Abstract
In this paper we prove that, ifK is a closed subset ofW 1,p0 (Ω,R m) with 1<p<+∞ andm≥1, thenK is stable under convex combinations withC 1 coefficients if and only if there exists a closed and convex valued multifunction from Ω toR m such that
The casem=1 has already been studied by using truncation arguments which rely on the order structure ofR (see [2]). In the casem>1 a different approach is needed, and new techniques involving suitable Lipschitz projections onto convex sets are developed.
Our results are used to prove the stability, with respect to the convergence in the sense of Mosco, of the class of convex sets of the form (*); this property may be useful in the study of the limit behaviour of a sequence of variational problems of obstacle type.
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Maso, G.D., Defranceschi, A. & Vitali, E. A characterization ofC 1-convex sets in Sobolev spaces. Manuscripta Math 75, 247–272 (1992). https://doi.org/10.1007/BF02567083
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DOI: https://doi.org/10.1007/BF02567083