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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 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.
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 26 (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
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
Mathematical analysis of the optimal habitat configurations for species persistence
, 2006
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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 21 (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.
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.
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 17 (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
DOMAIN DEFORMATIONS AND EIGENVALUES OF THE DIRICHLET LAPLACIAN IN A RIEMANNIAN MANIFOLD
, 2007
"... For any bounded regular domain Ω of a real analytic Riemannian manifold M, we denote by λk(Ω) the kth eigenvalue of the Dirichlet Laplacian of Ω. In this paper, we consider λk and as a functional upon the set of domains of fixed volume in M. We introduce and investigate a natural notion of critical ..."
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Cited by 15 (1 self)
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For any bounded regular domain Ω of a real analytic Riemannian manifold M, we denote by λk(Ω) the kth eigenvalue of the Dirichlet Laplacian of Ω. In this paper, we consider λk and as a functional upon the set of domains of fixed volume in M. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for λk. These results rely on Hadamard type variational formulae that we establish in this general setting. As an application, we obtain a characterization of critical domains of the trace of the heat kernel under Dirichlet boundary conditions.