On the first eigenvalue of the normalized p-Laplacian
Abstract
We prove that, if $\Omega$ is an open bounded domain with smooth and connected boundary, for every $p \in (1, + \infty)$ the first Dirichlet eigenvalue of the normalized $p$-Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (non-optimal) lower bound for the eigenvalue in terms of the measure of $\Omega$, and we address the open problem of proving a Faber-Krahn type inequality with balls as optimal domains.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2018
- DOI:
- 10.48550/arXiv.1811.10024
- arXiv:
- arXiv:1811.10024
- Bibcode:
- 2018arXiv181110024C
- Keywords:
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- Mathematics - Analysis of PDEs;
- 49K20;
- 35J60;
- 47J10
- E-Print:
- 13 pages, 2 figures