The Steklov and Laplacian spectra of Riemannian manifolds with boundary
The Steklov and Laplacian spectra of Riemannian manifolds with boundary
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DOI：
10.1016/j.jfa.2019.108409
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发表时间：
2018-10
影响因子：
1.7
通讯作者：
B. Colbois;A. Girouard;Asma Hassannezhad
中科院分区：
文献类型：
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作者：
B. Colbois;A. Girouard;Asma Hassannezhad
给定两个紧致黎曼流形M1和M2，使得它们各自的边界Ω 1和Ω 2允许等距的邻域Ω 1和Ω 2，我们证明了存在常数C，使得|σ k（M 1）− σ k（M 2）|对于每个k∈ N，≤ C。常数C仅取决于Ω 1 <$Ω 2的几何形状。这是由紧致黎曼流形M的Steklov本征值σ k与其边界上拉普拉斯算子的本征值λ k之间的定量关系得出的。我们的主要结果表明，|σ k− λ k|是由一个常数，这取决于几何的M只在其边界的邻域以上有界。证明是基于Pohozaev身份和比较几何的主曲率的平行超曲面。在几种情况下，常数C是根据Ω 1 <$Ω 2的几何界限明确给出的。
Given two compact Riemannian manifolds M 1 and M 2 such that their respective boundaries Σ 1 and Σ 2 admit neighbourhoods Ω 1 and Ω 2 which are isometric, we prove the existence of a constant C such that| σ k (M 1)− σ k (M 2)|≤ C for each k∈ N. The constant C depends only on the geometry of Ω 1≅ Ω 2. This follows from a quantitative relationship between the Steklov eigenvalues σ k of a compact Riemannian manifold M and the eigenvalues λ k of the Laplacian on its boundary. Our main result states that the difference| σ k− λ k| is bounded above by a constant which depends on the geometry of M only in a neighbourhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant C is given explicitly in terms of bounds on the geometry of Ω 1≅ Ω 2.