A Haar-Rado type theorem for minimizers in Sobolev spaces

Mariconda, Carlo  ; Treu, Giulia

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 1133-1143

Résumé

Let $u\in \varphi +W_0^{1,1}\left(\Omega \right)$ be a minimum for

$$I\left(v\right)={\int }_{\Omega }g(x,v\left(x\right))+f\left(\nabla v\left(x\right)\right)\mathrm{d}x$$

where f is convex, $v\mapsto g(x,v)$ is convex for a.e. x. We prove that u shares the same modulus of continuity of ϕ whenever Ω is sufficiently regular, the right derivative of g satisfies a suitable monotonicity assumption and the following inequality holds

$$\forall \gamma \in \partial \Omega \left|u\right(x)-\varphi (\gamma \left)\right|\le \omega \left(\right|x-\gamma \left|\right)\text{a.e.}x\in \Omega .$$

This result generalizes the classical Haar-Rado theorem for Lipschitz functions.

Détail

DOI : 10.1051/cocv/2010038

Classification : 49K20
Keywords: Hölder, regularity, Lipschitz