Regularity for degenerate two-phase free boundary problems

We provide a rather complete description of the sharp regularity theory to a family of heterogeneous, two-phase free boundary problems, Jγ→minJγ→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 JγJγ becomes singular along the free interface {u=0}{u=0}. The degree of singularity is, in turn, dimmed by the parameter γ∈[0,1]γ∈[0,1]. For 0<γ<10<γ<1 we show that local minima are locally of class C1,αC1,α for a sharp α that depends on dimension, p and γ  . For γ=0γ=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.