Regularity of the free boundary for two-phase problems governed by divergence form equations and applications

TL;DR

This paper proves that Lipschitz or flat free boundaries in certain two-phase elliptic problems are smooth ($C^{1,eta}$), with applications to fluid dynamics models like the Prandtl-Bachelor problem.

Contribution

It establishes regularity results for free boundaries in divergence form elliptic equations, extending understanding of their smoothness in two-phase problems.

Findings

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

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

Results apply to classical fluid dynamics models

Abstract

We study a class of two-phase inhomogeneous free boundary problems governed by elliptic equations in divergence form. In particular we prove that Lipschitz or flat free boundaries are $C^{1,\gamma}$. Our results apply to the classical Prandtl-Bachelor model in fluiddynamics.