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Extremal domains and P\'{o}lya-type inequalities for the Robin Laplacian on rectangles and unions of rectangles

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arxiv 1805.10075 v1 pith:CN52LUGK submitted 2018-05-25 math.AP math.SP

Pedro Freitas , James Kennedy This is my paper · ORCID
classification math.APmath.SP
keywords rectanglesalphacorrespondingdomainsextremalinequalitieslaplacianlya-type
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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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  1. Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles

    math.SP 2019-08 accept novelty 7.0 of 10

    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.

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