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

Let uϕ+W 0 1,1 (Ω) be a minimum for I(v)= Ω g(x,v(x))+f(v(x))dx where f is convex, vg(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 γΩ|u(x)-ϕ(γ)|ω(|x-γ|)a.e.xΩ. This result generalizes the classical Haar-Rado theorem for Lipschitz functions.

DOI : https://doi.org/10.1051/cocv/2010038
Classification:  49K20
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {4},
     year = {2011},
     pages = {1133-1143},
     doi = {10.1051/cocv/2010038},
     zbl = {1239.49031},
     mrnumber = {2859868},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_4_1133_0}
}
Mariconda, Carlo; Treu, Giulia. A Haar-Rado type theorem for minimizers in Sobolev spaces. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, pp. 1133-1143. doi : 10.1051/cocv/2010038. http://www.numdam.org/item/COCV_2011__17_4_1133_0/

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