Lipschitz Continuity for Elliptic Free Boundary Problems with Dini Mean Oscillation Coefficients

TL;DR

This paper proves local Lipschitz continuity for solutions to certain free boundary elliptic problems with Dini mean oscillation coefficients, even allowing discontinuities in most variables, advancing regularity theory in elliptic PDEs.

Contribution

It establishes Lipschitz regularity for free boundary elliptic problems with coefficients having Dini mean oscillation in at least one direction, accommodating discontinuities in other variables.

Findings

Solutions are locally Lipschitz continuous under Dini mean oscillation conditions.

Regularity holds despite discontinuities in all but one variable.

Advances understanding of elliptic PDEs with irregular coefficients.

Abstract

We establish local interior Lipschitz continuity of the solutions of a class of free boundary elliptic problems assuming the coefficients of the equation of Dini mean oscillation in at least one direction. The novelty in this regularity result lies in the fact that it allows discontinuous coefficients in all but one variable.