Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles

D. Buoso; P. Freitas

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arXiv · 1908.06483

Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles

D. Buoso·P. Freitas

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Abstract

We study the behaviour of extremal eigenvalues of the Dirichlet biharmonic operator over rectangles with a given fixed area. We begin by proving that the principal eigenvalue is minimal for a rectangle for which the ratio between the longest and the shortest side lengths does not exceed $1.066459$. We then consider the sequence formed by the minimal $k^{\rm th}$ eigenvalue and show that the corresponding sequence of minimising rectangles converges to the square as $k$ goes to infinity.

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D. Buoso, P. Freitas. 2019-08-18. Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles. https://arxiv.org/abs/1908.06483

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