Tight frames and rotations: sharp bounds on eigenvalues of the Laplacian

@inproceedings{Laugesen2013TightFA,
  title={Tight frames and rotations: sharp bounds on eigenvalues of the Laplacian},
  author={Richard S. Laugesen},
  year={2013}
}
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 (λ1 + · · ·+ λn) A I (where A denotes area and I is moment of inertia about the centroid… CONTINUE READING

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