Free boundary on a cone

TL;DR

This paper investigates free boundary problems on a 2D cone, showing that the free boundary's behavior depends on the length of the generating curve, with the vertex being avoided or included in the boundary.

Contribution

It establishes a clear relationship between the length of the generating curve and the free boundary's location on the cone.

Findings

For length(c3)<2c0, the free boundary avoids the vertex.

For length(c3)e;2c0, examples show the free boundary can include the vertex.

The behavior of minimizers depends on the geometric property of the cone.

Abstract

We study two phase problems posed over a two dimensional cone generated by a smooth curve $\gamma$ on the unit sphere. We show that when $length(\gamma)<2\pi$ the free boundary avoids the vertex of the cone. When $length(\gamma) \geq 2\pi$ we provide examples of minimizers such that the vertex belongs to the free boundary.