Singular Perturbation Problem in Boundary/Fractional Combustion

Motivated by a nonlocal free boundary problem, we study uniform properties of solutions to a singular perturbation problem for a boundary-reaction-diffusion equation, where the reaction term is of combustion type. This boundary problem is related to the fractional Laplacian. After an optimal uniform H\"older regularity is shown, we pass to the limit to study the free boundary problem it leads to

An epiperimetric inequality approach to the regularity of the free boundary in the Signorini problem with variable coefficients

In this paper we establish the $C^{1,\beta}$ regularity of the regular part of the free boundary in the Signorini problem for elliptic operators with variable Lipschitz coefficients. This work is a continuation of the recent paper [GSVG14], where two of us established the interior optimal regularity of the solution. Two of the central results of the present work are a new monotonicity formula and a new epiperimetric inequality.Comment: 40 page