We provide a rather complete description of the sharp regularity theory for a family of heterogeneous, two-phase variational free boundary problems, J γ → 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 γ becomes singular along the free interface {u = 0}. The degree of singularity is, in turn, dimed by the parameter γ ∈ [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.KEYWORDS: free boundary problems, degenerate elliptic operators, regularity theory. MSC2000: 35R35, 35J70, 35J75, 35J20.

We study regularity properties of solutions to reaction-diffusion equations ruled by the infinity laplacian operator. We focus our analysis in models presenting plateaus, i.e. regions where a non-negative solution vanishes identically. We obtain sharp geometric regularity estimates for solutions along the boundary of plateaus sets. In particular we show that the (n − ε)-Hausdorff measure of the plateaus boundary is finite, for a universal number ε > 0.

Abstract. We consider fully nonlinear integro-differential equations governed by kernels that have different homogeneities in different directions. We prove a nonlocal version of the ABP estimate, a Harnack inequality and the interior C 1,γ regularity, extending the results of [4] to the anisotropic case.

We consider an one-phase free boundary problem for a degenerate fully non-linear elliptic operators with non-zero right hand side. We use the approach present in [DeS] to prove that flat free boundaries and Lipschitz free boundaries are C 1,γ . keywords: free boundary problems, degenerate fully non-linear elliptic operators, regularity theory.

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