Maximising Neumann eigenvalues on rectangles

Abstract We obtain results for the spectral optimisation of Neumann eigenvalues on rectangles in $ {\mathbb R}^2$ with a measure or perimeter constraint. We show that the rectangle with measure 1 that maximises the $k$th Neumann eigenvalue converges to the unit square in the Hausdorff metric as $k\r...

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Bibliographic Details
Published in:The Bulletin of the London Mathematical Society 2016-10, Vol.48 (5), p.877-894
Main Authors: van den Berg, M., Bucur, D., Gittins, K.
Format: Article
Language:English
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Summary:Abstract We obtain results for the spectral optimisation of Neumann eigenvalues on rectangles in $ {\mathbb R}^2$ with a measure or perimeter constraint. We show that the rectangle with measure 1 that maximises the $k$th Neumann eigenvalue converges to the unit square in the Hausdorff metric as $k\rightarrow \infty $. Furthermore, we determine the unique maximiser of the $k$th Neumann eigenvalue on a rectangle with given perimeter.
ISSN:0024-6093
1469-2120
DOI:10.1112/blms/bdw049