Two-phase problems with distributed source: regularity of the free boundary

TL;DR

This paper studies the regularity of free boundaries in two-phase problems with distributed sources, proving that Lipschitz or flat free boundaries are smoothly differentiable, thus confirming viscosity solutions are classical.

Contribution

It establishes the $C^{1,eta}$ regularity of free boundaries in a broad class of two-phase problems with non-zero sources, advancing understanding of boundary smoothness.

Findings

Lipschitz free boundaries are $C^{1,eta}$.

Flat free boundaries are $C^{1,eta}$.

Viscosity solutions are classical under these conditions.

Abstract

We investigate the regularity of the free boundary for a general class of two-phase free boundary problems with non-zero right hand side. We prove that Lipschitz or flat free boundaries are $C^{1,\gamma}$. In particular, viscosity solutions are indeed classical.