Weak solutions and regularity of the interface in an inhomogeneous free boundary problem for the $p(x)$-Laplacian

Abstract

In this paper we study a one phase free boundary problem for the $p(x)$-Laplacian with non-zero right hand side. We prove that the free boundary of a weak solution is a $C^{1,\alpha }$ surface in a neighborhood of every „flat" free boundary point. We also obtain further regularity results on the free boundary, under further regularity assumptions on the data. We apply these results to limit functions of an inhomogeneous singular perturbation problem for the $p(x)$-Laplacian that we studied in [25].