35J70, 35P30, 35R35.

In this paper, using the Hausdorff topology in the space of open sets under some capacity constraints on geometrical domains we prove the strong continuity with respect to the moving domain of the solutions of a p-Laplacian Dirichlet problem. We are also interested in the minimization of the first eigenvalue of the p-Laplacian with Dirichlet boundary conditions among open sets and quasi open sets of given measure.;