Monotonicity formulas for obstacle problems with Lipschitz coefficients

Abstract: We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a H\"older continuous linear term. With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli, Weiss and Monneau.