  We consider a one-Laplace equation perturbed by $p$-Laplacian with
$1<p<\infty$. We prove that a weak solution is continuously differentiable
($C^{1}$) if it is convex. Note that similar result fails to hold for the
unperturbed one-Laplace equation. The main difficulty is to show
$C^{1}$-regularity of the solution at the boundary of a facet where the
gradient of the solution vanishes. For this purpose we blow-up the solution and
prove that its limit is a constant function by establishing a Liouville-type
result, which is proved by showing a strong maximum principle. Our argument is
rather elementary since we assume that the solution is convex. A few
generalization is also discussed.
