Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of ∆u at the origin is Dini (in Lp average), then the origin is somehow a Lebesgue point of continuity (still in Lp average) for the second derivatives D2u. We extend this pointwise regularity result to the obtacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a kind of Taylor expansion up to the order 2 (in the Lp average). Moreover we get a quatitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quatitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula. AMS Classification: 35R35.