We prove quasi-monotonicity formulae for classical obstacle-type problems with quadratic energies with coefficients in fractional Sobolev spaces, and a linear term with a Dini-type continuity property. These formulae are used to obtain the regularity of free-boundary points following the approaches by Caffarelli, Monneau and Weiss.DiMaI, Università degli studi di Firenze

Dedicated to Nicola Fusco, a mentor and a friend.Abstract. We develop the complete free boundary analysis for solutions to classical obstacle problems related to nondegenerate nonlinear variational energies. The key tools are optimal C 1,1 regularity, which we review more generally for solutions to variational inequalities driven by nonlinear coercive smooth vector fields, and the results in [16] concerning the obstacle problem for quadratic energies with Lipschitz coefficients.Furthermore, we highlight similar conclusions for locally coercive vector fields having in mind applications to the area functional, or more generally to area-type functionals, as well.

We establish Weiss' and Monneau's type quasi-monotonicity formulas for quadratic energies having matrix of coefficients in a Sobolev space W 1,p , p > n, and provide an application to the corresponding free boundary analysis for the related classical obstacle problems.

In this paper we give a proof of an epiperimetric inequality in the setting of the lower dimensional obstacle problem. The inequality was introduced by Weiss (Invent. Math., 138 (1999), no. 1, 23-50) for the classical obstacle problem and has striking consequences concerning the regularity of the free-boundary. Our proof follows the approach of Focardi and Spadaro (Adv. Differential Equations 21 (2015), no 1-2, 153-200.) which uses an homogeneity approach and a Γ-convergence analysis.

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