We study the regularity of the regular and of the singular set of some free boundary problems in any dimension. To this end we introduce a new tool, which we call logarithmic epiperimetric inequality, which works also at singular points. It allows to study the regularity of the singular set and yields an explicit logarithmic modulus of continuity on the $C^1$ regularity (which is optimal in general), thus improving the known regularity and providing a fully alternative method. 

In turn, the logarithmic epiperimetric inequality follows from a suitable Lojasiewicz inequality for a (non-analytic) constrained functional associated to the obstacle problem.

The talk is based on joint work with Luca Spolaor and Bozhidar Velichkov.