Free boundary problems involving singular weights

TL;DR

This paper studies free boundary minimization problems with singular weights, establishing existence, boundedness, and a novel sharp regularity result at free boundary points, advancing understanding of singular operators in such problems.

Contribution

It introduces the first sharp $C^{1+eta}$ regularity result for solutions at singular free boundary points involving $A_2$ weights.

Findings

Existence and boundedness of minimizers.

Sharp $C^{1+eta}$ regularity at free boundary points.

Non-degeneracy estimates for solutions.

Abstract

In this paper we initiate the investigation of free boundary minimization problems ruled by general singular operators with $A_2$ weights. We show existence and boundedness of minimizers. The key novelty is a sharp $C^{1+\gamma}$ regularity result for solutions at their singular free boundary points. We also show a corresponding non-degeneracy estimate.