Solutions to overdetermined elliptic problems in nontrivial exterior domains

TL;DR

This paper constructs specific exterior domains in all dimensions where a certain overdetermined elliptic boundary value problem admits positive solutions, providing counterexamples to a longstanding conjecture in dimension 2.

Contribution

It presents the first exterior domain counterexamples to the Berestycki-Caffarelli-Nirenberg conjecture in dimension 2, expanding understanding of overdetermined elliptic problems.

Findings

Counterexamples in dimension 2 for the conjecture

Existence of positive solutions in constructed exterior domains

First exterior domain counterexamples in the literature

Abstract

In this paper we construct nontrivial exterior domains $\Omega \subset \mathbb{R}^N$, for all $N\geq 2$, such that the problem $$\left\{ {ll} -\Delta u +u -u^p=0,\ u >0 & \mbox{in }\; \Omega, {1mm] \ u= 0 & \mbox{on }\; \partial \Omega, [1mm] \ \frac{\partial u}{\partial \nu} = \mbox{cte} & \mbox{on }\; \partial \Omega, \right.$$ admits a positive bounded solution. This result gives a negative answer to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems in dimension 2, the only dimension in which the conjecture was still open. For higher dimensions, different counterexamples have been found in the literature; however, our example is the first one in the form of an exterior domain.