Convex planar Schiffer counterexample

Determine whether there exists a bounded convex non-ball domain in the plane with smooth or real-analytic boundary that admits a nonconstant solution of the Schiffer overdetermined problem Δu+μu=0 in Ω, with u constant and ∂νu=0 on ∂Ω.

Background

The paper constructs convex non-ball counterexamples to the Schiffer and Pompeiu conjectures in dimensions 3, 4, 6, 8, 10, and 14. It also notes that earlier planar counterexamples are noncircular but do not establish convexity, leaving unresolved whether the Schiffer phenomenon can occur for a convex planar domain.

The planar Schiffer problem asks whether a bounded domain in the plane supporting a nonconstant Helmholtz solution with constant Dirichlet data and vanishing normal derivative must be a disc. The stated open problem concerns the existence of a convex planar counterexample to this rigidity assertion.