A Free Boundary Problem on Cones
The one phase free boundary problem shares a well-known connection to area-minimizing surfaces. In this talk we review this connection and then discuss the one-phase problem on rough surfaces, and in particular cones. After reviewing results of the author with Chang Lara for the one-phase problem on two-dimensional cones, we revisit the connection to area-minimizing surfaces to gain insight into the problem on higher dimensional cones. We then present new results on when the free boundary is allowed to pass through the vertex of a three-dimensional cone as well as results for higher dimensional cones.