Isoperimetric Inequality for the Second Eigenvalue of a Sphere

By Nikolai Nadirashvili. 2002

81 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/14bdbf7e74c8e56479c047c736a2c0cf327c77fe

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In Applied Mathematics

In Geometry and Topology

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.”
Maximization of the second positive Neumann eigenvalue for planar domains (2008, methodology): “as proved in [CE, Corollary 1] that (1.2.2) λc k(S n,[g0]) := sup g∈[g0] λ(Sn,g)Vol(Sn,g)2n ≥n(kωn)2/n, The number λc k(S n,[g0]) is called the k-th conformal eigenvalue of (Sn,[g0]). It was shown in [Na] that for k= 2 and n= 2 the inequality in (1.2.2) is an equality, and the supremum is attained b”
Maximization of the second non-trivial Neumann eigenvalue (2018, methodology): “Their proof is based on a combination of topological and analytical arguments and relies on a folding and rearrangement technique introduced by Nadirashvili in [17], taking advantage on the use of conformal mappings.”
An isoperimetric inequality for Laplace eigenvalues on the sphere (2017, unknown): “2 As was mentioned in the introduction, it is known that Λk(S 2) = 8πk for k = 1 ([20]), k = 2 ([37, 49]) and k = 3 ([41]).”
Shape optimization for low Neumann and Steklov eigenvalues (2008, methodology): “3 we introduce hyperbolic caps and folded measures (this idea goes back to [25]), which are used to define test functions in section 3.”
Extremal Eigenvalues of the Laplacian in a Conformal Class of Metrics: The `Conformal Spectrum' (2003, unknown): “e of the n-sphere Sn endowed with its standard conformal class [gs], the equality holds in this inequality for k = 1. The equality also holds for k = 2 on S2 as it was recently proven by Nadirashvili [N2], i.e. λc 2(S 2,[g s]) = 8π. 4 Combined with the Korevaar estimate quoted above, Corollary 1 giv”