On generalized Sobolev algebras and their applications.

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Séverine Bernard, Silvère Paul Nuiro

To cite this version:

Séverine Bernard, Silvère Paul Nuiro. On generalized Sobolev algebras and their applications.. Topo- logical Methods in Nonlinear Analysis, Juliusz Schauder University Centre for Nonlinear Studies, 2005, 25 (2), pp.375-390. �hal-00004903�

ccsd-00004903, version 1 - 11 May 2005

Séverine Bernard and Silvère P. Nuiro

In thelasttwodeades, manyalgebras ofgeneralizedfun-

tionshave been onstruted,partiularly theso-alled gener-

alized Sobolev algebras. Our goal is to study the latter and

some of their main properties. In this framework, we pose

and solvea nonlinear degeneratedDirihlet problem with ir-

regular datasuh as Dirageneralized funtions.

Key words : nonlinear degenerate Dirihlet problem, generalized solu-

tion, Sobolev algebra,non positive solution.

2000 MSC:35J70, 46F30,46E35, 35D05, 35B50.

1. Introdution

A theoretial study of most of thewell-known algebras of generalized fun-

tions haspointed out two fundamental strutures. Therstone is thealge-

brai struture of a solidfator ring C ^of generalized numbers. The seond one is the topologial struture dened by a family P ^{of seminorms, on}

a loally onvex linear spae E^{, whih is also an algebra. These algebras}

have been denoted by A(C, E,P) ^{and one speaks of} (C, E,P)^{-algebras of}

generalized objets. The denition overs most of the well-known algebras

of generalized funtions, as for example, the Colombeau simplied algebra

[3℄, Goursat algebras[13℄ and asymptoti algebras [4℄. Ontheother hand,

speial hoies for E^, P ^and C ^{also allow the} introdution of some new al- gebras. One of them is the so-alled Egorov extended algebra, beause of

thesimilaritywiththeEgorov [5℄algebra ofgeneralized funtions. We have

beeninterestedinworkingwithinthe frameworkoftheso-alledgeneralized

SobolevalgebrasbasedonthelassialSobolevspaes. AsE ^{isadierential}

algebra, the main interest of these algebras is to give a framework whih is

well suitable to solve many non linear dierential problems with irregular

data. Themethodisbasedontheextensionofamappingfrom(E₁,P1)^into (E₂,P2) ^{to a mapping from} A(C1, E₁,P1) ^into A(C2, E₂,P2)^{. This method}

hasbeenintrodued,intheframeworkofasymptotialgebras,byA.Delroix

and D.Sarpalezos [4℄,and used,intheframework of (C, E,P)^{-algebras, to}

solveanonlinearDirihletproblem[12℄andanonlinearNeumannproblem

[11℄,bothwithirregulardatabyJ.-A.Martiand S.P.Nuiro. Inthispaper,

our goal is to lift upthe generalized Sobolev algebras, bygiving more lear

denitions of all the statements and general results in this framework, in

order to work more easily with these algebras. We introdue the rst ex-

ample of ordered generalized Sobolev algebras, whihallows us to pose and

eventually solvean obstaleproblemwithirregular data. We alsopoint out

some suient properties for the existene of an embedding of some spae

into ageneralized Sobolev algebra. Intheframework ofgeneralized Sobolev

algebra,weareableto solveanonlineardegeneratedDirihletproblem[12℄

withweakerassumptions.

Consider Ω ^{an open bounded domain of} IR^{d (}d ∈ IN^{∗) with a lipshitz}

ontinuousboundary ∂Ω^{,we an state thisformalproblem :} (P)

−∆Φ (u) +u=f in Ω, u=g on ∂Ω,

wheref ^andg^{arenonsmoothfuntionsdenedon}Ω^and∂Ωrespetively,Φ

aninreasing real-valueddierentiablefuntion denedon IR^sothatΦ^{′ isa}

ontinuousboundedfuntionthatanvanishonanitesetofdisretepoints

of IR^{. This is a} quasilinear diusion type problem, withnon homogeneous Dirihletonditionontheboundary. Oneanremarkthattheformalseond

order dierential operator L = −div (Φ^′(.)∇x) +Id ^{is a} degenerated one, beauseΦ^{′anvanish. Thus,}(P)^{isaDirihletnonlinearellipti}degenerated problem. Inorder to solve this problem, we introdue an auxiliary problem

byusingan artiialvisosityregularization dependingon aparameter ε^.

2. Speial types of generalized algebras

2.1. Denitions.Letus,rst,statethatIK^isIR^orCl^,and1I = (1I_ε)_ε^where 1Iε= 1 ^{for all} ε.^Thegeneralized algebras onstrutedfromE^{,a normed}IK^-

algebra,are partiular aseof (C, E,P)^{-algebras [10℄, [12℄,[11℄, [13℄.}

Consider a subring A ^{of the ring} IK^{]0,1] so that} 1I ∈ A^{, and whih, as a}

ring, issolid(with ompatible lattie struture) inthefollowing sense:

Denition 1. A issaid tobe solid iffrom(s_ε)_ε∈A^and|tε| ≤ |sε|^foreah ε∈]0,1] ^{it follows that} (tε)ε∈A^.

We also onsideran idealIA^of A ^{whih is solidaswell,and sothat}

(1) ∀(rε)_ε ∈IA, lim

ε→0rε= 0.

Then, we introdue the fator ring C = A/IA^{, whih is alled a ring of}

generalized numbers.

Denition2. LetE ^{beanormedalgebra. Weshallall}N-generalizedalgebra all fator algebra

A(C, E) =HA(E)/IIA(E),

where

HA(E) ={(uε)_ε∈E^]0,1] / (kuεkE)_ε ∈A⁺}

and

IIA(E) ={(uε)ε∈E^]0,1] / (kuεkE)ε∈I_A⁺},

when k · kE ^{is the norm on} E^, A⁺ = {(rε)ε ∈ A / ∀ε > 0, rε ∈ IR₊} ^and I_A⁺ = {(rε)ε ∈ IA / ∀ε > 0, rε ∈ IR₊}^{. Its ring of} generalized numbers is dened asthe ring

HA(IK)/IIA(IK) =C=A/IA.

Remark 1. We remark that the notation is A(C, E) ^{instead of} A(C, E,P)

sine the family P ^{isredued toone single element. The algebra} A(C, E) ^is

also a vetor spae on the eld IK^.

Example 1. With

IA=n

r = (rε)ε∈IR^]0,1]| ∀k∈IN^∗, |rε|=O ε^ko

and

A=n

r= (rε)ε∈IR^]0,1]| ∃k∈ZZ, |rε|=O ε^ko

,

we obtain a polynomialgrowthtype N-generalized algebra.

Example 2. We take

IA=n

r= (rε)ε∈IR^]0,1]| ∃ε₀ ∈]0,1], ∀ε∈]0, ε₀], rε= 0o ,

and A = IR^{]0,1]. With suh} A ^and IA^{, we obtain another} N-generalized algebra.

Example 3. When E ^{is a Sobolev algebra (that is, for example, on the}

form W^m+1,p(Ω)∩W^m,∞(Ω)^{, with}m ∈]0,+∞[^, p∈[1,+∞[^andΩ ^{an open}

subset of IR^{d (}d∈IN^∗)), respetively a Banah algebra, we will speak about generalized Sobolev algebra, respetively generalized Banah algebra, instead

of N-generalized algebra.

2.2. Embeddings and weak equalities.In the following paragraph, we

aregoingto show awayto embed E ^into A(C, E)^.

Proposition 1. The mapping i₀ ^{dened on} E^{, by :}

∀u∈E, i₀(u) =cl(u1I_ε)_ε,

islinear and one-to-one fromE ^intoA(C, E)^.

Proof.For every u∈E^{,we have :} (ku1Iεk_E)_ε=kuk_E1I. ^Furthermore, as

kuk_E ∈IK ^and 1I∈A^{,there exists}λ∈IN ^sothat

∀ε, kuεk_E ≤λ1I_ε,

andobviouslyλ1I∈A^{+. Asaonsequeneofthesolidpropertywhihimplies}

that (u_ε)_ε ∈ HA(E)^{,we have} i₀(u)∈ A(C, E)^{. It an easily be proved that} i₀ ^{islinear and}one-to-one.

Denition 3. The mapping i₀ ^from E ^into A(C, E)^{, dened in} proposition 1, will be the so-alled trivial embedding of E ^into A(C, E)^.

WeanalsoembedsometopologialvetorspaeintoA(C, E)^{. Let}(G,T)

be a Hausdor topologial vetor spae so that there exists a ontinuous

linearmapping j ^from (E,k.k_E) ^into (G,T)^.

Denition 4. T ∈G ^andU =cl(uε)_ε∈ A(C, E) ^are (G,T)^{-assoiated if} j(u_ε)→T ⁱⁿ (G,T) ^as ε→0.

It will be denoted by U ^G,T∼ T.

Remark 2. Thisdenition does not dependon the hosen representative of

U^{. Indeed, let} (eε)_ε ∈ IIA(E). ^Therefore, lim

ε→0keεk_E = 0, ^{whih means that} eε →0 ⁱⁿ (E,k.k_E) ^as ε→0. Consequently, we have j(eε) →0 ⁱⁿ (G,T)

as ε→0.

Denition 5. Assume that U =cl(uε)_ε, V =cl(vε)_ε ∈ A(C, E)^{. We shall}

say that U ^andV ^are (G,T)^{-weaklyequals if} (U −V)^G,T∼ 0.

It will be denoted by U ^G,T≃ V.

Proposition 2. Assumethat for every T ∈G^{,there exists}(uε)_ε∈ HA(E)^,

so that

j(uε)→T ⁱⁿ (G,T), ^as ε→0.

Then,thereexists,atleast,anembeddingiG^from(G,T)^intotheN-generalized algebraA(C, E)^. Furthermore, if,for allv∈E^{,there exists}(u_ε)_ε∈ HA(E)^,

so that (uε−v)_ε∈ IIA(E)^,then

(2) ∀u∈E, (i_G◦j) (u)^G,T≃ i₀(u).

Proof.For every T ∈G^{, thereexists}(uε)_ε∈ HA(E)^,sothat j(u_ε)→T ⁱⁿ (G,T) ^asε→0.

Let us state iG(T) = cl(uε)_ε^{. The mapping} iG ^from G ^into A(C, E) ^is

obviouslylinear. Letus provethatiG ^isone-to-one. IfiG(T) = 0ⁱⁿA(C, E)

then

iG(T) =cl(eε)_ε ^for (eε)_ε∈ IIA(E).

Wehave eε→0 ⁱⁿ(E,k.k_E)^{whihimpliesthat}j(eε)→0ⁱⁿ (G,T)^,when-

ever ε→0^{. This leads to} T = 0ⁱⁿG^{, beause} (G,T) ^{isa Hausdorspae.}

Theseond propertyis obvious.

Remark 3. If there exists anothersuh embedding i^′_G ^from(G,T) ^{into the}

N-generalized algebra A(C, E) ^then

∀T ∈G, i_G(T)^G,T≃ i^′_G(T).

Example 4. Let j ^{be the anonial embedding of}

L^∞(Ω),k.k_L∞(Ω)

in

H⁻²(Ω), σ H⁻²(Ω), H₀²(Ω)

, where σ H⁻²(Ω), H₀²(Ω)

denotes the weak

topology on H⁻²(Ω)^{. We will say that} T ∈ H⁻²(Ω) ^and U = cl(uε)ε ∈ A(C, L^∞(Ω)) ^are H⁻²(Ω)^{-assoiated if}

j(uε)→T ⁱⁿ H⁻²(Ω), σ H⁻²(Ω), H₀²(Ω)

, ^as ε→0,

and we will denote U ∼² T^{. Moreover, we will say that} U,V ∈ A(C, L^∞(Ω))

are H⁻²(Ω)^{-weakly equals if}U − V ∼² 0 ^{and we will denote} U ≃ V^{2 .}

2.3. Mappingon N-generalizedalgebra.Theideaofextensionofmap-

ping has been introdued by A. Delroix and D. Sarpalezos [4℄, in the

framework of asymptoti algebras. But it is, in fat, a partiular ase of

denitionof mapping onA(C, E)^-algebras.

If θ = (θε)_ε ^{is a family of mappings from a normed algebra} (E,k.k_E)

into a normed algebra (F,k.k_F)^{, one an view} θ ^{asa mapping from theN-}

generalized algebra A(C, E) ^{into the} N-generalized algebra A(D, F)^,where

we have set C=A/IA ^and D=B/IB ^when A, IA, B ^and IB ^{areasin Ÿ2.1.}

One remarksthatthe extension theoremof A.Delroix and D. Sarpalezos

[4℄ deals withthe asewhere θ= (θ)_ε^.

Theorem 1. Let E ^and F ^{be two normed algebras and} (θε)ε ^{a family of}

appliationsof E ⁱⁿ F^{. We assumethat}

1) A⊂B ^andI_A⊂I_B^,

2) there existsafamilyof polynomialfuntions(Ψε)ε ^{ofonevariable with}

oeients in A₊ ^{so that}

∀ε >0 , ∀x∈E , kθε(x)kF ≤Ψε(kxkE),

3) there exists two families of polynomial funtions (Ψ¹_ε)ε ^and (Ψ²_ε)ε ^of

one variable with oeients in A₊ ^{so that} Ψ²_ε(0) = 0 ^{for all} ε > 0^,

and

∀ε >0 , ∀x, ξ∈E , kθε(x+ξ)−θε(x)kF ≤Ψ¹_ε(kxkE)Ψ²_ε(kξkE).

Then there exists an appliation Θ ^: A(C, E) →A(D, F)^{, whih assoiates} cl(θε(xε))ε ^with cl(xε)ε^.

Proof.First, let (xε)ε ^{be in} HA(E) ^{and let us show that} (θε(xε))ε ^{is in}

HB(F)^{. We have} (kxεkE)_εⁱⁿA₊ ^so(Ψ_ε(kxεkE))_ε ^{isalso in}A₊^,sine(Ψ_ε)_ε

has oeients in A₊^{. Thus} (kθε(xε)kF)ε) ^{belongs to} A₊ ⊂ B₊^{, due to}

(1) and (2), whih implieswhat we want. Then, let (iε)ε ^{be in}IIA(E) ^and

let us show that (θ_ε(x_ε+i_ε)−θ_ε(x_ε))_ε ^{is in} IIB(F)^{. Sine} (kxεkE)_ε ^and

(kiεkE)ε ^arerespetivelyinA₊^and I_A^+then(Ψ¹_ε(kxεkE))ε ^and(Ψ²_ε(kiεkE))ε

arerespetivelyinA₊ ^andI_A^{+ ,sine,for} i∈ {1,2},(Ψⁱ_ε)ε ^hasoeientsin

A+^{. Then,} (Ψ¹_ε(kxεkE)Ψ²_ε(kiεkE))ε ^isinI_A^{+. Thus}(kθε(xε+iε)−θε(xε)kF)ε

belongsto I_A⁺⊂I_B^{+,due to (1)and (3), whih impliestherequired result.}

Asa onsequene,we obtain thefollowing result.

Proposition 3. Assume that A ⊂ B ^and I_A ⊂ I_B^{. If} (θ_ε)_ε ^{is a family of}

ontinuous linear mappings from a normed algebra E ^{into a normed algebra} F^{, then} (θ_ε)_ε ^{also denes a mapping} Θ^from A(C, E) ^into A(D, F)^.

Example 5. Let Ω ^{be an open subset of} IR^{d and} E = H¹(Ω)∩L^∞(Ω)

with kuk_E = kuk_L∞(Ω) +kuk_H1(Ω)^{. The anonial embedding} i : u 7→ u

is ontinuous as well as linear from the Banah algebra E ^{into the Banah}

algebra L^∞(Ω)^{. Obviously, themapping} i^veriesallthe assumptions of the previous proposition;this iswhy we an dene itsextension I ^{asa mapping}

fromA(C, E) ^intoA(C, L^∞(Ω))^.

Inthe same way,one an prove that:

Proposition 4. Assume that (θε)_ε ^{is a family of mappings from a normed}

algebra E ^{intothe topologial eld} (IK,|.|)^{,so that}

• ^{there existsafamilyof polynomialfuntions}(Ψε)ε ^{ofonevariable with}

oeients in A₊ ^{so that}

∀ε >0 , ∀x∈E , |θε(x)| ≤Ψ_ε(kxkE),

• ^{there exists two families of polynomial funtions} (Ψ¹_ε)ε ^and (Ψ²_ε)ε ^of

one variable with oeients in A₊ ^{so that} Ψ²_ε(0) = 0 ^{for all} ε > 0^,

and

∀ε >0 , ∀x, ξ∈E , |θε(x+ξ)−θε(x)| ≤Ψ¹_ε(kxkE)Ψ²_ε(kξkE).

ThenthereexistsanappliationΘ^: A(C, E)→ C^{,whihassoiates}cl(θε(xε))ε

withcl(xε)ε^.

Remark 4. If θ ^{is a ontinuous linear mapping from a normed algebra} (E,k.k_E) ^{into the topologial eld} (IK,|.|)^{, then} θ ^{also denes a mapping,}

denoted by Θ^{, from}A(C, E) ^{into the fatorring} C=A/IA^.

2.4. An example of ordered generalized Sobolev algebra.Consider

A ^and IA ^{as in Ÿ2.1, the Sobolev algebra} L^∞(Ω)^{, endowed with its usual}

topology,withΩ^{anopenbounded subsetof} IR^{d. Thus, we anonsiderthe}

algebra A(C, L^∞(Ω))^{. It is easy to prove, bymeans of theorem 1, thatthe}

mapping

p:L^∞(Ω) → L^∞(Ω)

u 7→ u⁺= sup{u,0}= ¹₂(u+|u|)

an be extendedasa mappingP ^fromA(C, L^∞(Ω))^{into itself,dened by:}

∀U =cl(uε)ε ∈ A(C, L^∞(Ω)), P(U) =cl(p(uε))ε,

due to the following relation :

∀r, s∈IR,

(r+s)⁺−r⁺ ≤ |s|.

We arenowableto state thefollowing result:

Proposition 5. The generalized Sobolev algebra A(C, L^∞(Ω)) ^{is partially}

ordered by the following binaryrelation :

∀U, V ∈ A(C, L^∞(Ω)), U ≤V ⇐⇒ P(U −V) = 0.

Proof.Obviously, the relation ≤^{is reexive, then we have to prove, for} U, V, W ∈ A(C, L^∞(Ω))^{,that :}

U ≤V ^andV ≤U ⇒ U =V;

(3)

U ≤V ^and V ≤W ⇒ U ≤W.

(4)

We state U =cl(u_ε)_ε^,V =cl(v_ε)_ε ^and W =cl(w_ε)_ε^.

Proof of property (3) : If U ≤ V ^and V ≤ U ^{then, there exists} (ϕε)_ε ^and (ψε)_ε ⁱⁿIIA(L^∞(Ω))^sothat(uε−vε)⁺ =ϕε ^and(vε−uε)⁺ =ψε^{. As,}

uε−vε = (uε−vε)⁺−(vε−uε)⁺=ϕε−ψε,

itfollows that(uε−vε)_ε= (ϕε−ψε)_ε∈ IIA(L^∞(Ω)),^whene U =V^.

Proof of property(4 ): IfU ≤V ^and V ≤W ^{thenwe have}

(uε−vε)⁺ L^∞(Ω)

ε∈IA ,

(vε−wε)⁺ L^∞(Ω)

ε ∈IA.

Bymeans of thesolidproperty, we dedue,from thefollowing inequality:

(u_ε−w_ε)⁺

L^∞(Ω) ≤

(u_ε−v_ε)⁺

L^∞(Ω)+

(v_ε−w_ε)⁺ L^∞(Ω),

that (uε−wε)⁺

ε ∈ IIA(L^∞(Ω)),^{whih yields} P(U −W) = 0^{,that is to}

sayU ≤W^.

Proposition 6. For all u, v ∈ L^∞(Ω)^{, we have} i0(u) ≤ i0(v) ^{if, and only}

if, u≤v ⁱⁿ L^∞(Ω)^{, that is} u≤v ^{almost everywhere in} Ω^.

Proof.If i₀(u) ≤ i₀(v) ^then P(i₀(u)−i₀(v)) = 0^. Consequently, there exists(ϕε)_ε,(eε)ε∈ IIA(L^∞(Ω))^sothat(u−v+eε)⁺ =ϕε^{,sine we have}

uε−vε=u−v+eε ^{for all}ε^{. Taking into aount that} ϕε→0 and eε →0 ⁱⁿ L^∞(Ω) ^,as ε→0,

itmaybeseenthat(u−v+eε)⁺=ϕε →0 ^{a.e. in} Ω,^whene(u−v)⁺= 0 ^{a.e. in}Ω^{,sine one aneasily prove that}

(u−v+eε)⁺→(u−v)^{+ in} L^∞(Ω) ^{, as} ε→0.

It meansthat u≤v ^{a.e. in}Ω.

Conversely, ifu ≤v ^{a.e. in}Ω ^then(u−v)⁺ = 0 ^{a.e. in}Ω^{. By denition}

of P ^andi₀^{,this leads to} i₀(u)≤i₀(v)^.

Proposition 7. Let u∈L^∞(Ω) ^andU ∈ A(C, L^∞(Ω))^{. If} U ^L1∼^(Ω)u ^(here L¹(Ω) ^{isendowed withits usual topology) and} U ≤0 ^then u≤0 ^{a.e. in} Ω^.

Proof.We set U = cl(uε)ε^{. Sine} U ^L1∼^(Ω) u ^{then, as} ε ^{goes to} 0^, uε →u ⁱⁿL¹(Ω)^,whihgivesu⁺_ε →u^{+ in}L¹(Ω)^{,bymeansoftheLebesgue}

dominated onvergene theorem. Sine U ≤ 0 ^then P(U) = 0^{. Conse-}

quently, there exists a sequene of funtions (ϕε)_ε ∈ IIA(L^∞(Ω)) ^{so that} u⁺_ε =ϕε ^forall ε^{. Taking into aount that}

ϕ_ε→0 ⁱⁿ L^∞(Ω) ^{, as}ε→0,

we nd that u⁺_ε = ϕε → 0 ^{a.e. in} Ω, ^whene u⁺ = 0 ^{a.e. in} Ω, ^whih

impliesu≤0 ^{a.e. in}Ω.

3. Solution of the nonlinear degenerate Dirihlet problem

After having solved the auxiliary problem by using an artiial visosity

regularization depending on aparameter ε^{,we solve our main problem}(P)

(see setion 1), ina generalized Sobolev algebra withthe lassial equality

and with the weak one dened in example 4. Then we perform a little

qualitative study ofthe solution.

3.1. The regularized Dirihlet problem.Letus set

V⁺_A=

(rε)ε ∈A⁺ / ∀ε >0, rε∈]0,1]; lim

ε→0rε= 0 ; 1

rε

ε

∈A⁺

.

Assumethat V⁺_A6=∅ ^{andthen, for all} (r_ε)_ε ⁱⁿV⁺_A^,set Φ_ε = Φ +r_εId^{. This}

setiononsistsinproving thefollowing proposition:

Proposition 8. If f ∈L^∞(Ω) ^andg ∈L^∞(∂Ω) ^{then there exists one, and}

onlyone, funtion u∈H¹(Ω)∩L^∞(Ω)^{solutionof the} regularized problem

(Pε)

−∆Φε(u) +u=f in Ω, u=g on ∂Ω.

Proof.Thisproofgoesinthree steps.

1)Maximum's priniple

We aregoing toprove that ifu∈H¹(Ω)^{isa solutionof thisproblem then} m≤u≤M ^{a.e. in}Ω,

with m = min{inf

Ω f , inf

∂Ωg} ^and M = max{sup

Ω

f , sup

∂Ω

g}, ^{whih means}

thatu ^belongsto L^∞(Ω)^.

Indeed,for suh a u^{,we have, for all}v ⁱⁿH₀¹(Ω) Z

Ω

∇Φε(u)∇vdx+ Z

Ω

uvdx= Z

Ω

f vdx,

where dx^{denotes the Lebesgue measureon} Ω^{. Letus onsider thefuntion} v= (Φε(u)−Φε(M))^{+ then}v ^isinH₀¹(Ω)^,so

Z

Ω

(∇(Φε(u)−Φε(M))⁺)²dx+ Z

Ω

u(Φε(u)−Φε(M))⁺dx=

Z

Ω

f(Φε(u)−Φε(M))⁺dx,

sine Φ_ε(M) ^{isa onstant.} Consequently,

k(Φε(u)−Φ_ε(M))⁺k²_H1 0(Ω)=

Z

Ω

(f −M)(Φ_ε(u)−Φ_ε(M))⁺dx

− Z

Ω

(u−M)(Φε(u)−Φε(M))⁺dx.

By denitionof M^{,the rstintegral isnegative and, sine thefuntions}Id

and Φε ^{areinreasing,the seond oneis non negative. Then}

k(Φε(u)−Φε(M))⁺k²_H1

0(Ω)≤0,

thatisΦ_ε(u)≤Φ_ε(M) ^{a.e. in}Ω^{,whihimpliestherstpartoftherequired}

result, sine Φε ^{is an inreasing funtion. For the seond part, we use a}

similarmethod bytakingv= (Φε(u)−Φε(m))^−.

2)Existene ofa solutioninH¹(Ω)

This resultis obtained byusing the Shauder's xed point theoremrelated

to aweaklysequentiallyontinuous mapping froma reexiveand separable

Banah spaeinto itself. Letusonsiderw₀ ∈H¹(Ω)^{theunique solutionof}

thefollowing linearDirihlet problem:

−∆w0 = 0 ⁱⁿΩ, w₀ = g ^on∂Ω.

Then a solution of the regularized problem is of the form w₀ +w^{, with} w∈H₀¹(Ω)^{and for all}v ⁱⁿH₀¹(Ω)^,onehas

Z

Ω

Φ^′_ε(w0+w)∇w0∇vdx+

Z

Ω

Φ^′_ε(w0+w)∇w∇vdx+

Z

Ω

(w0+w)vdx= Z

Ω

f vdx.