Preprint

Inserted: 14 apr 2025
Last Updated: 14 apr 2025

Year: 2025

Abstract:

In this paper, we study the regularity of the minimizer in the Alt-Caffarelli type minimum problem for the "weighted" $p-$Laplace operator ($1<p<\infty$) with free boundary:

$\min \bigg\{\int_\Omega (w|\nabla u|^p + \psi \, \chi_{\{u>0\}}) \,:\,u \in W^{1,p}(\Omega),\,\,u \geq 0,\,\,u=g \,\,\,\mbox{on}\,\,\,\partial\Omega \bigg\},$

where $w$ and $\psi$ are two given nonnegative functions on $\Omega$ and $g$ is a nonnegative boundary datum. More precisely, under the assumptions that $w$ is a $C^2$ function with $w \geq w_{\min}>0$ and $\psi$ belongs to $L^{q}_{loc}(\Omega)$ for some $q>\frac{N}{p}$, we will show that a minimizer $u$ is locally $\alpha-$Holderian with $\alpha=1-\frac{N}{pq}$.

If $\psi$ belongs to $L^\infty_{loc}(\Omega)$ and is bounded away from zero then, thanks to the Lipschitz regularity of $u$, we will be able also to prove that the free boundary $\partial\{u>0\}$ is locally of finite perimeter.

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• Regularity of the solution and the free boundary in a weighted p-Laplacian problem