A weak solution to a perturbed one-Laplace system by $p$-Laplacian is continuously differentiable

In this paper we aim to show continuous differentiability of weak solutions to a one-Laplace system perturbed by $p$-Laplacian with $1<p<\mathrm{\infty }$. 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 Hölder 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.