## ESAIM: Control, Optimisation and Calculus of Variations

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

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 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.