Monotonicity formulas for obstacle problems with Lipschitz coefficients

We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a Hölder 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 (J Fourier Anal Appl 4(4–5), 383–402, 1998), Monneau (J Geom Anal 13(2), 359–389, 2003), and Weiss (Invent Math 138(1), 23–50, 1999)