Limits of Relaxed Dirichlet Problems Involving a non Symmetric Dirichlet Form

In this paper we study the convergence of solutions of a sequence of relaxed Dirichlet problems relative to non-symmetric Dirichlet forms. The techniques rely on the study of the behaviour of the solutions of the adjoint problems, as suggested by G. Dal Maso and A. Garroni in [16] in the case of linear elliptic operators of second order with bounded measurable coefficients. In particular we prove a compactness result due to Mosco [31] in the symmetric case.