Submitted Paper
Inserted: 24 feb 2026
Last Updated: 25 feb 2026
Year: 2026
Abstract:
In this paper, we study the regularity of the free boundary $\partial\{u > 0\} \cap \Omega$ in the following Alt-Caffarelli type minimum problem for the weighted p-Laplace operator (where $1<p<\infty$):
$ \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\}.
$
More precisely, we will show under some appropriate assumptions on the weights $w$ and $\psi$ that the free boundary is $C^{1,\alpha}$, except possibly at a set of Hausdorff measure zero. This is a generalization of the pioneer papers $
[2,6
]$ to the case of nonuniform weights $w$ and $\psi$. In addition, this paper builds upon our prior work $
[8
]$, extending the analysis of this problem to the regularity of the free boundary.
Keywords: viscosity solution, free boundary, Weighted p-Laplacian
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