Lipschitz regularity of almost minimizers in one-phase problems driven by the \(p\)-Laplace operator

We prove that, given~\(p>\max\left\{\frac{2n}{n+2},1\right\}\), the nonnegative almost minimizers of the nonlinear free boundary functional $$ J_p(u,\Omega):=\int_{\Omega}\Big( |\nabla u(x)|^p+\chi_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz continuous.

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Bibliographic Details
Published in:arXiv.org 2022-06
Main Authors: Dipierro, Serena, Ferrari, Fausto, cillo, Nicolò, Valdinoci, Enrico
Format: Article
Language:English
Subjects:
Free boundaries
Online Access:Get full text
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Summary:We prove that, given~\(p>\max\left\{\frac{2n}{n+2},1\right\}\), the nonnegative almost minimizers of the nonlinear free boundary functional $$ J_p(u,\Omega):=\int_{\Omega}\Big( |\nabla u(x)|^p+\chi_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz continuous.
ISSN:2331-8422