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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The first eigenvalue of a scalene triangle


Authors: Robert Brooks and Peter Waksman
Journal: Proc. Amer. Math. Soc. 100 (1987), 175-182
MSC: Primary 58G25; Secondary 35P15
DOI: https://doi.org/10.1090/S0002-9939-1987-0883424-1
MathSciNet review: 883424
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Abstract: In this paper, we prove the lower bound

$\displaystyle {\lambda _1}(T) \geq \frac{{(L + \sqrt {4\pi A{)^2}} }}{{16{A^2}}}$

for a triangle $ T$ with area $ A$ and perimeter $ L$, where $ {\lambda _1}$ is the first eigenvalue for the Laplace operator with Dirichlet boundary conditions. We also present analogous estimates for an arbitrary convex polygon.

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DOI: https://doi.org/10.1090/S0002-9939-1987-0883424-1
Article copyright: © Copyright 1987 American Mathematical Society