Optimizing Riesz means of Robin Laplace operators on cuboids in a semiclassical limit

Abstract

We study asymptotic shape optimization for Riesz means of Robin Laplacian eigenvalues among cuboids of fixed measure. Our focus is the regime where the Robin parameter is proportional to the square root of the spectral parameter defining the Riesz means. Here, a transition emerges based on the precise ratio between the two parameters: as the spectral parameter tends to infinity, sequences of maximizers shift from converging to the unit cube to lacking convergent subsequences entirely. Key tools include two-term spectral asymptotics and uniform inequalities for the Riesz means. Notably, the transition point governing the behavior of optimizers may differ from the point at which the second asymptotic term changes sign. This shows that heuristics based solely on asymptotics for a fixed domain fail to accurately predict the asymptotic behavior of maximizers.