The generalised Pólya conjecture for the Dirichlet eigenvalues

Abstract

In this paper, we prove the Generalized Pólya conjecture for the Dirichlet eigenvalues. In other words, we show that ${\lambda }_k(\alpha )\ge \frac{(2\pi {)}^{\alpha }{k}^{\alpha /n}}{({\omega }_n\cdot vol(\Omega ){)}^{\alpha /n}},fork=1,2,3,....$ where ${\lambda }_k(\alpha )$ is the $k$-th Dirichlet eigenvalue for the fractional Laplacian $(-\Delta {)}^{\alpha /2}$ with $\alpha \in (0,2]$ in a bounded domain $\Omega \subset {\mathbb{R}}^n$.

6 pages. This paper has been withdrawn by the author due to an error in page 4

References in corpus (1)

• From the long jump random walk to the fractional Laplacian