Maximization of the second positive Neumann eigenvalue for planar domains

By A. Girouard, Nikolai Nadirashvili, Iosif Polterovich. 2008

85 citations recorded by the source corpus.

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 Polya 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e3e85bc71b939334e4b86fec20094947629d54dc

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In Applied Mathematics

1. Iosif Polterovich
2. Mateusz Kwaśnicki
3. A. Girouard
4. Dorin Bucur
5. Antoine Henrot
6. Tadeusz Kulczycki
7. Bartlomiej Siudeja
8. Alexander I. Nazarov

In Computational Theory and Mathematics

1. Iosif Polterovich
2. A. Girouard
3. Dorin Bucur
4. Antoine Henrot
5. Tadeusz Kulczycki
6. Bartlomiej Siudeja
7. Mateusz Kwaśnicki
8. Alexander I. Nazarov

In Geometry and Topology

1. Iosif Polterovich
2. A. Girouard
3. Bartlomiej Siudeja
4. Antoine Henrot
5. Tadeusz Kulczycki
6. Dorin Bucur

Citations in the corpus

• On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues (2008, methodology): “In Section 3, the “folding and rearrangement” technique, introduced in [25] and developed in [14], is adapted to the Steklov problem.”
• SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “In fact, Weinstock’s proof is the simplest application of the center of mass renormalization (also known as Hersch’s lemma, see [59, 96, 45, 48]).”
• The legacy of Vladimir Andreevich Steklov (2014, background): “For Neumann eigenvalues, a similar result holds for n = 2 [15], but the situation is quite different for n á 3 (see [1] and [48]).”
• Maximization of the second non-trivial Neumann eigenvalue (2018, result): “r numerical approximations of the (presumably) optimal sets for k10, but there is no proof of the existence of those sets. We refer the reader to the result of Girouard, Nadirashvili and Polterovich [11] where the authors prove that in R2 the union of two equal (and disjoint) disks gives a larger se”
• Shape optimization for low Neumann and Steklov eigenvalues (2008, methodology): “Remark 2.5.3. This argument is motivated by [ 12, section 2.7 ] and is a modification of the proof given in [26, p. 138].”