Geometry and Topology of some overdetermined elliptic problems

TL;DR

This paper explores geometric and topological conditions of domains in the plane that allow positive solutions to overdetermined elliptic problems, using tools from constant mean curvature surface theory, and extends some results to higher dimensions.

Contribution

It provides new necessary conditions on domain geometry and topology for solutions, partially answers a longstanding question, and generalizes some results to higher dimensions.

Findings

Identifies geometric constraints for solutions in 

Provides partial answer to Berestycki, Caffarelli, Nirenberg question

Extends some results to higher-dimensional domains

Abstract

We study necessary conditions on the geometry and the topology of domains in $\mathbb{R}^2$ that support a positive solution to a classical overdetermined elliptic problem. The ideas and tools we use come from constant mean curvature surface theory. In particular, we obtain a partial answer to a question posed by H. Berestycki, L. Caffarelli and L. Nirenberg in 1997. We investigate also some boundedness properties of the solution $u$. Some of our results generalize to higher dimensions.