Convergence of Convex Sets with Gradient Constraint

Given a bounded open subset of $\R^N$, we study the convergence of a sequence $(\K_n)_{n\in\N}$ of closed convex subsets of $\wump$ ($p\in]1,\infty[$) with gradient constraint, to a convex set $\K$, in the Mosco sense. A particular case of the problem studied is when $\K_n=\{v\in \wump: F_n(x,\gd v(x))\le g_n(x)\text{ for a.e. $x$ in }\Omega\}$. Some examples of non-convergence are presented. We also present an improvement of a result of existence of a solution of a quasivariational inequality, as an application of this Mosco convergence result