A Haar-Rado type theorem for minimizers in Sobolev spaces

Carlo Mariconda; Giulia Treu

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

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

Abstract

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Let be a minimum for where f is convex, 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 This result generalizes the classical Haar-Rado theorem for Lipschitz functions.

How to cite

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Mariconda, 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 -

References

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  10. C. Mariconda and G. Treu, Lipschitz regularity for minima without strict convexity of the Lagrangian. J. Differ. Equ.243 (2007) 388–413.  
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  12. C. Mariconda and G. Treu, Hölder regularity for a classical problem of the calculus of variations. Adv. Calc. Var.2 (2009) 311–320.  
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