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Zbl 0724.49008
Vitali, Enrico
Convergence of unilateral convex sets in higher order Sobolev spaces. (English)
[J] Ann. Fac. Sci. Toulouse, V. Sér., Math. 11, No.3, 93-149 (1990). ISSN 0240-2955

Summary: We establish a characterization of the Mosco convergence for the class of unilateral convex sets in $W\sb 0\sp{m,p}(\Omega)$ (m$\ge 1)$ of the form $\{u\in W\sb 0\sp{m,p}(\Omega)\vert$ $u\ge \psi$ on $\Omega\}$. For given obstacles $(\psi\sb h)\sb h$ and $\psi$ this characterization is expressed in terms of the Mosco convergence of the unilateral sets determined by the characteristic functions of the level sets $\{\psi\sb h>t\}$ and $\{\psi >t\}$. In the case $p=2$ we further develop this result obtaining necessary and sufficient conditions in terms of the convergence of the (m,2)-capacities of the intersections $\{\psi\sb h>t\}\cap A$ of the level sets with a suitable family of open sets A.

MSC 2000:: *49J45 Optimal control problems inv. semicontinuity and convergence

Keywords: Mosco convergence

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