Regularity for degenerate two-phase free boundary problems

We provide a rather complete description of the sharp regularity theory for a family of heterogeneous, two-phase variational free boundary problems, $\mathcal{J}_γ\to$ 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 $\mathcal{J}_γ$ becomes singular along the free interface $\{u= 0\}$. The degree of singularity is, in turn, dimed by the parameter $γ\in [0,1]$. For $0< γ< 1$ we show local minima is locally of class $C^{1,α}$ for a sharp $α$ that depends on dimension, $p$ and $γ$. For $γ= 0$ 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.

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