Asymptotic shape of optimizers for spectral Riesz mean problems

Establish that, in the limit λ→0, optimizing sets Ω_λ (among a given class C of open sets in R^d with |Ω|=1) for the supremum sup_{Ω∈C} Tr((−Δ_D)_Ω−λ)^γ and the infimum inf_{Ω∈C} Tr((−Δ_N)_Ω−λ)^γ converge, in an appropriate sense of shape convergence, to a ball of unit measure.

Background

The paper introduces spectral shape optimization problems that maximize (Dirichlet) or minimize (Neumann) Riesz means of Laplace spectra over sets of fixed volume. Heuristics based on two-term Weyl asymptotics suggest that the ball should be the asymptotic optimizer when the class C contains a ball, reducing the problem to the classical isoperimetric inequality.

While the authors prove results in convex classes and outline approaches using their main theorems, the general convergence of optimizers to a ball under broad classes C remains conjectural and is identified as an open problem in the introduction.