A Haar-Rado type theorem for minimizers in Sobolev spaces
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 1133-1143.

Let $u\in \varphi +{W}_{0}^{1,1}\left(\Omega \right)$ be a minimum for

 $I\left(v\right)={\int }_{\Omega }g\left(x,v\left(x\right)\right)+f\left(\nabla v\left(x\right)\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$
where f is convex, $v↦g\left(x,v\right)$ 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
 $\forall \gamma \in \partial \Omega \phantom{\rule{2em}{0ex}}|u\left(x\right)-\varphi \left(\gamma \right)|\le \omega \left(|x-\gamma |\right)\phantom{\rule{1em}{0ex}}\text{a.e.}\phantom{\rule{4pt}{0ex}}x\in \Omega .$
This result generalizes the classical Haar-Rado theorem for Lipschitz functions.

DOI : https://doi.org/10.1051/cocv/2010038
Classification : 49K20
Mots clés : Hölder, regularity, Lipschitz
@article{COCV_2011__17_4_1133_0,
author = {Mariconda, Carlo and Treu, Giulia},
title = {A Haar-Rado type theorem for minimizers in Sobolev spaces},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1133--1143},
publisher = {EDP-Sciences},
volume = {17},
number = {4},
year = {2011},
doi = {10.1051/cocv/2010038},
zbl = {1239.49031},
mrnumber = {2859868},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2010038/}
}
Mariconda, Carlo; Treu, Giulia. A Haar-Rado type theorem for minimizers in Sobolev spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 1133-1143. doi : 10.1051/cocv/2010038. http://archive.numdam.org/articles/10.1051/cocv/2010038/

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