Regularity for degenerate two-phase free boundary problems

Abstract

We provide a rather complete description of the sharp regularity theory to a family of heterogeneous, two-phase 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, dimmed by the parameter $\gamma \in [0,1]$. For $0<\gamma <1$ we show that local minima are 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.