An epiperimetric inequality approach to the regularity of the free boundary in the thin and fractional obstacle problems

We will discuss generalizations of Weiss's homogeneity improvement approach to the thin and fractional obstacle problems. The main ingredients are an epiperimetric inequality and a monotonicity formula, which give a powerful combination in the analysis of free boundaries and establish the $C^{1,\alpha}$ regularity of the regular set. The advantage of this method is that it is purely energy based and allows generalization to the case of thin obstacles living on codimension one $C^{1,1}$ manifolds, or more generally, the thin obstacle problem for the divergence form operators with Lipschitz coefficients. The method can also be used in the study of the obstacle problem for the fractional Laplacian with drift, when the fractional order is greater that one half.

Based on joint works with Nicola Garofalo, Camelia Pop, and Mariana Smit Vega Garcia.