A universal bound for lower Neumann eigenvalues of the Laplacian

Lu, Wei, Mao, Jing, and Wu, Chuanxi. "A universal bound for lower Neumann eigenvalues of the Laplacian." Czechoslovak Mathematical Journal 70.2 (2020): 473-482. <http://eudml.org/doc/297236>.

@article{Lu2020,
abstract = {Let $M$ be an $n$-dimensional ($n\ge 2$) simply connected Hadamard manifold. If the radial Ricci curvature of $M$ is bounded from below by $(n-1)k(t)$ with respect to some point $p\in M$, where $t=d(\cdot ,p)$ is the Riemannian distance on $M$ to $p$, $k(t)$ is a nonpositive continuous function on $(0,\infty )$, then the first $n$ nonzero Neumann eigenvalues of the Laplacian on the geodesic ball $B(p,l)$, with center $p$ and radius $0<l<\infty $, satisfy \[ \frac\{1\}\{\mu \_1\}+\frac\{1\}\{\mu \_2\}+\cdots +\frac\{1\}\{\mu \_n\}\ge \frac\{l^\{n+2\}\}\{(n+2)\int \_\{0\}^\{l\}f^\{n-1\}(t)\{\rm d\}t\}, \] where $f(t)$ is the solution to \[ \{\left\lbrace \begin\{array\}\{ll\} f^\{\prime \prime \}(t)+k(t)f(t)=0 \quad \text\{on\} \ (0,\infty ),\\ f(0)=0, \ f^\{\prime \}(0)=1. \end\{array\}\right.\} \]},
author = {Lu, Wei, Mao, Jing, Wu, Chuanxi},
journal = {Czechoslovak Mathematical Journal},
keywords = {Hadamard manifold; Neumann eigenvalue; radial Ricci curvature},
language = {eng},
number = {2},
pages = {473-482},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A universal bound for lower Neumann eigenvalues of the Laplacian},
url = {http://eudml.org/doc/297236},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Lu, Wei
AU - Mao, Jing
AU - Wu, Chuanxi
TI - A universal bound for lower Neumann eigenvalues of the Laplacian
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 2
SP - 473
EP - 482
AB - Let $M$ be an $n$-dimensional ($n\ge 2$) simply connected Hadamard manifold. If the radial Ricci curvature of $M$ is bounded from below by $(n-1)k(t)$ with respect to some point $p\in M$, where $t=d(\cdot ,p)$ is the Riemannian distance on $M$ to $p$, $k(t)$ is a nonpositive continuous function on $(0,\infty )$, then the first $n$ nonzero Neumann eigenvalues of the Laplacian on the geodesic ball $B(p,l)$, with center $p$ and radius $0<l<\infty $, satisfy \[ \frac{1}{\mu _1}+\frac{1}{\mu _2}+\cdots +\frac{1}{\mu _n}\ge \frac{l^{n+2}}{(n+2)\int _{0}^{l}f^{n-1}(t){\rm d}t}, \] where $f(t)$ is the solution to \[ {\left\lbrace \begin{array}{ll} f^{\prime \prime }(t)+k(t)f(t)=0 \quad \text{on} \ (0,\infty ),\\ f(0)=0, \ f^{\prime }(0)=1. \end{array}\right.} \]
LA - eng
KW - Hadamard manifold; Neumann eigenvalue; radial Ricci curvature
UR - http://eudml.org/doc/297236
ER -