The first eigenvalue of a scalene triangle
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- by Robert Brooks and Peter Waksman PDF
- Proc. Amer. Math. Soc. 100 (1987), 175-182 Request permission
Abstract:
In this paper, we prove the lower bound \[ {\lambda _1}(T) \geq \frac {{(L + \sqrt {4\pi A{)^2}} }}{{16{A^2}}}\] 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
- Robert Brooks, The fundamental group and the spectrum of the Laplacian, Comment. Math. Helv. 56 (1981), no. 4, 581–598. MR 656213, DOI 10.1007/BF02566228
- Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Additional Information
- © Copyright 1987 American Mathematical Society
- 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