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

If on a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$ there is a nonconstant Neumann eigenfunction $u$ that is locally constant on the boundary, must $\mathrm{\Omega }$ 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 $\mathrm{\partial }\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ with the above property. As a consequence, a certain linear combination of the indicator functions of the domains $\mathrm{\Omega }$ and of the bounded component of the complement ${\mathbb{R}}^2\mathrm{\backslash }\bar{\mathrm{\Omega }}$ 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.