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18
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
Estimates for Green Function and the Poisson Kernels of higher order Dirichlet boundary value problems
 J. Differential Equations
"... Pointwise estimates are derived for the kernels associated to the polyharmonic Dirichlet problem on bounded smooth domains. As a consequence one obtains optimal weighted L pL qregularity estimates for weights involving the distance function. ..."
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Cited by 15 (2 self)
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Pointwise estimates are derived for the kernels associated to the polyharmonic Dirichlet problem on bounded smooth domains. As a consequence one obtains optimal weighted L pL qregularity estimates for weights involving the distance function.
Sharp Estimates for Iterated Green Functions
 PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH A 132 (2002), 91–120.
, 2002
"... Optimal pointwise estimates from above and below are obtained for iterated (poly)harmonic Green functions corresponding to zero Dirichlet boundary conditions. For second order elliptic operators these estimates hold true on bounded C 1,1 domains. For higher order elliptic operators we have to restri ..."
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Cited by 10 (1 self)
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Optimal pointwise estimates from above and below are obtained for iterated (poly)harmonic Green functions corresponding to zero Dirichlet boundary conditions. For second order elliptic operators these estimates hold true on bounded C 1,1 domains. For higher order elliptic operators we have to restrict ourselves to the polyharmonic operator on balls. We will also consider applications to noncooperatively coupled elliptic systems and to the lifetime of conditioned Brownian motion.
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
An efficient direct solver for a class of mixed finite element problems
 SCHOOL OF COMPUTER STUDIES RESEARCH REPORT 99.03 (UNIVERSITY OF LEEDS
, 1999
"... In this paper we present an efficient, accurate and parallelizable direct method for the solution of the (indefinite) linear algebraic systems that arise in the solution of fourth order partial differential equations (PDEs) using mixed finite element approximations. The method is intended particular ..."
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Cited by 3 (1 self)
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In this paper we present an efficient, accurate and parallelizable direct method for the solution of the (indefinite) linear algebraic systems that arise in the solution of fourth order partial differential equations (PDEs) using mixed finite element approximations. The method is intended particularly for use when multiple righthand sides occur, and when high accuracy is required in these solutions. The algorithm is described in some detail and its performance is illustrated through the numerical solution of a biharmonic eigenvalue problem where the smallest eigenpair is approximated using inverse iteration after discretization via the CiarletRaviart mixed finite element method.
On the accurate finite element solution of a class of fourth order eigenvalue problems, preprint
, 1999
"... This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation ..."
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Cited by 2 (0 self)
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This paper is concerned with the accurate numerical approximation of the spectral properties of the biharmonic operator on various domains in two dimensions. A number of analytic results concerning the eigenfunctions of this operator are summarized and their implications for numerical approximation are discussed. In particular, the asymptotic behaviour of the first eigenfunction is studied since it is known that this has an unbounded number of oscillations when approaching certain types of corner on domain boundaries. Recent computational results of Bjørstad and Tjøstheim [4], using a highly accurate spectral LegendreGalerkin method, have demonstrated that a number of these sign changes may be accurately computed on a square domain provided sufficient care is taken with the numerical method. We demonstrate that similar accuracy is also achieved using an unstructured finite element solver which may be applied to problems on domains with arbitrary geometries. A number of results obtained from this mixed finite element approach are then presented for a variety of domains. These include a family of circular sector regions, for which the oscillatory behaviour is studied as a function of the internal angle, and another family of (symmetric and nonconvex) domains, for which the parity of the least eigenfunction is investigated. The paper not only verifies existing asymptotic theory, but also allows us to make a new conjecture concerning the eigenfunctions of the biharmonic operator. 1
Spectral properties and norm estimates associated to the C(k)c functional calculus
 J. Operator Th
"... Abstract. We show that the Davies functional calculus and the AC(ν)calculus coincide under common hypotheses. Then we apply the calculus to operators on Banach spaces, to investigate spectral invariance and norm estimates linked to abstract Cauchy equations. This extends some previous results in th ..."
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Abstract. We show that the Davies functional calculus and the AC(ν)calculus coincide under common hypotheses. Then we apply the calculus to operators on Banach spaces, to investigate spectral invariance and norm estimates linked to abstract Cauchy equations. This extends some previous results in the area, and unifies diverse approaches. The theory is applied to operators related to multiplier theory on Lie groups.
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"... quadrature formula for diffusion polynomials corresponding to a generalized heat kernel ..."
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quadrature formula for diffusion polynomials corresponding to a generalized heat kernel
A SURVEY ON THE KREIN–VON NEUMANN EXTENSION, THE CORRESPONDING ABSTRACT BUCKLING PROBLEM, AND WEYLTYPE SPECTRAL ASYMPTOTICS FOR PERTURBED KREIN LAPLACIANS IN NONSMOOTH DOMAINS
, 2012
"... In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, S ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buc ..."
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In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein–von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, S ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where S = −∆  C ∞ 0 (Ω) in L 2 (Ω;d n x) for Ω ⊂ R n an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein–von Neumann extension of S), SKv = λv, λ ̸ = 0, is in onetoone correspondence with the problem of the buckling of a clamped plate, where u and v are related via the pair of formulas (−∆) 2 u = λ(−∆)u in Ω, λ ̸ = 0, u ∈ H 2 0(Ω), u = S −1 F (−∆)v, v = λ−1 (−∆)u,