# A free boundary problem on three-dimensional cones

## TL;DR

This paper investigates a free boundary problem on three-dimensional cones, identifying conditions under which the free boundary passes through the vertex, and establishes the critical dimension for this behavior.

## Contribution

It determines the critical dimension for free boundary passage through the vertex of cones, linking geometric analysis with free boundary problem behavior.

## Key findings

- For large c, the free boundary avoids the vertex.
- For small positive c, the free boundary passes through the vertex.
- Identifies 3 as the critical dimension for this phenomenon.

## 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.