Quasi-Monotonicity Formulas for Classical Obstacle Problems with Sobolev Coefficients and Applications

Abstract

We establish Weiss and Monneau’s type quasi-monotonicity formulas for quadratic energies having a matrix of coefficients in a Sobolev space with summability exponent larger than the space dimension. We provide applications to the free boundary analysis for the corresponding classical obstacle problems first, and then for nonlinear classical obstacle problems thanks to a linearization argument