Abstract
Minimization of the Dirichlet eigenvalues of the Laplacian among sets of prescribed measure is a standard problem in shape optimization. The main result of this paper is that in the Euclidean plane, apart from the first four, no Dirichlet eigenvalue can be minimized by disks or disjoint unions of disks.
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Acknowledgments
The author was partially supported by the Swiss National Science Foundation (request 200020_149261). The author would also like to thank an anonymous referee for his thorough work and numerous interesting remarks.
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Berger, A. The eigenvalues of the Laplacian with Dirichlet boundary condition in \(\mathbb {R}^2\) are almost never minimized by disks. Ann Glob Anal Geom 47, 285–304 (2015). https://doi.org/10.1007/s10455-014-9446-9
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DOI: https://doi.org/10.1007/s10455-014-9446-9