Epiconvergence method to a nonlinear value boundary problem with L 1 Data

Where Ω is a bounded domain in R with lipschitz boundary ∂Ω, ∆p is the pLaplacian operator defined on the sobolev space W 1,p 0 (Ω) to its dual W −1,p′(Ω), p > 1, p the conjugate of p and f ∈ L(Ω). Recently, many works is considered, in the literature of boundary problem with L data, where its authors have study the existence of renormalization solutions, for more details we can refer the reader to [2,3,4]. Now we interest to an other approach to study the existence of the weak solutions of the problem (1.1). Our approach is based to the truncation and epi-convergence method. This paper is organized in the following way. In section 2, we express the problem to study with truncation step, we define functional space for this study and we study the problem (2.1). The section 3 is reserved to the determination of the limit problem.


Introduction
In this present paper, our goal is to study the problem of the existence of the weak solutions of the following value boundary problem Where Ω is a bounded domain in R 3 with lipschitz boundary ∂Ω, ∆ p is the p-Laplacian operator defined on the sobolev space W 1,p 0 (Ω) to its dual W −1,p ′ (Ω), p > 1, p ′ the conjugate of p and f ∈ L 1 (Ω).Recently, many works is considered, in the literature of boundary problem with L 1 data, where its authors have study the existence of renormalization solutions, for more details we can refer the reader to [2,3,4].Now we interest to an other approach to study the existence of the weak solutions of the problem (1.1).Our approach is based to the truncation and epi-convergence method.
This paper is organized in the following way.In section 2, we express the problem to study with truncation step, we define functional space for this study and we study the problem (2.1).The section 3 is reserved to the determination of the limit problem.

Statement of the problem
In order to study the existence of solutions of the problem (1.1), we propose our method, based on two steps: truncation step and the second step is reserved to the asymptotic behaviour of solutions.

Truncation step
It is known, in [5], that there exists a sequence when ε close to 0, for more details we refer the reader to [5].Now we consider the following problem where V ∞ a closed subset of W 1,p 0 (Ω) defined by it's clear to see that V ∞ , endowed with the norm .W 1,p 0 (Ω) , is a banach and reflexive space.In the sequel, we denote .V ∞ = .W 1,p 0 (Ω) and we study the existence and uniqueness of solution of the problem (2.2).
Proposition 2.1.The problem (2.2) possess an unique bounded solution u ε in V ∞ .Moreover there exists a subsequence of u ε (still denoted by u ε ) and Proof: Let F ε be an operator defined on V ∞ with real values such that Epiconvergence method to a nonlinear value boundary problem 37 thanks to Young and Poincaré inequality, there exists C > 0 such that (1) and we can show easily that F ε is weakly lower semicontinuous on V ∞ , so by applying the classical result ( see [6, theorem 1.1 p.48]), and by strict convexity of F ε , then the minimization problem (2.2) possess an unique solution in particular for v = u ε and according to Poincaré, Hôlder and Young inequality (see for example [5]), there exists a constant c > 0 such that thus there exists a constant M > 0 such that Since V ∞ is a reflexive space, so there exists a subsequence of (u ε ) ε>0 ), still denoted by (u ε ) ε>0 ), and u * ∈ V ∞ such that u ε ⇀ u * in V ∞ , to prove the strong convergence in V ∞ , we will use the fact that −∆ p is of type (S+) for more details see for example [5].First, we prove that 1 For more details about the coercivity definition see [6].

J. Messaho
We have , so there exists a constant A > 0 such that u ε − u * L ∞ (Ω) ≤ A, so there exists B > 0 and we have By passing to the limit, we obtain

Limit study
In this section, we will interest to the asymptotic behaviour of the solution u ε of the problem (2.2), and this behaviour will be derived with the epi-convergence method, (see definition 4.1).Now we will determinate the epi-limit of the energy functional, linked to the minimization problem (2.2), defined by We design by τ the strong topology on the space V ∞ .In the sequel, we shall characterize, the epi-limit of the energy functional given by (3.1) in the following theorem.
Theorem 3.1.There exists a functional F : where F is given by Proof: (a) We are now in position to determine the upper epi-limit.Let u ∈ V ∞ , let us u ε = u, so u ε → u in V ∞ when ε → 0, so we obtain (b) We are now in position to determine the lower epi-limit.Let u ∈ V ∞ and We have According to (3.2), we obtain We have Since f ε → f in L 1 (Ω) when ε close to 0, so for a small enough ε there exists a constant c > 0 such that f ε (x) ≤ c, a.e x ∈ Ω, from the Hôlder inequality we obtain so by passing to the limit when ε → 0, we obtain From to (3.4) and (3.5), (3.3) becomes In the sequel, we will determine the limit problem linked to (2.2), when ε approaches to zero.Thanks to the epi-convergence results, (see Annex, theorem 4.2) and the theorem 3.1, we have F ε τ -epiconverges to F in V ∞ .

J. Messaho
Proposition 3.2.There exists an unique element u * ∈ V ∞ satisfying: Proof: Thanks to proposition 2.1, the solution of the problem (2.2), (u ε ) ε>0 , possess a subsequence still denoted by (u ε ) ε>0 converges strongly to an element u * ∈ V ∞ .And thanks to a classical epi-convergence result, theorem 4.2, it follows that u * is a solution of the following limit problem According to the convexity strict of F, so the problem (3.7) admits an unique solution in V ∞ .✷

Conclusion
We showed that the τ -cluster point of the solution of the problem (2.2) is the solution of the problem (1.1).
• For all x ∈ X and all x ε with x ε τ → x, lim inf ε→0 This epi-convergence is a special case of the Γ−convergence introduced by De Giorgi (1979) [7].It is well suited to the asymptotic analysis of sequences of minimization problems since one has the following fundamental result.Then every cluster point u of the sequence (u ε ) minimizes F on X and if (u ε ′ ) ε ′ denotes the subsequence of (u ε ) ε which converges to u.