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Musielak−Orlicz−Sobolev spaces on arbitrary metrique space

Akdim YoussefNoureddine AissaouiMy Cherif Hassib — 2016

Commentationes Mathematicae

In this article we define Musielak−Orlicz−Sobolev spaces on arbitrary metric spaces with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a pointwise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds, and that the Lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application,...

Existence of a renormalized solution of nonlinear degenerate elliptic problems

Youssef AkdimChakir Allalou — 2014

Applicationes Mathematicae

We study a general class of nonlinear elliptic problems associated with the differential inclusion β ( u ) - d i v ( a ( x , D u ) + F ( u ) ) f in Ω where f L ( Ω ) . The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general L -data.

Existence of solutions of degenerated unilateral problems with L 1 data

Lahsen AharouchYoussef Akdim — 2004

Annales mathématiques Blaise Pascal

In this paper, we shall be concerned with the existence result of the Degenerated unilateral problem associated to the equation of the type A u + g ( x , u , u ) = f - div F , where A is a Leray-Lions operator and g is a Carathéodory function having natural growth with respect to | u | and satisfying the sign condition. The second term is such that, f L 1 ( Ω ) and F Π i = 1 N L p ( Ω , w i 1 - p ) .

Existence results for quasilinear degenerated equations via strong convergence of truncations.

Youssef AkdimElhoussine AzroulAbdelmoujib Benkirane — 2004

Revista Matemática Complutense

In this paper we study the existence of solutions for quasilinear degenerated elliptic operators A(u) + g(x,u,∇u) = f, where A is a Leray-Lions operator from W (Ω,ω) into its dual, while g(x,s,ξ) is a nonlinear term which has a growth condition with respect to ξ and no growth with respect to s, but it satisfies a sign condition on s. The right hand side f is assumed to belong either to W(Ω,ω*) or to L(Ω).

Existence of Solution for Quasilinear Degenerated Elliptic Unilateral Problems

Youssef AkdimElhoussine AzroulAbdelmoujib Benkirane — 2003

Annales mathématiques Blaise Pascal

An existence theorem is proved, for a quasilinear degenerated elliptic inequality involving nonlinear operators of the form A u + g ( x , u , u ) , where A is a Leray-Lions operator from W 0 1 , p ( Ω , w ) into its dual, while g ( x , s , ξ ) is a nonlinear term which has a growth condition with respect to ξ and no growth with respect to s , but it satisfies a sign condition on s , the second term belongs to W - 1 , p ( Ω , w * ) .

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