Regularity of flat free boundaries for a $p(x)$-Laplacian problem with right hand side

TL;DR

This paper proves that flat free boundaries in a $p(x)$-Laplacian free boundary problem are $C^{1,eta}$ smooth, extending regularity results and introducing new findings for the operator with a non-zero right hand side.

Contribution

It establishes $C^{1,eta}$ regularity of flat free boundaries for the $p(x)$-Laplacian with right hand side and presents new insights into the operator's properties.

Findings

Flat free boundaries are $C^{1,eta}$ smooth.

New results on the $p(x)$-Laplacian operator.

Extension of regularity theory to variable exponent problems.

Abstract

We consider viscosity solutions to a one-phase free boundary problem for the $p(x)$-Laplacian with non-zero right hand side. We apply the tools developed in \cite{D} to prove that flat free boundaries are $C^{1,\alpha}$. Moreover, we obtain some new results for the operator under consideration that are of independent interest.