For the clamped plate on rectangles of fixed area, the first eigenvalue has a minimizer with aspect ratio below 1.066459, and the k-th minimizing rectangle tends to the square as k grows.
Extremal domains and P\'{o}lya-type inequalities for the Robin Laplacian on rectangles and unions of rectangles
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abstract
We show that eigenvalues of the Robin Laplacian with a positive boundary parameter $\alpha$ on rectangles and unions of rectangtes satisfy P\'{o}lya-type inequalities, albeit with an exponent smaller than that of the corresponding Weyl asympotics for a fixed domain. We determine the optimal exponents in either case, showing that they are different in the two situations. Our approach to proving these results includes a characterisation of the corresponding extremal domains for the $k$th eigenvalue in regions of the $(k,\alpha)$-plane.
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2019 1verdicts
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Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles
For the clamped plate on rectangles of fixed area, the first eigenvalue has a minimizer with aspect ratio below 1.066459, and the k-th minimizing rectangle tends to the square as k grows.