On eigenvalue problems related to the laplacian in a class of doubly connected domains
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n eigenvalue problems related to the laplacian in a class of doubly connected domains Sheela Verma1 · G. Santhanam2 Received: 20 April 2020 / Accepted: 15 September 2020 / Published online: 22 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We study eigenvalue problems in some specific class of doubly connected domains. In particular, we prove the following. 1. Let B1 be an open ball in Rn , n > 2 and B0 be an open ball contained in B1 . Then the first eigenvalue of the problem Δu = 0 in B1 \ B 0 , u = 0 on ∂ B0 , ∂u ∂ν = τ u on ∂ B1 , attains its maximum if and only if B0 and B1 are concentric. Here ν is the outward unit normal on ∂ B1 and τ is a real number. 2. Let B0 ⊂ M be a geodesic ball of radius r centered at a point p ∈ M, where M denote either a non-compact rank-1 symmetric space (M, ds 2 ) with curvature −4 ≤ K M ≤ −1 or M = Rm . Let D ⊂ M be a domain of fixed volume which is geodesically symmetric with respect to the point p ∈ M such that B0 ⊂ D. Then the first non-zero eigenvalue of Δu = μu in D \ B 0 , ∂u on ∂(D \ B 0 ), ∂ν = 0 attains its maximum if and only if D is a geodesic ball centered at p. Here ν represents the outward unit normal on ∂(D \ B 0 ) and μ is a real number. Keywords Laplacian · Neumann eigenvalue problem · Steklov–Dirichlet eigenvalue problem · Doubly connected domain · Non-compact rank-1 symmetric space · Geodesically symmetric domain Mathematics Subject Classification 35P15 · 58J50
Communicated by Adrian Constantin. Extended author information available on the last page of the article
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1 Introduction In the last few years, the study of an eigenvalue problem on a punctured domain has been a topic of interest. Several interesting results have been proved in this area by considering various boundary conditions on a punctured domain. In [9], Ramm and Shivakumar considered Dirichlet boundary condition on a punctured ball in R2 (a ball of smaller radius is removed from a ball) and proved that the first eigenvalue of this problem attains its maximum if and only if the balls are concentric. In [8], Kesavan proved the above result for higher dimensions. Later this result was extended to a wider class of domains [1,4–7]. In [3], Banuelos et al. proved some classical inequalities between the eigenvalues of the mixed Steklov–Dirichlet and mixed Steklov–Neumann eigenvalue problems on a bounded domain in Rn . In this paper, we consider the mixed Steklov–Dirichlet problem and Neumann eigenvalue problem on some specific class of punctured domains and prove that the first non-zero eigenvalue of both problems is maximal for annular domain. In between, we also give the characterization of the first non-zero eigenvalue of the Neumann problem on an annulus. To the best of our knowledge, this is the first attempt to consider these problems on such domains. Now we state the main result related to the mixed Steklov–Dirichlet problem. For n > 2, let B1 be an open ball in Rn of radius R1 and B2 be an open ball in Rn of radius R2 such that B
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