# A Haar-Rado type theorem for minimizers in Sobolev spaces

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 4, page 1133-1143
- ISSN: 1292-8119

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topMariconda, Carlo, and Treu, Giulia. "A Haar-Rado type theorem for minimizers in Sobolev spaces." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 1133-1143. <http://eudml.org/doc/276330>.

@article{Mariconda2011,

abstract = {
Let $u\in\phi+ W_0^\{1,1\}(\Omega)$ be a minimum for $\[I(v)=\int_\{\Omega\}g(x,v(x))+f(\nabla v(x))\,\{\rm d\}x\]$
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
$\forall \gamma\in\partial\Omega\qquad |u(x)-\phi(\gamma)|\le
\omega(|x-\gamma|) \quad\text\{a.e. \}x\in\Omega.$
This result generalizes the classical Haar-Rado theorem for
Lipschitz functions.
},

author = {Mariconda, Carlo, Treu, Giulia},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Hölder; regularity; Lipschitz; Hölder functions; Lipschitz functions},

language = {eng},

month = {11},

number = {4},

pages = {1133-1143},

publisher = {EDP Sciences},

title = {A Haar-Rado type theorem for minimizers in Sobolev spaces},

url = {http://eudml.org/doc/276330},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Mariconda, Carlo

AU - Treu, Giulia

TI - A Haar-Rado type theorem for minimizers in Sobolev spaces

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2011/11//

PB - EDP Sciences

VL - 17

IS - 4

SP - 1133

EP - 1143

AB -
Let $u\in\phi+ W_0^{1,1}(\Omega)$ be a minimum for $\[I(v)=\int_{\Omega}g(x,v(x))+f(\nabla v(x))\,{\rm d}x\]$
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
$\forall \gamma\in\partial\Omega\qquad |u(x)-\phi(\gamma)|\le
\omega(|x-\gamma|) \quad\text{a.e. }x\in\Omega.$
This result generalizes the classical Haar-Rado theorem for
Lipschitz functions.

LA - eng

KW - Hölder; regularity; Lipschitz; Hölder functions; Lipschitz functions

UR - http://eudml.org/doc/276330

ER -

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