# Weak Solutions and Regularity of the Interface in an Inhomogeneous Free Boundary Problem for the p(x)-Laplacian

## TL;DR

This paper investigates the regularity of the free boundary in a p(x)-Laplacian problem with non-zero right side, proving it is a C^1 surface near each free boundary point and exploring further regularity under additional data assumptions.

## Contribution

It establishes the C^1 regularity of the free boundary for weak solutions of the inhomogeneous p(x)-Laplacian problem, extending previous results to variable exponent settings.

## Key findings

- Free boundary is a C^1 surface near every free boundary point.
- Additional regularity results are obtained under stronger data assumptions.
- Applications to limit functions in inhomogeneous singular perturbation problems.

## 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 surface in a neighborhood of every 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 Lederman, C., & Wolanski, N. An inhomogeneous singular perturbation problem for the p(x)-Laplacian, Non- linear Anal. 138 (2016), 300-325.