Asymptotic behaviour of cuboids optimising Laplacian eigenvalues

2017/03/29 by Katie Gittins, Gittins, Katie, Simon Larson +1
Computer Science · Mathematics · #35J20 #35P99 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1703.10249

openalex publication_date 2017/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that in dimension n ≥ 2, within the collection of unit measure cuboids in ℝⁿ (i.e. domains of the form ∏_i=1ⁿ(0, a_n)), any sequence of minimising domains R_k^D for the Dirichlet eigenvalues λ_k converges to the unit cube as k → ∞. Correspondingly we also prove that any sequence of maximising domains R_k^N for the Neumann eigenvalues μ_k within the same collection of domains converges to the unit cube as k→ ∞. For n=2 this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues and van den Berg, Bucur and Gittins for the Neumann eigenvalues. The Dirichlet case for n=3 was recently treated by van den Berg and Gittins. In addition we obtain stability results for the optimal eigenvalues as k → ∞. We also obtain corresponding shape optimisation results for the Riesz means of eigenvalues in the same collection of cuboids. For the Dirichlet case this allows us to address the shape optimisation of the average of the first k eigenvalues.

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