Optimization of the second Robin eigenvalue on a planar exterior domain under fixed area constraint

Abstract

We consider the Robin Laplacian on a planar exterior domain with a negative boundary parameter. For a negative boundary parameter of a sufficiently large absolute value this operator has at least two negative eigenvalues below the bottom of the essential spectrum. We prove that under a fixed area constraint and convexity assumption on the bounded complement of the exterior domain the second Robin eigenvalue is maximized by the exterior of the disk. The inequality is strict for every exterior of a non-circular domain whenever the comparison disk has a negative second eigenvalue. The proof modifies the parallel-coordinate method by allowing the angular phase of a trial function to vary with the distance from the boundary. This result resolves the first item in a 2024 conjecture by the authors under an additional convexity assumption. Relaxing the convexity and the perimeter constraint remain open problems.