On the Cheeger inequality in Carnot-Carath\'eodory spaces

Martijn Kluitenberg

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arXiv · 2312.13058

On the Cheeger inequality in Carnot-Carath\'eodory spaces

Martijn Kluitenberg

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Abstract

We generalize the Cheeger inequality, a lower bound on the first nontrivial eigenvalue of a Laplacian, to the case of geometric sub-Laplacians on rank-varying Carnot-Carath\'eodory spaces and we describe a concrete method to lower bound the Cheeger constant. The proof is geometric, and works for Dirichlet, Neumann and mixed boundary conditions. One of the main technical tools in the proof is a generalization of Courant's nodal domain theorem, which is proven from scratch for Neumann and mixed boundary conditions. Carnot groups and the Baouendi-Grushin cylinder are treated as examples.

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BibTeX RIS

Martijn Kluitenberg. 2023-12-20. On the Cheeger inequality in Carnot-Carath\'eodory spaces. https://arxiv.org/abs/2312.13058

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