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.