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, Jγ→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 Jγ becomes singular along the free interface {u=0}. The degree of singularity is, in turn, dimmed by the parameter γ∈[0,1]. For 0<γ<1 we show that local minima are locally of class C1,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.