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MINIMIZING DIRICHLET EIGENVALUES ON CUBOIDS OF UNIT MEASURE

Published online by Cambridge University Press:  13 March 2017

M. van den Berg
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, U.K. email mamvdb@bristol.ac.uk
K. Gittins
Affiliation:
Institut de mathématiques, Rue Emile-Argand 11, 2000 Neuchâtel, Switzerland email katie.gittins@unine.ch
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Abstract

We consider the minimization of Dirichlet eigenvalues $\unicode[STIX]{x1D706}_{k}$, $k\in \mathbb{N}$, of the Laplacian on cuboids of unit measure in $\mathbb{R}^{3}$. We prove that any sequence of optimal cuboids in $\mathbb{R}^{3}$ converges to a cube of unit measure in the sense of Hausdorff as $k\rightarrow \infty$. We also obtain an upper bound for that rate of convergence.

Type
Research Article
Copyright
Copyright © University College London 2017 

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