Taiwanese Journal of Mathematics

INEQUALITIES BETWEEN DIRICHLET AND NEUMANN EIGENVALUES FOR DOMAINS IN SPHERES

Yi-Jung Hsu and Tai-Ho Wang

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Abstract

Let $M$ be a domain in the unit $n$-sphere with smooth boundary. The purpose of this paper is to describe some inequalities between Dirichlet and Neumann eigenvalues for $M$ under certain convex restrictions on the boundary. We prove that if the mean curvature of the boundary is nonpositive, then the $k$th nonzero Neumann eigenvalue is less than or equal to the $k$th Dirichlet eigenvalue for $k=1, 2,\cdots$. Furthermore, if the second fundamental form of the boundary is nonpositive, then the $(k+\left[\frac{n-1}{2}\right])$th nonzero Neumann eigenvalue is less than or equal to the $k$th Dirichlet eigenvalue for $k=1,2,\cdots$.

Article information

Source
Taiwanese J. Math., Volume 5, Number 4 (2001), 755-766.

Dates
First available in Project Euclid: 20 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500574993

Digital Object Identifier
doi:10.11650/twjm/1500574993

Mathematical Reviews number (MathSciNet)
MR1870045

Zentralblatt MATH identifier
1115.35361

Subjects
Primary: 35P15: Estimation of eigenvalues, upper and lower bounds 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60]

Keywords
Laplace-Beltrami operator dirichlet eigenvalue Neumann eigenvalue

Citation

Hsu, Yi-Jung; Wang, Tai-Ho. INEQUALITIES BETWEEN DIRICHLET AND NEUMANN EIGENVALUES FOR DOMAINS IN SPHERES. Taiwanese J. Math. 5 (2001), no. 4, 755--766. doi:10.11650/twjm/1500574993. https://projecteuclid.org/euclid.twjm/1500574993


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