On Eigenvalue Problems Related to the Laplacian in a Class of Doubly Connected Domains

2018/03/15 by Sheela Verma, Verma, Sheela · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics #math.DG

paper · pdf · doi:10.48550/arxiv.1803.05750

arxiv created 2019/09/24 · arxiv updated 2019/09/25

Abstract

We consider two eigenvalue problems for Laplacian on some specific doubly connected domain. In particular, we study the following two eigenvalue problems. Let B₁ be an open ball in ℝⁿ and B₀ be a ball contained in B₁. Let ν be the outward unit normal on ∂ B₁. Then the first eigenvalue of the problem Δu · = · 0 · in B₁ ∖ B₀ ,
u · = · 0 · on ∂ B₀,
(∂ u)/(∂ ν) · = · τ u · on ∂ B₁, attains maximum if and only if B₀ and B₁ are concentric. Let D be a domain in a non-compact rank-1 symmetric space (\mathbbM, ds²), geodesically symmetric with respect to the point p∈ \mathbbM. Let B₀ be a ball in \mathbbM centered at p such that B₀⊂ D and ν be the outward unit normal on ∂ (D ∖ B₀). Then the first non-zero eigenvalue of Δu · = · μ u · in D ∖ B₀,
(∂ u)/(∂ ν) · = · 0 · on ∂ (D ∖ B₀), attains maximum if and only if D is a geodesic ball centered at p.

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