Alt-Caffarelli-Friedman Monotonicity formulas

In this thesis, the main topic is the study of the Alt-Caffarelli-Friedman monotonicity formula, its generalizations and its relation with a free boundary problem. The principal interest of the work is to provide a detailed proof of the first elliptic ACF formula. Then, it is studied the parabolic counterpart and some extensions for variable coefficients operators. The last part of thesis focuses on the existence of the ACF formula in Carnot groups. After recalling the results of Ferrari and Forcillo on the Heisenberg group, it is obtained a necessary condition on the existence of an ACF formula on Carnot groups exploiting the lack of orthogonality of intrinsic harmonic homogeneous polynomials. The necessary condition is applied to case of the 3-step Engel group and is given an explicit counterexample.