The Classical Obstacle Problem for nonlinear variational energies and related problems

In this thesis we investigate the classical obstacal problem for nonlinear variational energies and related problems. 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 type-Dini continuity property. These formulae are used to obtain the regularity of free boundary points following the approaches by Caffarelli, Monneau and Weiss. 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 Focardi et al. (2015) 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 prove also an epiperimetric inequality for the fractional obstacle problem thus extending the pioneering results by Weiss (1999) on the classical obstacle problem and the results of Focardi and Spadaro (2016) in the thin obstacle problem. We deduce the regularity of a suitable subset of the free boundary as a consequence of a decay estimate of a boundary adjusted energy “à la Weiss”, the non degeneracy of the solution and the uniqueness of the limits of suitable rescaled funciotns.