Abstract

We study the regularity of the free boundary $\partial \{u>0\}\cap \mathrm{\Omega }$ in the following Alt–Caffarelli type minimum problem for the “weighted” $p$-Laplace operator (where $1<p<\infty$):

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 pioneering work by Alt, Caffarelli and Friedman (1984) and Danielli and Petrosyan (2005) to the case of nonuniform weights $w$ and $\psi$. In addition, this paper builds upon our prior work (2025), extending the analysis of the above problem to the regularity of the free boundary.

Keywords

Mathematical Subject Classification

Milestones

Authors