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VOL. 45 | 2013 Tight frames and rotations: sharp bounds on eigenvalues of the Laplacian
R. S. Laugesen

Editor(s) Xuan Duong, Jeff Hogan, Chris Meaney, Adam Sikora

Abstract

Isoperimetric estimates stretch back for thousands of years in geometry, and for more than a hundred years in harmonic analysis and mathematical physics. We will touch on some of these highlights before describing recent progress that uses rotational symmetry to prove sharp upper bounds on sums of eigenvalues of the Laplacian. For example, we prove in 2 dimensions that the scale-normalized eigenvalue sum \[ (\lambda_1 + \cdots + \lambda_n) \frac{A^3}{I} \] (where $A$ denotes area and $I$ is moment of inertia about the centroid) is maximized among triangles by the equilateral triangle, for each $n \geq 1$. This theorem, which is due to the author and B. A. Siudeja, generalizes a result of Pólya for the fundamental tone.

Numerous related problems will be discussed, such as the inverse spectral and spectral gap problems for triangular domains.

Information

Published: 1 January 2013
First available in Project Euclid: 3 December 2014

zbMATH: 1338.42044
MathSciNet: MR3424868

Rights: Copyright © 2013, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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