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24
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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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.
A unified approach to universal inequalities for eigenvalues of elliptic operators
"... We present an abstract approach to universal inequalities for the discrete spectrum of a selfadjoint operator, based on commutator algebra, the Rayleigh–Ritz principle, and one set of “auxiliary ” operators. The new proof unifies classical inequalities of Payne–Pólya–Weinberger, Hile–Protter, and H ..."
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We present an abstract approach to universal inequalities for the discrete spectrum of a selfadjoint operator, based on commutator algebra, the Rayleigh–Ritz principle, and one set of “auxiliary ” operators. The new proof unifies classical inequalities of Payne–Pólya–Weinberger, Hile–Protter, and H.C. Yang and provides a Yang type strengthening of Hook’s bounds for various elliptic operators with Dirichlet boundary conditions. The proof avoids the introduction of the “free parameters ” of many previous authors and relies on earlier works of Ashbaugh and Benguria, and, especially, Harrell (alone and with Michel), in addition to those of the other authors listed above. The Yang type inequality is proved to be stronger under general conditions on the operator and the auxiliary operators. This approach provides an alternative route to recent results obtained by Harrell and Stubbe. 1.
Commutators, spectral trace identities, and universal estimates for eigenvalues
 J. Funct. Analysis
"... Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract selfadjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs o ..."
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Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract selfadjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the eigenvalue gaps. As particular examples, we present simple proofs of the classical universal estimates for eigenvalues of the Dirichlet Laplacian, as well as of some known and new results for other differential operators and systems. We also suggest an extension of the methods to the case of nonselfadjoint operators. 1
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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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
Minimizing Neumann fundamental tones of triangles: an optimal Poincaré inequality
"... Abstract. The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincaré inequality for triangles is derived. The proof relies on symmetry of the Neumann fundamental mode for i ..."
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Abstract. The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincaré inequality for triangles is derived. The proof relies on symmetry of the Neumann fundamental mode for isosceles triangles with aperture less than π/3. Antisymmetry is proved for apertures greater than π/3. 1.
Range of the first three eigenvalues of the planar Dirichlet Laplacian
"... We conduct extensive numerical experiments aimed at nding the admissible range of the ratios of the rst three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M Ashbaugh and R Benguria. We also prove some properties of a maximizer o ..."
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Cited by 5 (1 self)
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We conduct extensive numerical experiments aimed at nding the admissible range of the ratios of the rst three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M Ashbaugh and R Benguria. We also prove some properties of a maximizer of the ratio 3 = 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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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
Universal bounds for eigenvalues of a buckling problem
 Commn. Math. Phys
"... Abstract. In this paper, we investigate universal estimates for eigenvalues of a buckling problem. For a bounded domain in a Euclidean space, we solve partially a conjecture proposed in [7]. For a domain in the unit sphere, we give an important improvement on the results of Wang and Xia [16]. 1. ..."
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Abstract. In this paper, we investigate universal estimates for eigenvalues of a buckling problem. For a bounded domain in a Euclidean space, we solve partially a conjecture proposed in [7]. For a domain in the unit sphere, we give an important improvement on the results of Wang and Xia [16]. 1.
Maximizing Neumann fundamental tones of triangles
 J. Math. Phys
"... Abstract. We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar re ..."
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Abstract. We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proved for the harmonic and arithmetic means of the first two nonzero eigenvalues. 1.
Trace identities and universal estimates for eigenvalues of linear pencils
 in Elliptic and Parabolic Problems, Proceedings of the IV European Conference (Rolduc/Gaeta 2001), edited by J Bemelmans, etc, World Scientific
, 2002
"... We describe the method of constructing the spectral trace identities and the estimates of eigenvalue gaps for the linear selfadjoint operator pencils A − λB. ..."
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We describe the method of constructing the spectral trace identities and the estimates of eigenvalue gaps for the linear selfadjoint operator pencils A − λB.