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 Richard Laugesen

Asymptotically optimal shapes: drums with least n-th frequency, and generalized ellipses enclosing most lattice points

Abstract:What shape of domain minimizes the n-th eigenvalue (frequency) of the Laplacian, for large n? Does the minimizer approach a disk as n tends to infinity? This asymptotic optimality conjecture is supported by the discovery of Antunes and Freitas that among rectangular drums, the one minimizing the n-th frequency converges to a square as n tends to infinity. Their proof relies on lattice point counting in ellipses.

We extend to lattice point counting inside concave and convex curves such as p-ellipses with p ≠ 1, and allow both positive and (some) negative translations of the lattice. A natural open problem then arises about right triangles or, equivalently, about asymptotically optimal harmonic oscillators.

[Joint with Shiya Liu (U. of Illinois) and Sinan Ariturk (Pontificia U. Catolica do Rio de Janeiro, Brazil).]