A Schiffer-type problem for annuli with applications to stationary planar Euler flows

TL;DR

This paper constructs nontrivial doubly connected domains with special Neumann eigenfunctions, challenging a classical rigidity conjecture and demonstrating new stationary solutions to 2D Euler flows that are not locally radial.

Contribution

It provides a negative answer to a Schiffer-type problem for annuli, constructing explicit counterexamples and revealing new stationary Euler flows.

Findings

Existence of nontrivial doubly connected domains with locally constant boundary eigenfunctions

Failure of the Pompeiu property for certain domain indicator functions

Construction of non-radial stationary solutions to 2D Euler equations

Abstract

If on a smooth bounded domain $\Omega\subset\mathbb{R}^2$ there is a nonconstant Neumann eigenfunction $u$ that is locally constant on the boundary, must $\Omega$ be a disk or an annulus? This question can be understood as a weaker analog of the well known Schiffer conjecture, in that the function $u$ is allowed to take a different constant value on each connected component of $\partial \Omega$ yet many of the known rigidity properties of the original problem are essentially preserved. Our main result provides a negative answer by constructing a family of nontrivial doubly connected domains $\Omega$ with the above property. As a consequence, a certain linear combination of the indicator functions of the domains $\Omega$ and of the bounded component of the complement $\mathbb{R}^2\backslash\overline{\Omega}$ fails to have the Pompeiu property. Furthermore, our construction implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial.