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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
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.

References [Enhancements On Off] (What's this?)

  • [1] R. Brooks, The fundamental group and the spectrum of the Laplacian, Comment. Math. Helv. 56 (1981), 581-598. MR 656213 (84j:58131)
  • [2] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis, Princeton Univ. Press, Princeton, N. J., 1970, pp. 195-199. MR 0402831 (53:6645)
  • [3] H. Federer, Geometric measure theory, Springer-Verlag, Berlin and New York, 1969. MR 0257325 (41:1976)

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Article copyright: © Copyright 1987 American Mathematical Society

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