Sharp bounds for eigenvalues of triangles

Details Sharp bounds for eigenvalues of triangles Sharp bounds for eigenvalues of triangles

2006-03-27

arxiv.org/abs/math/0603630v1

Mathematics

Spectral Theory

preprint

Scientific paper

We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle
in the plane is bounded above by $\pi^2 L^2\over 9A^2$, where $L$ is the
perimeter and $A$ is the area of this triangle. We show that the \mbox{constant
9} is optimal and that the optimal constant for the lower bound of the same
form is 16. This gives a positive answer to a conjecture made by P. Freitas.

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