Upper bounds for higher-order Poincaré constants
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- by Kei Funano and Yohei Sakurai PDF
- Trans. Amer. Math. Soc. 373 (2020), 4415-4436 Request permission
Abstract:
Here we introduce higher-order Poincaré constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the $p$-Laplacian. In the case of the closed eigenvalue problem and the Neumann eigenvalue problem these are related to the estimates obtained by Chung-Grigor’yan-Yau and Gozlan-Herry. We also obtain similar upper bounds for Dirichlet eigenvalues and multi-way isoperimetric constants. As an application, for manifolds with boundary of nonnegative dimensional weighted Ricci curvature, we give upper bounds for inscribed radii in terms of dimension and the first Dirichlet Poincaré constant.References
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Additional Information
- Kei Funano
- Affiliation: Division of Mathematics & Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, Japan
- MR Author ID: 822229
- Email: kfunano@tohoku.ac.jp
- Yohei Sakurai
- Affiliation: Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan
- MR Author ID: 1205408
- Email: yohei.sakurai.e2@tohoku.ac.jp
- Received by editor(s): July 14, 2019
- Received by editor(s) in revised form: September 5, 2019, October 16, 2019, and October 27, 2019
- Published electronically: March 9, 2020
- Additional Notes: The first author was partially supported by JSPS KAKENHI (17K14179).
The second author was partially supported by JPSJ Grant-in-Aid for Scientific Research on Innovative Areas “Discrete Geometric Analysis for Materials Design” (17H06460). - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 4415-4436
- MSC (2010): Primary 35P15, 53C23, 58J50
- DOI: https://doi.org/10.1090/tran/8049
- MathSciNet review: 4105528