Degenerate elliptic equations with nonlinear boundary conditions and measures data

Degenerate elliptic equations with nonlinear boundary conditions and measures data

FUENSANTAANDREU, NOUREDDINE IGBIDA, JOSE´ M. MAZON´

ANDJULIAN´ TOLEDO

Dedicated to our friend Lucio Boccardo on the occasion of his 60th birthday.

Abstract. In this paper we study the questions of existence and uniqueness of solutions for equations of type−diva(x,Du)+γ (u) µ1, posed in an open bounded subsetofR^N, with nonlinear boundary conditions of the form a(x,Du)·η+β(u)µ2. The nonlinear elliptic operator diva(x,Du)is modeled on thep-Laplacian operatorp(u)=div(|Du|^p−2Du), with p> 1,γ andβ are maximal monotone graphs inR²such that 0∈γ (0)∩β(0)and the dataµ1 andµ2are measures.

Mathematics Subject Classification (2000): 35J60 (primary); 35D05 (sec- ondary).

1.

Introduction

The purpose of this paper is to establish existence and uniqueness of solutions for a degenerate elliptic problem with nonlinear boundary condition of the form

(S_µ^γ,β

1,µ2)





−diva(x,Du)+γ (u)µ1 in a(x,Du)·η+β(u)µ2 on∂,

where is a bounded domain in

R

^N with smooth boundary ∂, the function a : ×

R

^{N →}

R

^N is a Carath´eodory function with growth of order p − 1 (p > 1) with respect to the gradient, satisfying the classical Leray-Lions con- ditions, η is the unit outward normal on ∂andµ1, µ2 are measures such that µ1 = µ1 , µ2 = µ2 ∂and µ1 +µ2 is a diffuse measure (it does not charge sets of zero p-capacity). The nonlinearitiesγ andβare maximal monotone Received October 14, 2008; accepted in revised form March 3, 2009.

graphs in

R

^2(see, e.g., [16]), 0 ∈ γ (0)∩β(0), satisfying rather general assump- tions. In particular, they may be multivalued and this allows to include the Dirichlet condition (takingβto be the monotone graphDdefined by D(0)=

R

) and the non homogeneous Neumann boundary condition (taking β to be the monotone graph N defined byN(r) = 0 for allr ∈

R

) as well as many other nonlinear fluxes on the boundary that occur in some problems in Mechanics and Physics (see,e.g., [25]

or [15]). Note also that, sinceγ may be multivalued, problems of type(S_µ^γ,β_1,µ2)ap- pear in various phenomena with changes of state like the multiphase Stefan problem (cf.[21]) and in the weak formulation of the mathematical model of the so called Hele-Shaw problem (cf.[23] and [26]).

In the particular casea(x, ξ)=ξ,the problem(S^γ,β_µ1,µ2)reads (L^γ,β_µ1,µ2)





−u+γ (u)µ1 in

∂_ηu+β(u)µ2 on∂,

where ∂_ηu simply denotes the outward normal derivative of u. For this kind of problems in the homogeneous case,µ2≡0, the pioneering works are the paper by H. Brezis [15], in which problem(L^γ,β_µ

1,0)is studied forγ the identity,βa maximal monotone graph and µ1 ∈ L²(), and the paper by H. Brezis and W. Strauss [20], in which problem (L^γ,β_µ

1,0)is studied for µ1 ∈ L¹() andγ, β continuous nondecreasing functions from

R

^into

R

^{with γ ≥ >} 0. These works were extended by Ph. B´enilan, M. G. Crandall and P. Sacks [9] to the case of anyγ and βmaximal monotone graphs in

R

²such that 0∈γ (0)∩β(0).

In [1, 2] and [4], the results of [9] are extended by proving the existence and uniqueness of weak (or entropy/renormalized) solutions for the general nonhomo- geneous problem(S^γ,β_φ,ψ), withφ ∈L¹()andψ ∈L¹(∂), which is quite different from the homogeneous case. The arguments of the proofs are very connected to the nature of the nonlinearitiesγ andβ.More precisely, the following cases are studied separately,

(A)D(γ )=

R

^and,D(β)=

R

^{or diva}(x,Du)=p(u), (B)ψ ≡0 and, D(β)=

R

^{or diva(x,Du)=p(u),} (C)

R

^{= D(γ )⊂ D(β)}(the obstacle problem).

For the case where the dataφandψare Radon measures, the problem is again dif- ferent. Our aim in this paper is to extend some of the above results to this situation.

There is a large literature on elliptic problems with measure data, mainly for the homogeneous Dirichlet problem andγ ≡0, that is, for the problem

(S_µ,^0,D₀)





−diva(x,Du)=µ in

u=0 on∂.

In the linear case, existence and uniqueness of solutions of(S_µ,^0,D₀)was obtained by G. Stampacchia [32] by duality techniques. In the nonlinear case the first attempt to solve problem (S_µ,^0,D₀)was done by L. Boccardo and T. Gallou¨et, who proved in [11] and [12] the existence of weak solutions of (S_µ,^0,D₀)under the assumption p>2−_N¹. On the question of uniqueness, even for the particular caseµ∈L¹(), the definition of weak solution is not enough in order to get uniqueness. It was necessary to find some extra conditions on the distributional solutions of(S_µ,^0,D₀)in order to ensure both existence and uniqueness. This was done by Ph. B´enilanet al., for the case of measures in L¹(), by introducing the concept ofentropy solution in [6], and by P. L. Lions and F. Murat in an unpublished paper where the concept of renormalized solutionwas introduced. For diffuse measures, that is, for measures inL¹()+W^−1,p(), the problem was solved by L. Boccardo, T. Gallou¨et and L.

Orsina in [13] in the framework of entropy solutions, and for general measures by G. Dal Masoet al.in [22] in the framework of renormalized soltuions.

The study of the homogeneous Dirichlet problem for the Laplacian andγ ≡0 was initiated by Ph. B´enilan and H. Brezis in 1975 (see [7]) for the particular case γ (r)=g_p(r):= |r|^p−1r. They proved the existence of weak solutions of problem

(L^γ,_µ,^D₀)





−u+γ (u)=µ in

u=0 on∂,

for any measureµifp< _N^N₋₂(N ≥2), and non existence if p≥ _N^N₋₂(N ≥3) for µ=δa, witha∈. Problem (L^γ,_µ,^D₀) was also studied by P. Baras and M. Pierre [5].

Recently it has been studied by H. Brezis, M. Marcus and A. C. Ponce in [18], where the general case of a continuous nondecreasing nonlinearityγ (r),γ (0)=0, is dealt with (see also [10,33] for the particular caseγ (r)=e^r−1). The same problem has been studied by H. Brezis and A. C. Ponce [19] in the case Dom(γ )=

R

^closed.

The case Dom(γ )=

R

open has been studied by L. Dupaigne, A. C. Ponce and A.

Porretta [24].

The study of nonlinear equations involving measures as boundary condition was initiated by A. Gmira and L. Veron [27]. They proved the existence of weak solutions of problem

(G V)





−u+ |u|^q−1u=0 in

u=µ on∂,

for any Radon measure µon ∂in the subcritical case 1 < q < ^N_N⁺₋¹₁. In the supercritical case, q ≥ ^N_N⁺₋¹₁, existence of solutions no longer holds; for instance, the problem has no solution if the measureµis concentrated at a single point. M.

Marcus and L. Veron in [30] characterized the Radon measuresµon∂for which problem(G V)has solution in the supercritical case, these measures are those that

are absolutely continuous respect to the Bessel capacity C2

q,q on∂. In the last years an extensive study of removable singularities and boundary traces for this type of problems has been done by M. Marcus and L. Veron (see [31] and the references therein).

The study of reduced measures initiated in [18] by H. Brezis, M. Marcus and A. C. Ponce for problem (L^γ,_µ,^D₀) has been developed in [19] by H. Brezis and A. C.

Ponce for problems of the form





−u+γ (u)=0 in

u=µ on∂,

where γ :

R

^→

R

is a nondecreasing continuous function withγ (r) = 0 for all r ≤0. In that paper the authors make the observation that in all the above problems the equation in is nonlinear but the boundary conditions is the usual Dirichlet boundary condition. They also point out that it would be interesting to investigate problems with nonlinear boundary conditions of type

(L^g₀¹_,µ^,β)







−u+u=0 in

∂u

∂η+β(u)µ on∂,

whereβis a maximal monotone graph in

R

². Observe that this problem is a partic- ular case of our general problem.

Let us briefly summarize the contents of the paper. In Section 2 we fix the notation and give some preliminaries. Section 3 deals with the different concepts of solution we use. The next section is dedicated to establish the existence and uniqueness results. Finally, the last section is devoted to the particular case of Dirichlet boundary conditions.

ACKNOWLEDGEMENTS. This work has been performed during the visit of the sec- ond author to the Universitat de Val`encia and the visits of the first, third and fourth authors to the Universit´e de Picardie Jules Verne. They thank these institutions for their support and hospitality. The authors have been partially supported by the Spanish MEC and FEDER, project MTM2008-03176.

2.

Preliminaries

Throughout the paper, ⊂

R

is a bounded domain with smooth boundary ∂, p>1,γ andβare maximal monotone graphs in

R

²such that 0∈γ (0)∩β(0)and the Carath´eodory functiona:×

R

^{N →}

R

^{N satisfies}

(H₁) there exists >0 such thata(x, ξ)·ξ ≥|ξ|^pfora.e.x ∈and for all ξ ∈

R

^N,

(H₂) there existsσ >0 and∈ L^p()such that|a(x, ξ)| ≤σ((x)+ |ξ|^p−1) fora.e.x ∈and for allξ∈

R

^{N, where p=} p−^p1,

(H₃) (a(x, ξ1)−a(x, ξ2))·(ξ1 −ξ2) > 0 for a.e. x ∈ and for all ξ1, ξ2 ∈

R

^{N, ξ1=ξ2.}

The hypotheses(H₁)-(H₃)are classical in the study of nonlinear operators in diver- gence form (cf., [29]). The model example of a functionasatisfying these hypothe- ses isa(x, ξ)= |ξ|^p−2ξ. The corresponding operator is the p-Laplacian operator p(u)=div(|Du|^p−2Du).

For 1 ≤ p < +∞, L^p() and W^1,p() denote respectively the standard Lebesgue space and Sobolev space, and W₀^1,p() is the closure of

D

() in W^1,p(). Foru ∈ W^1,p(), we denote byuorτ(u)the trace ofuon∂in the usual sense and byW

1 p,p

(∂)the setτ(W^1,p()). Recall that Ker(τ)=W₀^1,p(). We denote by

L

^{N theN}-dimensional Lebesgue measure of

R

^{N and by}

H

^N−1 the(N −1)-dimensional Hausdorff measure.

For an open bounded set U of

R

^N, we define the p-capacity relative toU, C_p(.,U), in the following classical way. For any compact subsetK ofU,

C_p(K,U)=inf

U|Du|^p; u∈

C

c^∞(U), u≥χ_K ,

whereχ_K is the characteristic function ofK; we will use the convention that inf∅ = +∞. The p-capacity of any open subsetO ⊂U is defined by

C_p(O,U)=sup

C_p(K); K ⊂O compact . Finally, the p-capacity of any Borel setA⊂U is defined by

C_p(A,U)=inf

C_p(O); O ⊂ Aopen .

A function u defined on U is said to be cap_p-quasi-continuous in A ⊂ U if for every ε > 0, there exists an open set B ⊆ U with C_p(B,U) < ε such that the restriction of uto A\B is continuous. It is well known that every function in W^1,p(U)has a cap_p-quasi-continuous representative, whose values are defined cap_p-quasi everywhere inU, that is, up to a subset ofU of zero p-capacity. When we are dealing with the pointwise values of a function u ∈ W^1,p(U), we always identifyuwith its cap_p-quasi-continuous representative.

We denote

sign₀(r):=







1 if r >0, 0 if r =0,

−1 if r <0,

sign⁺₀(r):=

1 if r >0, 0 if r ≤0.

Fork>0,

T_k(r):=max{−k,min{r,k}}, r ∈

R

and

T_k⁺(r):=min{r⁺,k}, r ∈

R

^. In [6], the authors introduce the set

T

^1,p()= {u:−→

R

measurable such thatT_k(u)∈W^1,p() ∀k>0}.

They also prove that for a given u ∈

T

^1,p(), there exists a unique (up to a.e.

equivalence) measurable functionv:→

R

^{N such that} DT_k(u)=vχ_{|v|<k} ∀k >0.

This function v will be denoted by Du. It is clear that if u ∈ W^1,p(), then v∈L^p()andv=Duin the usual sense.

As in [1],

T

tr^1,p()denotes the set of functionsu in

T

^1,p() satisfying the following conditions, there exists a sequenceu_ninW^1,p()such that

(a) u_nconverges tou a.e.in,

(b) DT_k(u_n)converges toDT_k(u)inL¹()for allk>0,

(c) there exists a finite measurable functionvon∂, such thatu_n converges tov a.e.in∂.

The functionvis the trace ofu in the generalized sense introduced in [1]. In the sequel, the trace ofu∈

T

tr^1,p()on∂will be denoted by tr(u)oru. Let us recall that in the caseu∈W^1,p(), tr(u)coincides with the trace ofu,τ(u), in the usual sense, and

Ker(tr)=

T

0^1,p(),

the space introduced in [6] to study(S_φ,^γ,₀^D). Moreover, for everyu∈

T

tr^1,p()and k > 0,τ(T_k(u)) = T_k(tr(u)). Ifφ ∈ W^1,p()∩L^∞(), thenu−φ ∈

T

tr^1,p() and tr(u−φ)=tr(u)−τ(φ).

Let us remark that ifu ∈

T

^{1,p(), thenuhas a capp}-quasi-continuous rep- resentative, which will be denoted byu; the cap_p-quasi-continuous representative can be infinite on a set of positive p-capacity (see [22]). If in addition the function u∈

T

^1,p()is assumed to satisfy the estimate

|DT_k(u)|^pd x ≤C(k+1) ∀k>0,

whereC is independent ofk, then the cap_p-quasi-continuous representative ofuis cap_p-quasi every where finite (see [22]).

From now on,is assumed to be a bounded domain in

R

^{N with∂of class} C¹. Then, is an extension domain (see [17]), so we can fix an open bounded subset U of

R

^{N such that ⊂ U}, and there exists a bounded linear operator E :W^1,p()→W₀^1,p(U)for which

(i) E(u)=u a.e infor eachu∈W^1,p(),

(ii) E(u)_W^1,p

0 (U) ≤ CuW^1,p(),where C is a constant depending only on p and.

We call E(u)an extension ofutoU. Ifu∈W^1,p(), 1< p≤ ∞, it is possible to give a pointwise definition of the trace τ(u)ofu on∂in the following way (see [34]), as E(u)∈ W₀^1,p(U), every point ofU, except possibly a set of zero p-capacity, is a Lebesgue point of E(u). Since p> 1, the sets of zero p-capacity are of

H

^N−1-measure zero and thereforeE(u)is defined

H

^N−1-almost everywhere on∂, soτ(u)= E(u)on∂. This definition is independent of the open setU and also of the extensionE(u). We denoteτ(u)byuin the rest of the paper.

Lemma 2.1. Assumeis a bounded domain in

R

^{N with∂of class C1. Given} u∈

T

^1,p()there exists u∈

T

0^1,p(U)such that

T_k(u)= E[T_k(u)] for all k>0.

Proof. To prove this result we need to recall the construction of the extension op- erator E : W^1,p()→ W₀^1,p(U)given in [17]. Forx =(x₁, . . .x_N)∈

R

^{N, we} write

x=(x,x_N), with x∈

R

^{N−1, x=(x1, . . . ,xN}−1), and we set

|x| = _N−1

i=1

x_i^{2 1}₂

.

We denote

Q= {(x,x_N)∈

R

^N−1×

R

^{: |x|<1, |xN|<1},} Q₊= {(x,x_N)∈

R

^N−1×

R

^{: |x|<1, 0<xN <1}} and

Q₀= {(x,x_N)∈

R

^N−1×

R

^{: |x|<1, xN =0}.}

LetR: Q →Qthe reflection operator defined by R(x,x_N):=

(x,x_N) if x_N ≥0 (x,−x_N) if x_N <0.

Since∂is of classC¹, there exist open setsU_i ⊂U,i =1, ...,k, such that

∂⊂ k i=1

U_i,

and bijective functions G_i : Q → U_i such thatG_i ∈ C¹(Q), G_i⁻¹ ∈ C¹(U_i), G_i(Q₊) = U_i ∩ andG_i(Q₀) = U_i ∩∂. Moreover, there exists a partition

of unity {θi}i=0,1,...,k subordinate to ∂and U₁, . . .U_k (see [17]), that is, θi ∈ C^∞(

R

^N), 0≤θi ≤1,

k i=0

θi =1 in

R

^N,





supp(θi)is compact and supp(θi)⊂U_i, i =1, . . . ,k supp(θ0)⊂

R

^N \∂, and θ0| ∈C_c^∞().

Givenw ∈ W^1,p(), forx ∈ U, we have, settingU₀ = , F₀ = I and F_i = G_i ◦R◦G⁻_i ¹,i =1,2, ...,k,

E(w)(x)=

i∈{0,1,...,k}:x∈U_i

θi(x)w (F_i(x)) . (2.1)

Fixu∈

T

^1,p(). First, observe that by (2.1), we have

|E[T_h(u)](x)| ≤h ∀h>0. (2.2) Let us prove that

A:= {x ∈U : |E[T_h(u)](x)| ≥h, ∀h>0}

is an

L

^N-null set. Obviously, if

A := {x ∈ : |E[T_h(u)](x)| ≥h, ∀h>0},

we have

L

^N(A)=0. On the other hand, by (2.1) and (2.2), it is easy to see that A\A ⊂^k

i=1

(G_i◦R◦G⁻_i ¹)(A∩U_i).

Consequently,

L

^N(A) = 0. Therefore, we can define

L

^N-almost everywhere the functionu:U →

R

^by

u(x):=E[T_h(u)](x), ifxis such that |E[T_h(u)](x)|<h, which is well defined by (2.2) and verifies the lemma.

LetU be an open subset of

R

^N. We set by

M

b(U)the space of all Radon measures in U with bounded total variation. We recall that for a measure µ ∈

M

b(U)and a Borel set A ⊂ U, the measureµ Ais defined by(µ A)(B) = µ(B ∩ A) for any Borel set B ⊂ U. If a measure µ ∈

M

b(U) is such that µ = µ Afor a certain Borel set A, the measureµis said to be concentrated on A. Forµ∈

M

b(U),we denote byµ⁺, µ⁻and|µ|the positive part, negative part and the total variation of the measureµ,respectively. Byµ=µa+µswe denote

the Radon-Nikodym decomposition ofµrelatively to

L

^N. For simplicity, we write alsoµafor its density respect to

L

^N, that is, for the function f ∈ L¹(U)such that µa = f

L

^{N U.}

Let V be an open subset of

R

^N. For a given measureν ∈

M

^b(U) and a continuous function f : U → V, the push-forward measure f#ν is the Radon measure inV defined by

f#ν, ϕ :=

U

ϕ◦ f dν ∀ϕ∈C_c(V).

We denote by

M

b^p(U) the space of all diffuse Radon measures in U, i.e., mea- sures which do not charge sets of zero p-capacity. In [13] it is proved that µ ∈

M

^{b(U)belongs to}

M

b^p(U)if and only if it belongs toL¹(U)+W^−1,p(U), where W^−1,p(U) = [W₀^1,p(U)]^∗. Moreover, ifu ∈ W^1,p(U)andµ ∈

M

b^p(U), thenu is measurable with respect toµ. Ifufurther belongs to L^∞(U), thenubelongs to L^∞(U,dµ), hence toL¹(U,dµ).

We define

M

_b^p():=

µ∈

M

b^p(U) : µis concentrated on .

This definition is independent of the open setU. Note that for u ∈ W^1,p()∩ L^∞()andµ∈

M

_b^p(), we have

µ,E(u) =

u dµ+

∂u dµ;

on the other hand, there exists f ∈ L¹(U)andF ∈ (L^p(U))^N such thatµ = f +div(F), therefore, we also can write

µ,E(u) =

U

f E(u)d x−

U

F·D E(u)d x.

Note that, if f ∈ L¹() and g ∈ L¹(∂) then f

L

^N + g

H

^N−1 ∂ is a diffuse measure concentrated in. Now, if p > N −k, 1≤k < N −1, and

M

is ak-rectifiable subset of∂, then

H

^k

M

is a diffuse measure concentrated in

∂which is not an L¹function in∂(see, [28, Theorem 2.26] or [34, Theorem 2.6.16]).

Letϑ be a maximal monotone graph in

R

^×

R

^{. Forr ∈}

N

^{, the}Yosida approx- imationϑr ofϑ is given byϑr =r(I −(I + ¹_rϑ)⁻¹). The functionϑr is maximal monotone and Lipschitz. We recall the definition of themain sectionϑ⁰ofϑ

ϑ⁰(s):=











the element of minimal absolute value ofϑ(s) ifϑ(s)= ∅, +∞ if[s,+∞)∩Dom(ϑ)= ∅,

−∞ if(−∞,s] ∩Dom(ϑ)= ∅.

We have that|ϑr|is increasing inr, ifs ∈Dom(ϑ),ϑr(s)→ ϑ⁰(s)asr → +∞, and ifs ∈/Dom(θ),|ϑr(s)| → +∞asr → +∞.

We set

ϑ(r+):=infϑ(]r,+∞[), ϑ(r−):=supϑ(] − ∞,r[)

forr ∈

R

, where we use the conventions inf∅ = +∞and sup ∅ = −∞. It is easy to see that

ϑ(r)= [ϑ(r−), ϑ(r+)] ∩

R

^{for r ∈}

R

^. Moreover,

J(ϑ):= {θ ∈Dom(ϑ):ϑ(r−) < ϑ(r+)} (2.3) is a countable set.

We shall denoteϑ₋ := inf Ran(ϑ) andϑ₊ := sup Ran(ϑ). If 0 ∈ Dom(ϑ), j_ϑ(r) = _r

0 ϑ⁰(s)ds defines a convex lower semi-continuous function such that ϑ =∂j_ϑ. If j_ϑ^∗is the Legendre transformation of j_ϑthenϑ⁻¹=∂j_ϑ^∗.

To finish these preliminaries, let us recall some of the results obtained in [2]

for the case of integrable functions that will be used afterward.

We set V^1,p():=

φ∈L¹(): ∃M>0 such that

|φv|≤Mv_W1,p()∀v∈W^1,p()

and

V^1,p(∂) :=

ψ∈L¹(∂): ∃M >0 such that

∂|ψv|≤Mv_W1,p()∀v∈W^1,p()

.

V^1,p()is a Banach space endowed with the norm φ_V1,p():=inf

M>0:

|φv| ≤Mv_W1,p()∀v∈W^1,p()

,

andV^1,p(∂)is a Banach space endowed with the norm ψ_V^1,p_(∂):=inf

M >0:

∂|ψv| ≤Mv_W^1,p₍₎∀v∈W^1,p()

. Observe that, Sobolev embedding and Trace theorems imply, for 1≤ p< N,

L^p()⊂L^{(N p/(N−p))}()⊂V^1,p() and

L^p(∂)⊂L^{((N−1)p/(N−p))}(∂)⊂V^1,p(∂).

For the maximal monotone graphsγ andβ, we shall denote

R

⁺_γ,β ^:=γ+

L

^N()+β+

H

^N−1(∂),

R

⁻_γ,β ^:=γ−

L

^N()+β−

H

^N−1(∂).

We will suppose

R

⁻_γ,β <

R

⁺_γ,βand we will write

R

γ,β :=]

R

⁻_γ,β,

R

⁺_γ,β[.

Theorem 2.2 ([2]). AssumeDom(β)=

R

^{. For anyφ ∈V1,p()such thatφ ∈}

R

γ,β,there exists a weak solution[u,z, w] ∈W^1,p()×V^1,p()×V^1,p(∂)of (S_φ,^γ,β₀), that is,

a(x,Du)·Dv+

zv+

∂wv=

φv, ∀v∈W^1,p().

Moreover,

z^±_L¹₍₎+ w^±_L¹_(∂)≤ φ^±_L¹₍₎.

Theorem 2.3 ([2]). For any φ ∈ V^1,p() there exists a weak solution[u,z] ∈ W₀^1,p()×V^1,p()of(S_φ,^γ,₀^D), that is,

a(x,Du)·Dv+

zv=

φv, ∀v∈W₀^1,p(), and

z^±L¹()≤ φ^±L¹().

3.

The concepts of solution

We introduce the following concepts of solution for problem(S_µ^γ,β_1,µ2).

Definition 3.1. Letµ1, µ2measures, µ1 = µ1 andµ2 = µ2 ∂, such that µ1+µ2∈

M

_b^p₍₎. A triple of functions[u,z, w] ∈ W^1,p()×L¹()×L¹(∂) is a weak solution of problem (S_µ^γ,β_1,µ2) if z(x) ∈ γ (u(x)) a.e. in , w(x) ∈ β(u(x))a.e.in∂and

a(x,Du)·Dvd x+

zvd x+

∂wvd

H

^N−1=

vdµ1+

∂vdµ2

for allv∈W^1,p()∩L^∞().

Let us remark that the fact of being the weak solution in the energy space forces the measureµ1+µ2to belong to a dual space (see Theorem 4.5).

As we pointed out in the introduction, for this type of problems, the concept of weak solution is not enough in order to get uniqueness. It is necessary to find some extra conditions on the distributional solutions in order to ensure both existence and uniqueness. This was done, introducing the concepts of entropy and renormalized solutions (see [6]). For our problem these concepts are the following.

Definition 3.2. Letµ1, µ2measures, µ1 = µ1 andµ2 = µ2 ∂, such that µ1+µ2∈

M

_b^p(). A triple of functions[u,z, w] ∈

T

tr^1,p()×L¹()×L¹(∂)

is anentropy solution of problem(S_µ^γ,β_1,µ2) ifz(x) ∈ γ (u(x))a.e. in, w(x) ∈ β(u(x))a.e.in∂and

a(x,Du)·DTk(u−v)d x+

zTk(u−v)d x+

∂wTk(u−v)d

H

^N−1

≤

T_k(u−v)dµ1+

∂T_k(u−v)dµ2 ∀k>0,

(3.1)

for allv∈W^1,p()∩L^∞().

Definition 3.3. Letµ1, µ2measures, µ1 = µ1 andµ2 = µ2 ∂, such that µ1+µ2∈

M

_b^p(). A triple of functions[u,z, w] ∈

T

tr^1,p()×L¹()×L¹(∂) is arenormalized solutionof problem(S^γ,β_µ1,µ2)ifz(x)∈γ (u(x))a.e.in,w(x)∈ β(u(x))a.e.in∂, and the following conditions hold

(a) for everyh∈W^1,∞(

R

⁾with compact support we have

a(x,Du)·Du h(u)ϕd x+

a(x,Du)·Dϕh(u)d x +

z h(u)ϕd x+

∂wh(u)ϕd

H

^N−1

=

h(u)ϕdµ1+

∂ h(u)ϕdµ2 ∀k>0,

(3.2)

for allϕ∈W^1,p()∩L^∞()such thath(u)ϕ∈W^1,p(), (b)

n→+∞lim

{n≤|u|≤n+1}a(x,Du)·Du d x =0. (3.3) Remark 3.4. Every term in (3.2) is well defined. This is clear for the right hand side sinceh(u)ϕbelongs toL^∞(, µ1+µ2), and thus toL¹(, µ1+µ2). On the other hand, since supp(h)⊂ [−k,k]for somek >0, the two first terms of the left hand side can be written as

a(x,DT_k(u))·DT_k(u)h(u)ϕd x+

a(x,DT_k(u))·Dϕh(u)d x, and both integrals are well defined in view of(H₂), since bothϕandT_k(u)belong to W^1,p(). Moreover, it is not difficult to see that the productDT_k(u)h(u)coincides with the gradient of the composite function h(u) = h(T_k(u))almost everywhere (see [14]).

In the next result we will see that entropy and renormalized solutions coincide.

Theorem 3.5. Let µ1, µ2 measures,µ1 = µ1 andµ2 = µ2 ∂, such that µ1+µ2 ∈

M

_b^p(). Then,[u,z, w]is an entropy solution of problem(S_µ^γ,β_1,µ2)if and only if[u,z, w]is a renormalized solution of problem(S_µ^γ,β_1,µ2).

To prove the above theorem we firstly show the following lemma:

Lemma 3.6. Let µ1, µ2 measures,µ1 = µ1 andµ2 = µ2 ∂, such that µ1+µ2∈

M

_b^p₍₎. Let[u,z, w]be an entropy solution of problem(S_µ^γ,β_1,µ2). Then,

lim

h→+∞

{x∈:h<|u(x)|<h+k}|Du|^p =0, ∀k>0. (3.4) Proof. Let us writeµ1+ µ2 = f +divF in

D

(U), f ∈ L¹(U), and F ∈ (L^p(U))^N. First, let us see that for allh>0,

{x∈:h<|u(x)|<h+k}|Du|^p

≤M

k

{x∈U:|vh(x)|≥h}|f|+

{x∈U:|vh(x)|>h}|F|^p+

|T_k(u−T_h(u))|^p

, (3.5)

wherevh = E[T_k+h](u)is the extension ofT_k+h(u)toW₀^1,p(U)and the constant M is independent ofhandk. Indeed, takingT_h(u)as test function in (3.1), since E[T_k(u−T_h(u))] =T_k(vh−T_h(vh)), we have

a(x,Du)·DT_k(u−T_h(u))+

zT_k(u−T_h(u))+

∂wT_k(u−T_h(u))

≤

U

f T_k(vh−T_h(vh))−

U

DT_k(vh−T_h(vh))·F.

(3.6)

Now, since

U

|DT_k(vh−T_h(vh))|^p≤C

|DT_k(u−T_h(u))|^p+

|T_k(u−T_h(u))|^p

,

using Young’s inequality, we get that there exists a constant M₁independent ofh andksuch that

U

DT_k(vh−T_h(vh))·F ≤ M₁

{|vh|>h}|F|^p+ 2C

U|DT_k(vh−T_h(vh))|^p

≤M₁

{|vh|>h}|F|^p+ 2

|DT_k(u−T_h(u))|^p+

|T_k(u−T_h(u))|^p

. Then, by(H₁)and the positivity of the second and third terms in (3.6), it follows (3.5).

Letube the function obtained by Lemma 2.1. Sincevh = T_h+k(u), we have {|vh| ≥h} ⊂ {|u| ≥h}, which implies that

lim

h→+∞

L

^{N({|vh| ≥h})=0.} Consequently, from (3.5), (3.4) is deduced.