Inequations variationnelles avec obstacles et espaces fonctionnels en theorie du potentiei

Abstract

Nous etudions la stabilite par rapport a 1'obstacle d'une inequation variationnelle de WQ'P(Q) (p>l) avec contrainte de type obstacle. Utilisant de recents travaux de theorie du potentiei, nous obtenons une notion de convergence sur les obstacles qui implique la convergence des solutions d'inequations variationnelles unilaterales; cette notion est optimale iorsque I'obstacle limite est une fonction quasicontinue. Ainsi nous unifions differentes approches anterieures ([8], [4], [5], [9], [7] ...) The aim of this article is the study of the stability with respect to the obstacle of variational inequalities in W^1-P ₀(ω) (p>1) with constraints of obstacle type. Using recent works of potential theory. We obtain a notion of convergence on the obstacles which implies the convergence of the solutions of the variational unilateral inequalities; this notion is optimal when the obstacles are quasicontinuous functions. Hence, we unify the various anterior approaches of L.Carbone, C.Sbordone[8], H. Attouch[4], H. Attouch, C. Picard[5], L.Carbone, F.Colombini [9], L. Boccardo F.Murat[7]... In a first part following K. Hansson [12], D.R. Adams[1],[2],[3], D. Feyel, A. de la Pradelle[11]..., We study the Banach space L^p(C_1-p), a space of quasi continuous functions twith respect to the capieity C_1-p of W^1-P ₀(ω) we prove that W^1-P ₀(ω) L^P(C_1-p) with continuous injection and for every the injefction from W^1-P ₀(ω) is compact the dual of is the order dual of W^1-P ₀(ω) that is the vectorial space generated by In a second part we give we give a general condition of convergence of the obstacles implying the convergence of the solutions of the associated variational inequalities. (to the solution of the variational ineqality associated to the limit obstacie). This condition is satisfied for example if the sequence of obstacles g_n converge to g in L^p(C_1-p)In fact, what is important is the convergence of (g_n g) to zero in L^p(C_1-p)