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arXiv · 1411.2400

The generalised Pólya conjecture for the Dirichlet eigenvalues

Genqian Liu
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Abstract

In this paper, we prove the Generalized Pólya conjecture for the Dirichlet eigenvalues. In other words, we show that $λ_k(α) \ge \frac{(2π)^α k^{α/n}}{\big(ω_n \cdot {vol}(Ω)\big)^{α/n}}, \quad\, {for}\;\; k=1,2,3,....$ where $λ_k(α)$ is the $k$-th Dirichlet eigenvalue for the fractional Laplacian $(-Δ)^{α/2}$ with $α\in (0,2]$ in a bounded domain $Ω\subset {\Bbb R}^n$.

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Genqian Liu. 2015-02-13. The generalised Pólya conjecture for the Dirichlet eigenvalues. https://arxiv.org/abs/1411.2400

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