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arXiv · 1805.10075

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

Pedro Freitas·James Kennedy
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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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Pedro Freitas, James Kennedy. 2018-05-25. Extremal domains and P\'{o}lya-type inequalities for the Robin Laplacian on rectangles and unions of rectangles. https://arxiv.org/abs/1805.10075

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