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arXiv · 1704.05131

A free boundary problem on three-dimensional cones

Mark Allen
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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.

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Mark Allen. 2017-04-17. A free boundary problem on three-dimensional cones. https://arxiv.org/abs/1704.05131

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