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 limits in the sense of Kuratowski of a sequence of Sobolev spaces {W1, p0(Ωn)}n and compare them to a fixed spaceW1, p0(Ω). The results are expressed in terms of the behavior of the local capacity in balls of the complementary sets. If these two limits coeïncide, we obtain necessary and sufficient conditions for the convergence in the sense of Mosco of the Sobolev spaces, hence for the continuity of the solution of a Dirichlet problem associated to thep-Laplacian in terms of the geometric domain variation.