Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators

Abstract

Let λ k be the k-th Dirichlet eigenvalue of totally characteristic degenerate elliptic operator $ - \Delta _\mathbb{B} $ defined on a stretched cone $\mathbb{B}_0 \subseteq [0,1) \times X$ with boundary on {x 1 = 0}. More precisely, $\Delta _\mathbb{B} = (x_1 \partial _{x_1 } )^2 \partial _{x_2 }^2 + \cdots + \partial _{x_n }^2 $ is also called the cone Laplacian. In this paper, by using Mellin-Fourier transform, we prove that $\lambda _k \geqslant C_n k^{\tfrac{2} {n}} $ for any k ⩾ 1, where $C_n = (\tfrac{n} {{n + 2}})(2\pi )^2 (|\mathbb{B}_0 |B_n )^{ - \tfrac{2} {n}} $ , which gives the lower bounds of the Dirchlet eigenvalues of $ - \Delta _\mathbb{B} $ . On the other hand, by using the Rayleigh-Ritz inequality, we deduce the upper bounds of λ k , i.e., $\lambda _{k + 1} \leqslant (1 + \tfrac{4} {n})k^{2/n} \lambda _1 $ . Combining the lower and upper bounds of λ k , we can easily obtain the lower bound for the first Dirichlet eigenvalue $\lambda _1 \geqslant C_n (1 + \tfrac{4} {n})^{ - 1} 2^{\tfrac{n} {2}} $ .