Correctors results for relaxed Dirichlet problems

Abstract

We consider the asymptotic behaviour of the solutions of a sequence of elliptic problems with homogeneous Dirichlet boundary conditions, involving singular potentials which can be Borel measures with infinite values. Problems of this kind, introduced in (Dal Maso and Mosco, 1986) and (Dal Maso and Mosco, 1987) are called ‘relaxed Dirichlet problems’ since they form the smallest family of equations, stable under the convergence of solutions in L², which includes Dirichlet problems with zero boundary conditions on many small holes (see also (Cionarescu and Murat, 1982) and (Dal Maso, 1987)).

After extraction of a subsequence, the solutions converge weakly in H¹₀(Ω) to the solution of a limit problem of the same type. We determine a corrector, i.e. an explicit expression constructed from the limit of the solutions and from some generalized capacitary potentials, the difference of which with the solution tends strongly to zero in H¹₀(Ω). Under suitable regularity assumptions on the limit problem, we are able to prove a strong convergence results and an abstract error estimate. Moreover, we prove in the general case that there is no need of correctors in the singular set of the measure of the limit problem.