsthefollowingresult.

Corollary 3.6. Let

p

∈ [1, +∞]

, and assume that a fun tional

F

: L

p

_{(Ω) →}

[0, +∞]

is onvex,

1

-homogeneousandsatises

(

αk∇vk

p

≤ F (v) ≤ βk∇vk

p

if

v

∈ W

1,p

0

(Ω)

,

F(v) = +∞

otherwise

;

for some onstants

β > α >

0

. Then for any positive integer

k

onehas

inf

G∈Σ

k

p

sup

v∈G

F

(v) = min

G∈G

k

p

sup

v∈G

F

(v).

Proof. We just need to justify that the minimum on

G

k

p

is attained. To this end we noti e that the fun tional

J

k

asso iated with

F

is oer ive on

K

s

(Ω)

as an appli ationof Theorem3.3, and the

Γ

-liminfestimateof thisTheorem alsoyields that

J

k

is lower than its lower semi- ontinuous envelope, so that it is l.s. . on

K

s

(Ω)

. Thisfun tionalthusattainsitsminimumon

K

s

(Ω)

.



4. Limit as

p

→ ∞

of the eigenvalues of the

p

-Lapla ian

Inthisse tionwewillapplytheresultsofSe tion3totheapproximationofthe

so alled

∞

-eigenvalueproblem[20℄. Werstxsomenotations,andthen onsider some onsequen esoftheapproximationresults.

As wewill onsider thelimitas

p

→ ∞

ofthe

k

-th nonlineareigenvalueofthe

p

-Lapla iandenedby(2.3),throughoutthisse tion

k

willdenoteaxedpositive integer. Withoutlossofgenerality,wealsoassumethat

p

≥ N + 1

inthefollowing. Sin e

W

1,N +1

0

(Ω)

is ompa tly embedded in

C

0

(Ω)

, we shallalso assume that we work with the ontinuous representativeof everyfun tion

v

∈ W

1,p

0

(Ω)

. Finally, wedenoteby

K

s

(Ω)

thesetwhose elementsarethe ompa tsymmetri subsetsof

C

0

(Ω)

. For

p

∈ [N + 1, +∞]

,wenowdene

J

k

p

: K

s

(Ω) → [0, +∞]

by

J

_p

k

(G) =



sup

u∈G

k∇uk

p

if

G

∈ G

p

k

(Ω),

+∞

otherwise

.

As explained above,anyelement

G

∈ G

p

k

(Ω)

may be onsidered asan element of

K

s

(Ω)

sothat onemayrewrite(2.3)for

p

∈ [N + 1, +∞)

as

(λ

k

p

)

1

p

_{= min{J}

k

where theminimumisattained thankstoCorollary3.6. Inamoredetailed form, (4.1)reads

(λ

k

p

)

1

p

= min{sup

v∈G

k∇vk

p

: G ∈ K

s

(Ω),

G

⊂

W

1,p

0

(Ω) ∩ {v : kvk

p

= 1}, γ(G) ≥ k}.

Weshallstudythe

Γ

- onvergen eof thefamily

(J

k

p

)

p

in thespa e

(K

s

(Ω), d

H

)

: inwhatfollows,thedistan e

d

H

willdenotethatindu edon

K

s

(Ω)

bythesupnorm

k.k

∞

on

C

0

(Ω)

.

Noti ethatinthesettingofthepreviousse tion,thefamily

(J

k

p

)

p≥N +1

is asso- iatedwiththefamily

(F

ε

)

ε≤

1

N +1

where

F

ε

isgivenon

L

1

_(Ω)

by

F

ε

(v) :=



k∇vk

1

ε

if

v

∈ W

1,p

0

(Ω),

+∞

otherwise

.

Asa onsequen eoftheproofofLemma3.2weknowthatthegenusof

G

∈ G

p

k

(Ω)

asasubsetof

W

1,p

0

(Ω)

isthesameasitsgenusasasubsetof

C

0

(Ω)

,sothat from now on, we shall always assume that the genus is that omputed in

C

0

(Ω)

. It also followsfrom Theorem 3.3 that thefamily

(J

k

p

)

p∈[N +1,+∞]

is equi oer iveon

(K

s

(Ω), d

H

)

.

Wenowstudy the

Γ

-limitof thefamily

(J

k

p

)

p≥N +1

as

p

→ ∞

. Inthis asewe areableto provethefull

Γ

- onvergen eofthisfamilytothefun tional

J

k

∞

. Theorem4.1. Let

k

beapositiveinteger. Thenthefamilyoffun tionals

(J

k

p

)

p≥N +1

Γ

- onvergesto

J

k

∞

in

K

s

(Ω)

as

p

→ +∞

.

Proof. The

Γ − lim inf

inequalityfollowsfrom Theorem 3.3. Wethusturn to the proofofthe

Γ − lim sup

inequality: let

G

∞

∈ K

s

(Ω)

besu hthat

J

k

∞

(G

∞

) < +∞

, wehaveto deneafamily

G

p

d

H

→ G

∞

su hthat

lim sup

p→∞

J

_p

k

(G

p

) ≤ J

∞

k

(G

∞

).

Sin e

G

∞

isa ompa tsubsetof

{v : kvk

∞

= 1}

andthe

L

N

+1

-norm is

ontin-uouswithrespe tto the

L

∞

-normweinfer

0 < m := min{kuk

N+1

: u ∈ G

∞

} ≤ max{|Ω|, 1} min{kuk

p

: u ∈ G

∞

}

forany

p

≥ N + 1

. Thentheappli ation

ϕ

p

: G

∞

→ C

0

(Ω)

givenby

ϕ

p

(u) =

u

kuk p

is welldened,bije tiveand ontinuouson

G

∞

. For

p

≥ N + 1

weset

G

p

= ϕ

p

(G

∞

)

. Sin e

G

∞

is ompa tin

C

0

(Ω)

,sois

G

p

in

L

p

_(Ω)

and we on ludethat

γ(G

∞

) =

γ(G

p

)

. Asa onsequen e,

G

p

∈ G

k

p

(Ω)

andwehave

∀p ≥ N + 1

J

_p

k

(G

p

) ≤

|Ω|

p

1

min{kuk

p

: u ∈ G

∞

}

J

_∞

k

(G

∞

).

(4.2)

Takingthe

lim sup

as

p

→ ∞

onegets

lim sup

p→∞

J

_p

k

(G

p

) ≤ J

∞

k

(G

∞

).

It remains to provethat

G

p

d

H

→ G

∞

. We inferfrom the equi oer ivity of

J

k

p

and thepreviousargumentthatthefamily

(G

p

)

p>N+1

ispre ompa tin

K

s

. Itiseasily seenthat any luster point of

(G

p

)

p

as

p

→ +∞

ontains

G

∞

. Let

(u

p

)

p

besu h that

u

p

∈ G

p

forany

p

and assumethat asubsequen e

(u

p

u

∈ C

0

(Ω)

. Forany

n

there exists

v

n

∈ G

∞

su h that

u

p

n

= ϕ

p

n

(v

n

)

. Sin e

G

∞

isa ompa tsubsetof

C

0

(Ω)

, we anassumewithoutlossofgeneralitythat

(v

n

)

n

onvergesuniformlytosome

v

∈ G

∞

,but then

kv

n

k

p

n

→ kvk

∞

= 1

andit followsthat

u

= v

belongsto

G

∞

,so thatany lusterpointof

(G

p

)

p

in

K

s

isin ludedin

G

∞

,whi h on ludestheproof.



Remark 4.2. It results from Theorem 4.1 that the fun tional

J

k

∞

is oer iveand lowersemi- ontinuouson

K

s

(Ω)

.

As a onsequen eof the previousTheorem and the basi properties of the

Γ

- onvergen e,wegetthefollowing onvergen eresultforthe

k

-theigenvalues. Theorem4.3. Let

λ

k

p

be the

k

-theigenvalueof the

p

-Lapla ian operator,then (1)

lim

p→∞

(λ

k

p

)

1

p

= Λ

k

∞

where

Λ

k

∞

:=

min

G∈G

∞

k

(Ω)

sup

u∈G

k∇uk

∞

=

min



sup

u∈G

k∇uk

∞

: G ∈ K

s

(Ω),

G

⊂

W

1,∞

0

(Ω) ∩ {v : kvk

∞

= 1}, γ(G) ≥ k

o

.

(2) Let

(G

p

)

p

be a family in

K

s

(Ω)

su h that

G

p

∈ G

k

p

(Ω)

and

J

k

p

(G

p

) =

min{J

k

p

(G) : G ∈ K

s

(Ω)}

for any

p

. If for some sequen e

p

k

→ +∞

onehas

G

p

k

d

H

→ G

∞

,then

J

k

∞

(G

∞

) = Λ

k

∞

. Remark 4.4. Thepoint

(1)

alsoreads:

lim

p→∞

G∈G

min

k

p

(Ω)

sup

u∈G

k∇uk

p

=

min

G∈G

∞

k

(Ω)

sup

u∈G

k∇uk

∞

.

Moreover,noti ethatif

G

p

∈ G

k

p

(Ω)

issu hthat

J

k

p

(G

p

) = (λ

k

p

)

1

p

forany

p

,thenit followsfromTheorem 4.1and 4.3(1)that thefamily

(G

p

)

p≥N +1

ispre ompa tin

K

s

(Ω)

.

Proof. Werstnoti ethatthefamily

((λ

k

p

)

1

p

)

p

isboundedas

p

→ +∞

. Let

G

∞

∈

G

k

∞

be su h that

sup

u∈G

∞

k∇uk

∞

<

+∞

. With the notations of the proof of Theorem4.1andusing(4.2),weget

(λ

k

p

)

1

p

_{= min{J}

k

p

(G) : G ∈ K

s

(Ω)} ≤ J

p

k

(ϕ

p

(G

∞

)) ≤

max{|Ω|, 1}

m

J

k

∞

(G

∞

)

forany

p

≥ N + 1

. Sin etherighthandsidedoesnotdependon

p

,thisprovesthe laim.

As a onsequen e, the family

(min

K

s

J

_p

k

)

p

is bounded, so that (1) and (2) are straightforward onsequen esofTheorems2.1 and4.1.



Theorem 4.5. Forany

p

≥ N + 1

,let

u

p

∈ W

1,p

0

(Ω)

be adistributional solution of

−div(|∇u

p

|

p−2

∇u

p

) = λ

p

k

u

p

|u

p

|

p−2

in

Ω.

Assume that

ku

p

k

p

= 1

for any

p

≥ N + 1

, and that (up to subsequen es)

(u

p

)

p

onverges in

C

0

(Ω)

tosome

u

∞

∈ W

1,∞

0

(Ω)

as

p

→ ∞

. Then there exists

G

∞

∈

K

s

(Ω)

su hthat

u

∞

∈ G

∞

and

k∇u

∞

k

∞

= J

k

∞

(G

∞

) = Λ

k

∞

.

Remark 4.6. It follows from Lemma 5.2 in [20℄ that the luster point

u

∞

is a vis ositysolutionof







min{|∇u| − Λ

k

∞

u ,

−∆

∞

u} = 0

in

{u > 0},

−∆

∞

u

= 0

in

{u = 0},

max{−|∇u| − Λ

k

∞

u ,

−∆

∞

u} = 0

in

{u < 0},

(4.3)

and by thedenition givenin [20℄this meansthat

u

∞

is an eigenfun tionof the innite-Lapla ianforthe

∞−

eigenvalue

Λ

k

∞

.

Proof. The key point is to show that

k∇u

∞

k

∞

= Λ

k

∞

. To this end, we noti e that

ku

∞

k

∞

= 1

and assumewithoutlossof generalitythat

u

∞

(x

0

) = 1

for some

x

0

∈ Ω

. Sin e

k∇u

p

k

p

= (λ

k

p

)

1

p

→ Λ

k

∞

, we infer that

k∇u

∞

k

∞

≤ Λ

k

∞

. We show by ontradi tionthat

k∇u

∞

k

∞

≥ Λ

k

∞

: otherwise,one wouldhave

k∇u

∞

k

∞

< L <

Λ

k

∞

forsome

L >

0

. Let

ϕ

∈ W

1,∞

loc

(R

N

)

bedened by

ϕ(x) = −L|x − x

0

|

, then

u

∞

− ϕ

attainsastri tminimumon

Ω

at

x

0

. Forany

δ >

0

, onsider

φ

δ

:= ρ

δ

∗ ϕ

, where

ρ

δ

isamollier,i.e.

ρ

δ

(x) := δ

N

_ρ(

x

δ

)

forsomefun tion

ρ

su hthat

ρ

∈ C

∞

_(R

N

_,

_{[0, +∞[) , spt(ρ) ⊂ B(0, 1)}

and

Z

R

N

ρ(x)dx = 1 .

As

ϕ

δ

→ ϕ

uniformlyon

Ω

as

δ

→ 0

,weinferthat

u

− ϕ

δ

attainsalo alminimum on

Ω

at some

x

δ

for

δ

small enough, and that

x

δ

→ x

0

as

δ

→ 0

. Sin e

u

∞

is a vis osity solutionof (4.3), we on ludethat

|∇ϕ

δ

(x

δ

)| ≥ Λ

k

∞

u(x

δ

)

for

δ

small enough. It remainsto noti ethat

|∇ϕ(x

δ

)| =









Z

R

N

ρ(y)∇ϕ(x

δ

− y)dx









≤ L

and

u(x

δ

) → 1

toobtainthe ontradi tion

L

≥ Λ

k

∞

byletting

δ

goto

0

. Now,let

F

∞

∈ K

s

(Ω)

besu hthat

J

k

∞

(F

∞

) = Λ

k

∞

,andset

G

∞

:= F

∞

∪ {±u

∞

}

,

then

G

∞

satises thedesiredproperty.



Remark 4.7. Noti e that for agiven

G

∈ K

s

(Ω)

with

J

k

∞

(G) = Λ

k

∞

, afun tion

u

∈ G

su h that

k∇uk

∞

= Λ

k

∞

is not ne essarily a vis osity solution of (4.3). Indeed any fun tion

v

∈ W

1,∞

0

(Ω)

with

k∇vk

∞

= Λ

k

∞

and

kvk

∞

= 1

may be added to su h a set

G

by onsidering

G

∪ {±v}

, but in generalsu h afun tion an't beexpe ted to be asolution of (4.3). The above Theorem assertsthat the

limits of riti al points of the Rayleigh quotients are also in the vis osity sense

5. Homogenizationof nonlineareigenvalue problems for

p

-Lapla ian typeoperators

Inthefollowing,

p

isaxedrealnumberin

[1, +∞[

and

k

apositiveinteger. Let

f

hom

and thefamily

(f

ε

)

ε>0

be integrandson

Ω × R

N

satisfyingtheassumptions

(a1)to (a5) of Ÿ2.4for some ommon positive onstants

α, β

. Forany

ε >

0

, we denethefun tional

F

ε

: L

p

_{(Ω) → [0, +∞]}

by

F

ε

(v) :=







Z

Ω

f

ε

(x, ∇v(x))dx

if

v

∈ W

1,p

0

(Ω)

,

+∞

elsewhere

.

Forany

ε >

0

we onsiderthe

k

-theigenvalueproblemfor

F

ε

,whi h anbewritten thankstoCorollary3.6inthefollowingway

λ

k

_ε

:=

min

G∈G

p

_k

(Ω)

u∈G

sup

Z

Ω

f

ε

(x, ∇u)dx :=

min

G∈G

_k

p

(Ω)

u∈G

sup

F

ε

(u).

We are interested in the onvergen e of the family

(λ

k

ε

)

ε>0

as

ε

→ 0

. Following se tion Ÿ3, weshall denote by

K

s

(Ω)

bethe set whose elements are the ompa t symmetri subsetsof

L

p

_(Ω)

,equippedwith theHausdordistan e

d

H

indu edon bythenorm

k.k

p

. Wealsodene

J

k

ε

: K

s

(Ω) → [0, +∞]

asfollows:

J

_ε

k

(G) :=



sup{F

ε

(u)

1

p

: u ∈ G}

if

G

∈ G

k

p

(Ω),

+∞

otherwise

.

Thenotations

F

hom

,

λ

k

hom

and

J

k

hom

are dened inthesamewayasabove. Sin e anyelement

G

∈ G

p

k

(Ω)

maybe onsideredasanelementof

K

s

(Ω)

onehas

∀ε > 0

λ

k

ε

= min{J

ε

k

(G)

p

: G ∈ K

s

(Ω)}.

As adire t onsequen eofTheorem 3.3 wethen getthe following onvergen e

result.

Theorem5.1. Letthe above hypotheseshold,andassumethat thefamily of

fun -tionals

(F

ε

)

ε>0

Γ

- onvergesin

L

p

_(Ω)

to

F

hom

as

ε

→ 0

,thenforanypositiveinteger

k

onehas

λ

k

_ε

→ λ

k

hom

=

min

G∈G

p

k

(Ω)

sup

u∈G

F

hom

(u)

as

ε

goesto

0

.

In[5℄theauthors onsiderthehomogenizationfornon-lineareigenvalueproblems

relatedto afamilyofmonotoneellipti operatorsof theform

A

ε

:= −div(a

ε

(·, ·))

whi h

G−

onvergetoanoperator

A

hom

ofthesameform. Undertheassumptions (h1-5)theoperators

A

ε

arethesub-dierentialsofintegralfun tionalsofthetype

F

ε

whoseintegrandssatisfy(a1-5),seeŸ2.4. Theorem5.1thenimpliesthatforany positive integer

k

the generalizedsequen e of the

k

-th eigenvalues(as dened in se tionŸ2.4)oftheoperators

A

ε

onsideredin[5℄ onvergestothe

k

-theigenvalue ofthelimitoperator.

Remark 5.2. Theorem 3.3 also yields that

Γ − lim inf J

k

ε

≥ J

hom

k

on

K

s

(Ω)

, but unlike in the previousse tion one an't expe t the family

(J

k

ε

)

ε

to

Γ

- onvergeto

J

k

hom

. Indeed,inTheorem5.1itisonlyassumedthat

(F

ε

)

ε>0

Γ

- onvergesin

L

p

_(Ω)

to

F

hom

,whereasinse tionŸ4thefun tionals

F

ε

(v) = k∇vk

1

ε

notonly

Γ

- onverge butalsopointwise onvergeto

F

0

(v) = k∇vk

∞

on

C

0

(Ω)

,whi hallowstoobtainthe

Γ

-limsupestimatefor

(J

k

ε

)

ε

. Of oursetheproofofTheorem4.1 ouldbeadapted toget that

(J

k

ε

)

ε

Γ

- onvergesto

J

k

hom

when

(F

ε

)

ε>0

also onvergespointwise,but thislast hypothesisisnotusualinthe ontextofhomogenization.

A knowledgments The authors wish to thank G. Belletini, A. Garroni and

R.Mahadevanusefuldis ussionsandreferen es. LDPgratefullya knowledgesthe

support provided by the MIUR PRIN04 proje t Cal olo delle Variazioni. He

alsoa knowledgesthehospitalityoftheMathemati alS ien esResear hInstitute,

Berkeley,CA94720. TCalsoa knowledgesthehospitalityoftheCentrode

Mode-lamientoMatemáti o,UniversidaddeChile(Santiago).

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