no. 4
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∈ϕ+W 0 1,1 (Ω) be a minimum for

I(v)=∫ Ω g(x,v(x))+f(∇v(x))dx
where f is convex, v↦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
∀γ∈∂Ω|u(x)-ϕ(γ)|≤ω(|x-γ|)a.e.x∈Ω.
This result generalizes the classical Haar-Rado theorem for Lipschitz functions.

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},
     pages = {1133--1143},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {4},
     year = {2011},
     doi = {10.1051/cocv/2010038},
     mrnumber = {2859868},
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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. http://www.numdam.org/articles/10.1051/cocv/2010038/

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