Maximization of the second positive Neumann eigenvalue for planar domains
Girouard, Alexandre ; Nadirashvili, Nikolai ; Polterovich, Iosif
J. Differential Geom., Tome 81 (2009) no. 2, p. 637-662 / Harvested from Project Euclid
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We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of half this area. The estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Pólya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a by-product of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odd-dimensional spheres.
Publié le : 2009-11-15
Classification:  
@article{1264601037,
     author = {Girouard, Alexandre and Nadirashvili, Nikolai and Polterovich, Iosif},
     title = {Maximization of the second positive Neumann eigenvalue for planar domains},
     journal = {J. Differential Geom.},
     volume = {81},
     number = {2},
     year = {2009},
     pages = { 637-662},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1264601037}
}

Girouard, Alexandre; Nadirashvili, Nikolai; Polterovich, Iosif. Maximization of the second positive Neumann eigenvalue for planar domains. J. Differential Geom., Tome 81 (2009) no. 2, pp.  637-662. http://gdmltest.u-ga.fr/item/1264601037/