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

FERRARI, FAUSTO; LEDERMAN, CLAUDIA

JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES

GAUTHIER-VILLARS/EDITIONS ELSEVIER

Año: 2022 vol. 2023 p. 26 - 74

0021-7824

We continue our study in cite{FL} on viscosity solutions to a one-phase free boundary problem for the $p(x)$-Laplacian with non-zero right hand side. We first prove that viscosity solutions are locally Lipschitz continuous, which is the optimal regularity for the problem. Then we prove that Lipschitz free boundaries of viscosity solutions are $C^{1,alpha}$. We also present some applications of our results.Moreover, we obtain new results for the operator under consideration that are of independent interest, such as a Harnack inequality.