
Journal of Convex Analysis 11 (2004), No. 2, 285301 Copyright Heldermann Verlag 2004 Convergence of Convex Sets with Gradient Constraint Assis Azevedo Dept. of Mathematics, University of Minho, Braga, Portugal, assis@math.uminho.pt Lisa Santos Dept. of Mathematics, University of Minho, Braga, Portugal, lisa@math.uminho.pt [Abstractpdf] \newcommand{\gd}{\nabla} \newcommand{\K}{\mathbb K} \newcommand{\N}{\mathbb N} \newcommand{\R}{\mathbb{R}} \newcommand{\wump}{{\bf W}^{1,p}_0(\Omega)} 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)\mbox{ for a.e. $x$ in }\Omega\}$. Some examples of nonconvergence 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. FullTextpdf (475 KB) for subscribers only. 