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Courant-sharp eigenvalues of the three-dimensional square torus


Author: Corentin Léna
Journal: Proc. Amer. Math. Soc. 144 (2016), 3949-3958
MSC (2010): Primary 35P05; Secondary 35P15, 35P20, 58J50
DOI: https://doi.org/10.1090/proc/13148
Published electronically: April 27, 2016
MathSciNet review: 3513551
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Abstract: In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus $ (\mathbb{R}/\mathbb{Z})^3\,$, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domain theorem with equality (Courant-sharp situation). Following the strategy of Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber-Krahn-type inequality for domains in the torus, deduced, as in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. This inequality relies on the work of L. Hauswirth, J. Pérez, P. Romon, and A. Ros (2004) on the periodic isoperimetric problem.


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Additional Information

Corentin Léna
Affiliation: Department of Mathematics Guiseppe Peano, University of Turin, Via Carlo Alberto, 10, 10123 Turin, Italy
Email: clena@unito.it

DOI: https://doi.org/10.1090/proc/13148
Keywords: Nodal domains, Courant theorem, Pleijel theorem, isoperimetric problem, torus
Received by editor(s): November 12, 2015
Published electronically: April 27, 2016
Communicated by: Michael Hitrik
Article copyright: © Copyright 2016 American Mathematical Society