Minimization of the third eigenvalue of the Dirichlet Laplacian

By Antoine Henrot, Dorin Bucur. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2000

64 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5d1746c548b4a63177e340f4a8e7beefa9a7786d

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Citations in the corpus

Minimization problems for eigenvalues of the Laplacian (2003, methodology): “This theorem does not solve the general problem of existence of a minimizer for ¸k() since we assume to work with ”confined” sets (that is to say, sets included in a box D). In order to remove this assumption in [ 10 ], we used a ”concentration-compactness” argument together with the Wolff-Keller’s”
Geometrical Structure of Laplacian Eigenfunctions (2012, background): “Bucur and Henrot proved the existence of a minimizer for the third eigenvalue in the family of domains in R of given volume, although its shape remains unknown [95].”
Existence of minimizers for spectral problems (2011, background): “results are usually referred to as Faber{Krahn inequality and Krahn{Szego inequality respectively. Instead, the existence of a minimizer for 3 has been proved only in 2000 by Bucur and Henrot (see [5]), but it is still presently not known which set is the minimizer (and this is a major open problem”
Rearrangement inequalities and applications to isoperimetric problems for eigenvalues (2006, background): “Very few things seem to be known about optimization problems for the other eigenvalues, see [17, 25, 39, 40, 49].”
Minimization of the k-th eigenvalue of the Dirichlet Laplacian (2012, background): “For k = 3, the existence of a quasi-open set was proved in [6], while for k 4 it was proved only that a minimizer exists, provided that for every lower index a bounded minimizer exists.”