A Haar-Rado type theorem for minimizers in Sobolev spaces

Mariconda, Carlo; Treu, Giulia

doi:10.1051/cocv/2010038

Mariconda, Carlo ; Treu, Giulia

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

Résumé

Let $u \in ϕ + W_{0}^{1, 1} (Ω)$ be a minimum for

I (v) = \int_{Ω} g (x, v (x)) + f (\nabla v (x)) 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 γ \in \partial Ω | u (x) - ϕ (γ) | \leq ω (| x - γ |) a.e. x \in Ω .

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

MR Zbl

DOI : 10.1051/cocv/2010038

Classification : 49K20
Mots-clés : Hölder, regularity, Lipschitz

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     title = {A {Haar-Rado} type theorem for minimizers in {Sobolev} spaces},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
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     publisher = {EDP-Sciences},
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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. https://www.numdam.org/articles/10.1051/cocv/2010038/

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