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Behaviour of optimal shapes in spectral problems

IIIUL, Room B1-01
2013-05-14 14:30 .. 15:30
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Pedro Freitas (Faculty of Human Kinetics and Group of Mathematical Physics, University of Lisbon)

We consider the behaviour of optimal shapes for the kth eigenvalue of the Laplace operator under different boundary conditions and measure restrictions. This type of problems originated with Rayleigh's Theory of Sound and, some years later, the conjectures of Sommerfeld and Lorentz regarding the dependence of the number of high-frequencies on a given interval on the volume only. Among other things, these led, respectively, to the Rayleigh-Faber-Krahn inequality (1923) and Weyl's
law (1912).

Since then, progress has been very slow: Weyl's conjectured second term was only proven in the 1980's, while at the low-frequency end there has been no progress beyond the second eigenvalue - even this, for Robin and Neumann boundary conditions, was only proven in 2009.

By combining a rigorous analytic approach with numerical simulations, we study the structure of optimisers, shedding light on why progress should be expected to be difficult in the mid-frequency range, and solving the problem of determining the optimal asymptotic shape supporting the largest number of modes below a given frequency. This is based on joint work with P.R.S. Antunes, D. Bucur and J. Kennedy.