Certain min-max values related to the $p$-energy and packing radii of Riemannian manifolds and metric measure spaces
Abstract
Grosjean proved that the $(1/p)$-th power of the first eigenvalue of the $p$-Laplacian on a closed Riemannian manifold converges to the twice of the inverse of the diameter of the space, as $p \to \infty$. Before this, a corresponding result for the Dirichlet first eigenvalues was also obtained by Juutinen, Lindqvist and Manfredi. We extend those results for certain $k$-th min-max value related to the $p$-energy, where the corresponding limits are packing radii introduced by Grove-Markvorsen or its variant. Furthermore, we remark that our result holds for more singular setting.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2019
- DOI:
- 10.48550/arXiv.1912.01432
- arXiv:
- arXiv:1912.01432
- Bibcode:
- 2019arXiv191201432M
- Keywords:
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- Mathematics - Differential Geometry;
- 53C23;
- 53C20