ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - October 2011 - Volume 17, Issue 04  

Research Article

A Haar-Rado type theorem for minimizers in Sobolev spaces

Mariconda, Carloa1 and Treu, Giuliaa1

a1 Dipartimento di Matematica Pura ed Applicata – Università Degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy. maricond@math.unipd.it; treu@math.unipd.it

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.

(Received July 23 2009)

(Online publication October 28 2010)

Key Words:

  • Hölder;
  • regularity;
  • Lipschitz

Mathematics Subject Classification:

  • 49K20