The classical obstacle problem for nonlinear variational energies

TL;DR

This paper provides a comprehensive free boundary analysis for classical obstacle problems involving nonlinear variational energies, emphasizing regularity results and applications to area-type functionals.

Contribution

It extends regularity and free boundary analysis techniques to obstacle problems with nonlinear energies and vector fields, including applications to area functionals.

Findings

Achieved optimal $C^{1,1}$ regularity for solutions.

Extended analysis to nonlinear coercive vector fields.

Applied results to area and area-type functionals.

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 \cite{FocGelSp15} 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.