APPROXIMATION OF RELAXED DIRICHLET PROBLEMS BY BOUNDARY-VALUE-PROBLEMS IN PERFORATED DOMAINS

Given an elliptic operator L on a bounded domain Omega subset of or eq ual to R(n), and a positive Radon measure mu on Omega, not charging po lar sets, we discuss an explicit approximation procedure which leads t o a sequence of domains Omega(h) superset of or equal to Omega with th e following property: for every f is an element of H-1 (Omega) the seq uence u(h) of the solutions of the Dirichlet problems Lu-h = f in Omeg a(h), u(h) = 0 on partial derivative Omega(h), extended to 0 in Omega\ Omega(h), converses to the solution of the ''relaxed Dirichlet problem '' Lu + mu u = f in Omega, u = 0 on partial derivative Omega.