A free boundary problem on three-dimensional cones
Abstract
We consider a free boundary problem on cones depending on a parameter c and study when the free boundary is allowed to pass through the vertex of the cone. We show that 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. In view of the well-known connection between area-minimizing surfaces and the free boundary problem under consideration, our result is analogous to a result of Morgan that classifies when an area-minimizing surface on a cone passes through the vertex.
Cite
@article{arxiv.1704.05131,
title = {A free boundary problem on three-dimensional cones},
author = {Mark Allen},
journal= {arXiv preprint arXiv:1704.05131},
year = {2017}
}
Comments
23 pages