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首页> 外文期刊>Annales scientifiques de l'Ecole normale superieure >HIGHER ORDER CHEEGER INEQUALITIES FOR STEKLOV EIGENVALUES BY
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HIGHER ORDER CHEEGER INEQUALITIES FOR STEKLOV EIGENVALUES BY

机译：Steklov Eigenvalues的高阶Cheeger不等式

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摘要

We prove a lower bound for the k-th Steklov eigenvalues in terms of an isoperimetric constant called the k-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Steklov eigenvalues. In particular it extends the Cheeger type inequality for the first nonzero Steklov eigenvalue previously studied by Escobar in 1997 and by Jammes in 2015 to higher order Steklov eigenvalues. The technique we develop to get this lower bound is based on considering a family of accelerated Markov operators in the finite and measurable situations and of mass concentration deformations of the Laplace-Beltrami operator in the manifold setting which converges uniformly to the Steklov operator. As an intermediary step in the proof of the higher order Cheeger type inequality, we define the Dirichlet-Steklov connectivity spectrum and show that the Dirichlet connectivity spectra of this family of operators converges to (or is bounded by) the Dirichlet-Steklov spectrum uniformly. Moreover, we obtain bounds for the Steklov eigenvalues in terms of its Dirichlet-Steklov connectivity spectrum which is interesting in its own right and is more robust than the higher order Cheeger type inequalities. The Dirichlet-Steklov spectrum is closely related to the Cheeger-Steklov constants.
机译：我们在三种不同情况下称为K-Th Cheeger-Steklov常数的异常常数，我们证明了K-TH Steklov特征值的下限：有限空间，可测量的空间和黎曼歧管。这些下限可被认为是Steklov特征值的更高阶的Cheeger型不等式。特别是它延长了1997年以前通过Escobar研究的第一个非零斯蒂克洛夫特征值的Cheeger型不等式，并通过2015年的鼠所知到高阶Steklov特征值。我们开发获得这种下限的技术是基于在有限和可测量的情况下考虑一个加速马尔可夫运营商的系列，以及Laplace-Beltrami操作员在歧管设置中的Laplace-Beltrami操作员的质量浓度变形，这均匀地收敛到Steklov运算符。作为较高阶Cheeger型不等式证明的中间步骤，我们定义了Dirichlet-Steklov连接频谱，并表明该族的操作员的Dirichlet连接光谱均匀地收敛到Dirichlet-Steklov光谱的（或界定）。此外，我们在其独特 - Steklov连接光谱方面获得了Steklov特征值的界限，其自身右侧是有趣的，并且比高阶夹齿型不等式更强大。 Dirichlet-Steklov光谱与Cheeger-Steklov常数密切相关。

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