A characterization of \(C^ 1\)-convex sets in Sobolev spaces
A subset of \(W_ 0^{1,p}(\Omega,\mathbb{R}^ n)\) is said to be \(C^ 1\)- convex if for every pair of functions \(u,v\in{\mathcal K}\) the convex combination \(\alpha u+(1-\alpha)v\) belongs to \({\mathcal K}\) whenever \(\alpha\in C^ 1(\Omega)\cap W^{1,\infty}(\Omega)\) and \(0\leq\alpha\leq 1\). The authors prove the following theorem which is the main result of the paper: Let \(\Omega\) be a bounded open subset of \(\mathbb{R}^ n\) and let \({\mathcal K}\) be a closed subset of \(W_ 0^{1,p}(\Omega,\mathbb{R}^ n)\) with \(1<p<+\infty\). Then \({\mathcal K}\) is \(C^ 1\)-convex if and only if there exists a multifunction \(K:\Omega\to\mathbb{R}^ n\) with closed convex values such that \[ {\mathcal K}=\{u\in W_ 0^{1,p}(\Omega,\mathbb{R}^ n):\;u(x)\in K(x)\quad\text{for q.e. }x\varepsilon\Omega\}, \] where q.e. means quasi everywhere (with respect to the intrinsic \(W_ 0^{1,p}(\Omega)\)- capacity) and the pointwise value \(u(x)\) of \(u\) is defined for q.e. \(x\in\Omega\), as the limit, as \(r\to 0_ +\), of the average of \(u\) (with respect to the Lebesgue measure) on the ball \(B(x,r)\). Moreover, \(K\) can be of the form \(K(x)=\text{cl}\{u_ h(x):h\in\mathbb{N}\}\) for a suitable sequence \((u_ h)\) in \(W_ 0^{1,p}(\Omega,\mathbb{R}^ n)\), where cl denotes the closure in \(\mathbb{R}^ n\). This theorem generalizes the results from the paper of \textit{H. Attouch}, \textit{C. Picard} [Ann. Fac. Sci. Toulouse V Sér., Math. 1, 89-136 (1979; Zbl 0418.49012)].
scientific article; zbMATH DE number 91461
scientific article