Mark Allen
Brigham Young University, Provo, USA

We consider the two-phase thin free boundary problem which is the fractional Laplacian analogue of the two-phase Bernoulli free boundary problem. We consider the situation of almost-minimizers to the corresponding functional. We show that there is a separation of phases; consequently, the problem is reduced to the study of the one-phase problem. We adapt the techniques for the case in which $s=1/2$ to the full range 0<$s$<1 for almost minimizers to show that the free boundary is $C^{1,\alpha}$ in a neighborhood of flat points. This is joint work with Mariana Smit Vega Garcia.