2023/10/06 by Chee Han Tan, Tan, Chee Han, Robert Viator +1 · 2 citations
Computer Science · Mathematics · #35C20 #35P05 #41A58 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Asymptotic expansion #Ball (mathematics) #Eigenvalues and eigenvectors #FOS: Mathematics #Hermitian matrix #Isoperimetric inequality #Mathematical analysis #Mathematics #Optimization and Control (math.OC) #Perturbation (astronomy) #Physics #Pure mathematics #Quantum mechanics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2310.03960
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2023/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We consider Steklov eigenvalues of nearly hyperspherical domains in ℝd + 1 with d≥ 3. In previous work, treating such domains as perturbations of the ball, we proved that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term of the asymptotic expansion and show that the first-order perturbations are eigenvalues of a Hermitian matrix, whose entries can be written explicitly in terms of the Pochhammer's and Wigner 3j-symbols. We analyse the asymptotic expansion and show the following isoperimetric results among domains with fixed volume: (1) for an infinite subset of Steklov eigenvalues, the ball is not optimal, and (2) for a different infinite subset of Steklov eigenvalues, the ball is a stationary point.