Asymptotic Behaviour of Relaxed Dirichlet Problems Involving a Dirichlet-Poincaré Form

Abstract

We study the convergence of the solutions of a sequence of relaxed Dirichlet prob lems relative to Dirichlet forms to the solution of the $\Gamma$-limit problem. In particular we prove the strong convergence in $D_0^P[a,\Omega ](1\le p\le 2)$ and the existence of “correctors” for the strong convergence in $D_0[a,\Omega ]$. The above two results are generalizations to our framework of previous results proved in [10] in the usual uniformly elliptic setting.