Optimisation and monotonicity of the second Robin eigenvalue on a planar exterior domain
David Krejcirik, Vladimir Lotoreichik
TL;DR
This work analyzes the Robin Laplacian on planar exterior domains with negative boundary coupling, focusing on the second eigenvalue $\lambda_2^\alpha(\Omega^{\rm c})$ and its extremal properties. It develops a detailed spectral framework for exterior domains, obtains explicit results for the exterior of a disk via a fibre decomposition with Bessel functions, and proves monotonicity and isoelastic optimization results under geometric constraints. The main contributions are (i) a monotonicity result for $\lambda_2^\alpha(\Omega^{\rm c})$ under strict star-shaped and central symmetry assumptions, (ii) an isoelastic inequality showing $\lambda_2^\alpha(\Omega^{\rm c})$ is strictly reduced by imposing the same elastic energy as a disk when $\Omega$ is convex and not congruent to the disk, and (iii) a precise characterization of the disk case with explicit eigenpairs and the critical coupling constant $\alpha_\star$. Overall, the paper advances the understanding of spectral optimization for exterior Robin problems and highlights the role of geometry, symmetry, and elastic energy in shaping bound-state energies.
Abstract
We consider the Laplace operator in the exterior of a compact set in the plane, subject to Robin boundary conditions. If the boundary coupling is sufficiently negative, there are at least two discrete eigenvalues below the essential spectrum. We state a general conjecture that the second eigenvalue is maximised by the exterior of a disk under isochoric or isoperimetric constraints. We prove an isoelastic version of the conjecture for the exterior of convex domains. Finally, we establish a monotonicity result for the second eigenvalue under the condition that the compact set is strictly star-shaped and centrally symmetric.