Convergence of Convex Sets with Gradient Constraint

Given a bounded open subset of

, we study the convergence of a sequence

(\mathbb K_n)_{n\in{\mathbb N}} ( K n ) n ∈ N

of closed convex subsets of

{\bf W}^{1,p}_0(\Omega) W 0 1 , p ( Ω )

(

p\in]1,\infty[ p ∈ ] 1 , ∞ [

) with gradient constraint, to a convex set

\mathbb K K

, in the Mosco sense. A particular case of the problem studied is when

\def\gd{\nabla}\def\K{\mathbb K}\def\N{\mathbb N}\def\R{\mathbb{R}}\def\wump{{\bf W}^{1,p}_0(\Omega)}\K_n=\{v\in \wump: F_n(x,\gd v(x))\le g_n(x)\text{ for a.e. $x$ in }\Omega\} K n = { v ∈ W 0 1 , p ( Ω ) : F n ( x , ∇ v ( x ) ) ≤ g n ( x )  for a.e.  x  in  Ω }

. 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