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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 (4 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
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.
Generalized AlonBoppana theorems and errorcorrecting codes
 Journal of Discrete Mathematics
, 2002
"... In this paper we describe several theorems that give lower bounds on the second eigenvalue of any quotient of a given size of a fixed graph, G. These theorems generalize AlonBoppana type theorems, where G is a regular (infinite) tree. When G is a hypercube, our theorems give minimum distance upper ..."
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Cited by 13 (1 self)
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In this paper we describe several theorems that give lower bounds on the second eigenvalue of any quotient of a given size of a fixed graph, G. These theorems generalize AlonBoppana type theorems, where G is a regular (infinite) tree. When G is a hypercube, our theorems give minimum distance upper bounds on linear binary codes of a given size and information rate. Our bounds at best equal the current best bounds for codes, and only apply to linear codes. However, it is of interest to note that (1) one very simple AlonBoppana argument yields nontrivial code bound, and (2) our AlonBoppana argument that equals a current best bound for codes has some hope of improvement. We also improve the bound in sharpest known AlonBoppana theorem (i.e., when G is a regular tree). 1
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 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Lecture Notes On Geometric Analysis
, 1996
"... Contents 0 Introduction 1 First and Second Variational Formulas for Area 2 Bishop Comparison Theorem 3 BochnerWeitzenbock Formulas 4 Laplacian Comparison Theorem 5 Poincare Inequality and the First Eigenvalue 6 Gradient Estimate and Harnack Inequality 7 Mean Value Inequality 8 Rei ..."
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Cited by 10 (0 self)
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Contents 0 Introduction 1 First and Second Variational Formulas for Area 2 Bishop Comparison Theorem 3 BochnerWeitzenbock Formulas 4 Laplacian Comparison Theorem 5 Poincare Inequality and the First Eigenvalue 6 Gradient Estimate and Harnack Inequality 7 Mean Value Inequality 8 Reilly's Formula and Applications 9 Isoperimetric Inequalities and Sobolev Inequalities 10 Lower Bounds of Isoperimetric Inequalities 11 Harnack Inequality and Regularity Theory of De GiorgiNashMoser References 0 Introduction This set of lecture notes originated from a series of lectures given by the author at a Geometry Summer Program in 1990 at the Mathematical Sciences Research Institute in Berkeley. During the Fall quarter of 1990, the author also taught a course in G
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 (2 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 +.
On the Cases of Equality in Bobkov’s Inequality and Gaussian Rearrangement
"... Abstract We determine all of the cases of equality in a recent inequality of Bobkov that implies the isoperimetric inequality on Gauss space. As an application we determine all of the cases of equality in the Gauss space analog of the FaberKrahn inequality. ..."
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Cited by 7 (1 self)
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Abstract We determine all of the cases of equality in a recent inequality of Bobkov that implies the isoperimetric inequality on Gauss space. As an application we determine all of the cases of equality in the Gauss space analog of the FaberKrahn inequality.
Mathematical analysis of the optimal habitat configurations for species persistence
, 2006
"... ..."
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 7 (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.
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