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41
Isoperimetric and universal inequalities for eigenvalues, in Spectral Theory and Geometry
 London Mathematical Society Lecture Note Series
, 1999
"... PaynePólyaWeinberger conjecture, Sperner’s inequality, biharmonic operator, biLaplacian, clamped plate problem, Rayleigh’s conjecture, buckling problem, the PólyaSzegő conjecture, universal inequalities for eigenvalues, HileProtter inequality, H. C. Yang’s inequality. Short title: Isoperimetric ..."
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Cited by 50 (7 self)
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PaynePólyaWeinberger conjecture, Sperner’s inequality, biharmonic operator, biLaplacian, clamped plate problem, Rayleigh’s conjecture, buckling problem, the PólyaSzegő conjecture, universal inequalities for eigenvalues, HileProtter inequality, H. C. Yang’s inequality. Short title: Isoperimetric and Universal Inequalities This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and biLaplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with “Dirichlet ” boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to
Minimization problems for eigenvalues of the Laplacian, to appear in
 Journal of Evolution Equations special
"... This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study ..."
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Cited by 43 (2 self)
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This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues. AMS classification: 49Q10, 35P15, 49J20.
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 22 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Spectral theory for perturbed Krein Laplacians in nonsmooth domains
, 2010
"... We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all do ..."
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Cited by 20 (13 self)
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We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r> 1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula #{j ∈ N  λK,Ω,j ≤ λ} = (2π) −n vnΩ  λ n/2 + O ` λ (n−(1/2))/2 ´ as λ → ∞, where vn = πn/2 /Γ((n/2)+1) denotes the volume of the unit ball in Rn, and λK,Ω,j, j ∈ N, are the nonzero eigenvalues of HK,Ω, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von Neumann extension of − ∆ + V defined on C ∞ 0 (Ω)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980’s. Our work builds on that of Grubb in the early 1980’s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exteriortype domains Ω = Rn \K, n ≥ 3, with K ⊂ Rn compact and vanishing
Isoperimetric inequalities for eigenvalues of the Laplace operator, in Fourth summer school in analysis and mathematical physics: topics in spectral theory and quantum mechanics
 C. Villegas–Blas (Ed.), Contemporary Mathematics (AMS
, 2008
"... Abstract. The purpose of this manuscript is to present a series of lecture notes on isoperimetric inequalities for the Laplacian, for the Schrödinger operator, and related problems. 1. ..."
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Cited by 20 (2 self)
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Abstract. The purpose of this manuscript is to present a series of lecture notes on isoperimetric inequalities for the Laplacian, for the Schrödinger operator, and related problems. 1.
Optimal Eigenvalues For Some Laplacians And Schrödinger Operators Depending On Curvature
 Proceedings of QMath7 (Prague
, 1998
"... We consider Laplace operators and Schrödinger operators with potentials containing curvature on certain regions of nontrivial topology, especially closed curves, annular domains, and shells. Dirichlet boundary conditions are imposed on any boundaries. Under suitable assumptions we prove that the fun ..."
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Cited by 17 (3 self)
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We consider Laplace operators and Schrödinger operators with potentials containing curvature on certain regions of nontrivial topology, especially closed curves, annular domains, and shells. Dirichlet boundary conditions are imposed on any boundaries. Under suitable assumptions we prove that the fundamental eigenvalue is maximized when the geometry is round. We also comment on the use of coordinate transformations for these operators and mention some open problems.
Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions
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Minimization of λ2(Ω) with a perimeter constraint
, 2009
"... We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In N dimensions, we prove a ..."
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Cited by 12 (5 self)
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We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In N dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to set inclusion and γ lower semicontinuous.
Maximization of the second positive Neumann eigenvalue for planar domains
"... Abstract. We prove that the second positive Neumann eigenvalue of a bounded simplyconnected 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 ide ..."
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Cited by 12 (4 self)
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Abstract. We prove that the second positive Neumann eigenvalue of a bounded simplyconnected 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 byproduct of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odddimensional spheres. 1. Introduction and
Differential inequalities for Riesz means and Weyltype bounds for eigenvalues
, 2007
"... We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z): = � (z − λk) σ +. ..."
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Cited by 11 (1 self)
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We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z): = � (z − λk) σ +.