HIGHER ORDER CHEEGER INEQUALITIES FOR STEKLOV EIGENVALUES
HIGHER ORDER CHEEGER INEQUALITIES FOR STEKLOV EIGENVALUES
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DOI：
10.24033/asens.2417
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发表时间：
2020-01-01
影响因子：
1.9
通讯作者：
Miclo, Laurent
中科院分区：
文献类型：
--
作者：
Hassannezhad, Asma;Miclo, Laurent
我们证明了一个下界的第k次Steklov特征值的等周常数称为第k次Cheeger-Steklov常数在三种不同的情况下：有限空间，可测空间，黎曼流形。这些下界可以看作是Steklov特征值的高阶Cheeger型不等式。特别地，它将Escobar在1997年和Jammes在2015年研究的第一个非零Steklov特征值的Cheeger型不等式扩展到高阶Steklov特征值。我们开发的技术，以获得这个下限是基于考虑一个家庭的加速马尔可夫算子在有限的和可测量的情况下和质量浓度变形的拉普拉斯-贝尔特拉米算子在流形设置一致收敛到Steklov运营商。作为证明高阶Cheeger型不等式的中间步骤，我们定义了Dirichlet-Steklov连通谱，并证明了这类算子的Dirichlet连通谱一致收敛于（或有界于）Dirichlet-Steklov谱.此外，我们得到的Steklov特征值的Dirichlet-Steklov连通谱，这是有趣的，在其本身的权利，是更强大的比高阶Cheeger型不等式的界限。Dirichlet-Steklov谱与Cheeger-Steklov常数密切相关。
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.