Let $u\in\phi+ W_0^{1,1}(\Omega)$
be a minimum for
$\[I(v)=\int_{\Omega}g(x,v(x))+f(\nabla v(x))\,{\rm d}x\]$![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20161010014023098-0088:S1292811910000382:S1292811910000382_eqnU2.gif)
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
$\forall \gamma\in\partial\Omega\qquad |u(x)-\phi(\gamma)|\le\omega(|x-\gamma|) \quad\text{a.e. }x\in\Omega.$![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20161010014023098-0088:S1292811910000382:S1292811910000382_eqnU5.gif)
This result generalizes the classical Haar-Rado theorem forLipschitz functions.