Regularity for degenerate two-phase free boundary problems

TL;DR

This paper develops a comprehensive regularity theory for a family of heterogeneous, two-phase free boundary problems governed by nonlinear, degenerate elliptic operators, revealing new regularity results even for classical equations.

Contribution

It provides the first complete regularity description for a broad class of degenerate two-phase free boundary problems, including new results for classical cases.

Findings

Local minima are $C^{1,\alpha}$ for $0<\gamma<1$ with sharp $\alpha$.

Local minima are Log-Lipschitz continuous for $\gamma=0$.

Results extend to classical linear, nondegenerate equations.

Abstract

We provide a rather complete description of the sharp regularity theory for a family of heterogeneous, two-phase variational free boundary problems, $\mathcal{J}_\gamma \to $ min, ruled by nonlinear, $p$-degenerate elliptic operators. Included in such family are heterogeneous cavitation problems of Prandtl-Batchelor type; singular degenerate elliptic equations; and obstacle type systems. The Euler-Lagrange equation associated to $\mathcal{J}_\gamma$ becomes singular along the free interface $\{u= 0\}$. The degree of singularity is, in turn, dimed by the parameter $\gamma \in [0,1]$. For $0< \gamma < 1$ we show local minima is locally of class $C^{1,\alpha}$ for a sharp $\alpha$ that depends on dimension, $p$ and $\gamma$. For $\gamma = 0$ we obtain a quantitative, asymptotically optimal result, which assures that local minima are Log-Lipschitz continuous. The results proven in this article are new even in the classical context of linear, nondegenerate equations.