A free boundary problem on three-dimensional cones

This talk concerns the one-phase free boundary problem on three-dimensional cones. The one-phase problem has a well-known connection to the study of area-minimizing surfaces, and a result by Morgan classifies when an area minimizing surface is allowed to pass through the vertex of the cone. In this talk we present a result analogous to that of Morgan's. 
We consider  cones depending on a parameter c and study when the free boundary is allowed to pass through the vertex of the cone.  When the cone is three-dimensional and c is large enough, the free boundary avoids the vertex. We also show that when c is small enough but still positive, the free boundary is allowed to pass through the vertex. This establishes 3 as the critical dimension for which the free boundary may pass through the vertex of a right circular cone.