$\Gamma$-convergence of strongly local Dirichlet functionals

The aim of this paper is to give a variational motivation to an earlier work by M. Biroli and S. Marchi, in which the Authors proved the weak convergence of the solutions of relaxed Dirichlet problems with potentials described by a suitable non-negative measure. In this paper we prove the Γ-convergence of the strongly local Dirichlet functional associated to the above relaxed Dirichlet problem and the convergence of the related minima. We apply the same methods of Γ-convergence of functionals of the calculus of variations as Dal Maso and Mosco did in an earlier work of theirs.