Regularity for degenerate two-phase free boundary problems

Leitão, Raimundo; de Queiroz, Olivaine S.; Teixeira, Eduardo V.

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 4, pp. 741-762

We provide a rather complete description of the sharp regularity theory to a family of heterogeneous, two-phase free boundary problems, ${𝒥}_{\gamma }\to \min$, 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 ${𝒥}_{\gamma }$ 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 α that depends on dimension, p and γ. 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.

Classification : 35R35, 35J70, 35J75, 35J20
Keywords: Free boundary problems, Degenerate elliptic operators, Regularity theory

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