Characterization for the Kuratowski limits of a sequence of Sobolev spaces

The purpose of this paper is to study the strong lower and the weak upper l imits in the sense of Kuratowski of a sequence of Sobolev spaces (W-0(1,p)( Omega(n)))(n epsilon N) and compare them to a fixed space W-0(1,p)(Omega). The results are expressed in terms of the of the local capacity in balls of the complementary sets. If these two limits coeincide, we obtain necessary and sufficient conditions for the convergence in the sense of Mosco of the Subolev spaces, hence for the continuity of the solution of a Dirichlet pr oblem associated to the p-laplacian in terms of the geometric domain variat ion. (C) 1999 Academic Press.