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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Domain-independent upper bounds for eigenvalues of elliptic operators


Author: Stephen M. Hook
Journal: Trans. Amer. Math. Soc. 318 (1990), 615-642
MSC: Primary 35J25; Secondary 35P15, 47F05
MathSciNet review: 994167
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Abstract: Let $ \Omega \subseteq {\mathbb{R}^m}$ be a bounded open set, $ \partial \Omega $ its boundary and $ \Delta $ the Laplacian on $ {\mathbb{R}^m}$. Consider the elliptic differential equation:

(1)

$\displaystyle - \Delta u = \lambda u\quad {\text{in}}\;\Omega ;\qquad u = 0\quad {\text{on}}\;\partial \Omega .$

It is known that the eigenvalues, $ {\lambda _i}$, of (1) satisfy

(2)

$\displaystyle \sum\limits_{i = 1}^n {\frac{{{\lambda _i}}} {{{\lambda _{n + 1}} - {\lambda _i}}}} \geqslant \frac{{mn}} {4}$

provided that $ {\lambda _{n + 1}} > {\lambda _n}$.

In this paper we abstract the method used by Hile and Protter [2] to establish (2) and apply the method to a variety of second-order elliptic problems, in particular, to all constant coefficient problems. We then consider a variety of higher-order problems and establish an extension of (2) for problem (1) where the Laplacian is replaced by a more general operator in a Hilbert space.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0994167-2
PII: S 0002-9947(1990)0994167-2
Article copyright: © Copyright 1990 American Mathematical Society