Isoperimetric inequalities for eigenvalues of triangles

Details Isoperimetric inequalities for eigenvalues of triangles Isoperimetric inequalities for eigenvalues of triangles

2007-07-24

arxiv.org/abs/0707.3631v2

Mathematics

Spectral Theory

Scientific paper

Lower bounds estimates are proved for the first eigenvalue for the Dirichlet Laplacian on arbitrary triangles using various symmetrization techniques. These results can viewed as a generalization of P\'olya's isoperimetric bounds. It is also shown that amongst triangles, the equilateral triangle minimizes the spectral gap and (under additional assumption) the ratio of the first two eigenvalues. This last result resembles the Payne-P\'olya-Weinberger conjecture proved by Ashbaugh and Benguria.

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