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

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

Content Provider	Springer Nature Link
Author	Chen, Hua Qiao, RongHua Luo, Peng Xiao, DongYuan
Copyright Year	2014
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}} $ .
Starting Page	2235
Ending Page	2246
Page Count	12
File Format	PDF
ISSN	16747283
Journal	Science in China Series A: Mathematics
Volume Number	57
Issue Number	11
e-ISSN	18691862
Language	English
Publisher	Science China Press
Publisher Date	2014-09-04
Publisher Place	Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	cone Laplacian cone Sobolev spaces Dirichlet eigenvalues upper bounds of eigenvalues lower bounds of eigenvalues Applications of Mathematics
Content Type	Text
Resource Type	Article
Subject	Mathematics

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