  In this paper we aim to show continuous differentiability of weak solutions
to a one-Laplace system perturbed by $p$-Laplacian with $1<p<\infty$. The main
difficulty on this equation is that uniform ellipticity breaks near a facet,
the place where a gradient vanishes. We would like to prove that derivatives of
weak solutions are continuous even across the facets. This is possible by
estimating H\"{o}lder continuity of Jacobian matrices multiplied with its
modulus truncated near zero. To show this estimate, we consider an approximated
system, and use standard methods including De Giorgi's truncation and freezing
coefficient arguments.
