Validity of the Dirichlet-to-Neumann conjectures for simply connected non-convex planar domains

Determine whether Conjectures A and B for the Dirichlet-to-Neumann eigenvalues are valid for simply connected non-convex planar domains.

Background

The paper studies two conjectures concerning the Dirichlet-to-Neumann map for a bounded Lipschitz domain and non-positive Helmholtz parameter. Conjecture A asserts that, along every fixed real-analytic eigenvalue branch, the increase from the parameter value 0 to a parameter $\Lambda\leq0$ is at most $\sqrt{-\Lambda}$. Conjecture B is the corresponding ordered-eigenvalue inequality, asserting $$\sigma_k^{(\Lambda)}-\sigma_k^{(0)}\leq\sqrt{-\Lambda}$$ for every $k$.

The paper proves these conjectures for convex domains and obtains related results under weaker geometric assumptions, but constructs counterexamples for general non-convex domains, including annuli and Swiss-cheese domains. The authors explicitly leave unresolved whether either conjecture remains true in the narrower class of simply connected non-convex planar domains.