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

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 Lipsc...

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Bibliographic Details
Published in:arXiv.org 2023-05
Main Authors: Ferrari, Fausto, Lederman, Claudia
Format: Article
Language:English
Subjects:
Free boundaries
Regularity
Viscosity
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2305.07418