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24
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 23 (1 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.
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 20 (5 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
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 15 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
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 14 (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.
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 7 (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 +.
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 7 (7 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
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 5 (3 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
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 4 (2 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.
A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of
 S n , Trans. Amer. Math. Soc
, 2001
"... Abstract. For a domain Ω contained in a hemisphere of the n–dimensional sphere Sn we prove the optimal result λ2/λ1(Ω) ≤ λ2/λ1(Ω ⋆)fortheratio of its first two Dirichlet eigenvalues where Ω ⋆ , the symmetric rearrangement of Ω in Sn, is a geodesic ball in Sn having the same n–volume as Ω. We also s ..."
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Cited by 4 (2 self)
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Abstract. For a domain Ω contained in a hemisphere of the n–dimensional sphere Sn we prove the optimal result λ2/λ1(Ω) ≤ λ2/λ1(Ω ⋆)fortheratio of its first two Dirichlet eigenvalues where Ω ⋆ , the symmetric rearrangement of Ω in Sn, is a geodesic ball in Sn having the same n–volume as Ω. We also show that λ2/λ1 for geodesic balls of geodesic radius θ1 less than or equal to π/2 is an increasing function of θ1 which runs between the value (jn/2,1/jn/2−1,1) 2 for θ1 = 0 (this is the Euclidean value) and 2(n +1)/n for θ1 = π/2. Here jν,k denotes the kth positive zero of the Bessel function Jν(t). This result generalizes the Payne–Pólya–Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of Sn and having a fixed value of λ1 the one with the maximal value of λ2 is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for λ2/λ1. Various other results for λ1 and λ2 of geodesic balls in Sn are proved in the course of our work. 1.
Two new Weyltype bounds for the Dirichlet Laplacian
"... Abstract. In this paper, we prove two new Weyltype upper estimates for the eigenvalues of the Dirichlet Laplacian. As a consequence, we obtain the following lower bounds for its counting function. For λ ≥ λ1, one has N(λ)> 2 n + 2 ..."
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Cited by 3 (1 self)
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Abstract. In this paper, we prove two new Weyltype upper estimates for the eigenvalues of the Dirichlet Laplacian. As a consequence, we obtain the following lower bounds for its counting function. For λ ≥ λ1, one has N(λ)> 2 n + 2