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

If on a smooth bounded domain Ω⊂R2 there is a (nonconstant) Neumann Laplace eigenfunction u that is locally constant on the boundary, must Ω be a disk or an annulus? This question can be understood as a weaker analogue of the well-known Schiffer conjecture, in that the eigenfunction u is allowed to take a different constant value on each connected component of ∂Ω, 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 Ω with the above 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.