An Isoperimetric Inequality for Eigenvalues of the Stekloff Problem

By F. Brock. 2001

186 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/d20849689e3be51e9a2c5ad41b9cf2f1b80469e7

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization (2011, unknown): “From [4] we first recall that for the second order Steklov problem ∆u = 0 in Ω , uν = λu on ∂Ω , the first (nontrivial) eigenvalue satisfies λ1(Ω) ≤ λ1(Ω), where Ω∗ denotes a ball having the same measure as Ω.”
On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues (2008, background): “Various extensions of Weinstock’s inequality and related results can be found in [2], [18], [4], [10], and [11]; see also [1, Sec.”
Spectral optimization for the Stekloff--Laplacian: the stability issue (2012, background): “This further isoperimetric characterization of the ball is the cornerstone of Brock’s proof in [7]: then in order to derive (1.”
Minimization problems for eigenvalues of the Laplacian (2003, background): “Brock in [8] proved actually a sharper inequality, namely: Let Ω be a bounded domain in RN and R the radius of the ball Ω of same volume than Ω, then N+1”
The legacy of Vladimir Andreevich Steklov (2014, result): “Generalizations of this result were obtained in [6] and [4].”
Isoperimetric Inequalities for Eigenvalues of the Laplacian (2007, methodology): “Finally, for a generalization to n dimensions, we have the recent result of Brock [38]”
SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “Brock [16].”
A few shape optimization results for a biharmonic Steklov problem (2015, unknown): “, [10, 42, 44]).”
Shape optimization for low Neumann and Steklov eigenvalues (2008, unknown): “Many results were motivated by Weinstock’s inequality: see, for instance, [Ba3, Bro , Dit, Ed, HP, HPS, HePhSa].”
An extremal eigenvalue problem for the Wentzell-Laplace operator (2014, background): “It was first proved in the case d = 2 by Weinstock and extented by Brock to any dimension in [6] (Weinstock inequality is slightly stronger but restricted to simply-connected domains: he proved indeed that the disk maximizes λ1 among simply-connected sets of given perimeter).”
Rearrangement inequalities and applications to isoperimetric problems for eigenvalues (2006, background): “Bounds or optimization results for other eigenvalues of the Laplacian under Neumann boundary condition ([40, 44, 48], see also [10] for inhomogeneous problems), for Robin boundary condition ([15, 19, 20]) or for the Stekloff eigenvalue problem ([16]) have also been established.”
Eigenvalue inequalities for mixed Steklov problems (2009, background): “For some of this literature, see [3], [6], [1], [16], [17], [21], [12], [10].”
Upper bounds for Steklov eigenvalues on surfaces (2012, background): “Isoperimetric inequalities for Steklov eigenvalues have been actively studied for more than fifty years [21, 22, 3, 15, 12].”
On a fourth order Steklov eigenvalue problem (2005, unknown): “it is known (see [ 6, Theorem 3 ]) that the first (nontrivial) eigenvalue λ1 = λ1(�) satisfies λ1(�) ≤ λ1(� ∗) where”