We consider fully nonlinear obstacle-type problems of the formwhere Ω is an unknown open set and K > 0. In particular, structural conditions on F are presented which ensure that W 2,n (B1) solutions achieve the optimal C 1,1 (B 1/2 ) regularity when f is Hölder continuous. Moreover, if f is positive on B1, Lipschitz continuous, and {u = 0} ⊂ Ω, then we obtain local C 1 regularity of the free boundary under a uniform thickness assumption on {u = 0}. Lastly, we extend these results to the parabolic setting.

In the study of classical obstacle problems, it is well known that in many configurations the free boundary intersects the fixed boundary tangentially. The arguments involved in producing results of this type rely on the linear structure of the operator. In this paper we employ a different approach and prove tangential touch of free and fixed boundary in two dimensions for fully nonlinear elliptic operators. Along the way, several n-dimensional results of independent interest are obtained such as BMOestimates, C 1,1 regularity up to the fixed boundary, and a description of the behavior of blow-up solutions.

We study the semilinear Poisson equation(1)Our main results provide conditions on f which ensure that weak solutions of (1) belong to C 1,1 (B 1/2 ). In some configurations, the conditions are sharp.

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We use the mean value property in an asymptotic way to provide a notion of a pointwise Laplacian, called AMV Laplacian, that we study in several contexts including the Heisenberg group and weighted Lebesgue measures. We focus especially on a class of metric measure spaces including intersecting submanifolds of R n , a context in which our notion brings new insights; the Kirchhoff law appears as a special case. In the general case, we also prove a maximum and comparison principle, as well as a Green-type identity for a related operator.