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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
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
A Note on High Precision Solutions of Two Fourth Order Eigenvalue Problems
, 1998
"... We solve the biharmonic eigenvalue problem \Delta 2 u = u and the buckling plate problem \Delta 2 u = \Gamma\Deltau on the unit square using a highly accurate spectral LegendreGalerkin method. We study the nodal lines for the first eigenfunction near a corner for the two problems. Five sign cha ..."
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Cited by 3 (0 self)
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We solve the biharmonic eigenvalue problem \Delta 2 u = u and the buckling plate problem \Delta 2 u = \Gamma\Deltau on the unit square using a highly accurate spectral LegendreGalerkin method. We study the nodal lines for the first eigenfunction near a corner for the two problems. Five sign changes are computed and the results show that the eigenfunction exhibits a self similar pattern as one approaches the corner. The amplitudes of the extremal values and the coordinates of their location as measured from the corner are reduced by constant factors. These results are compared with the known asymptotic expansion of the solution near a corner. This comparison shows that the asymptotic expansion is highly accurate already from the first sign change as we have complete agreement between the numerical and the analytical results. Thus, we have an accurate description of the eigenfunction in the entire domain. AMS Subject Classification: 65N25. Key words: Biharmonic operator, eigenvalu...
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.
A parabolic free boundary problem modeling electrostatic
"... ABSTRACT. A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrowgap model is given by showing that ..."
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Cited by 3 (3 self)
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ABSTRACT. A free boundary problem describing small deformations in a membrane based model of electrostatically actuated MEMS is investigated. The existence of stationary solutions is established for small voltage values. A justification of the widely studied narrowgap model is given by showing that steady state solutions of the free boundary problem converge toward stationary solutions of the narrowgap model when the aspect ratio of the device tends to zero.
SIGNPRESERVING PROPERTY FOR SOME FOURTHORDER ELLIPTIC OPERATORS IN ONE DIMENSION AND RADIAL SYMMETRY
"... ABSTRACT. For a class of onedimensional linear elliptic fourthorder equations with homogeneous Dirichlet boundary conditions it is shown that a nonpositive and nonvanishing righthand side gives rise to a negative solution. A similar result is obtained for the same class of equations for radiall ..."
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Cited by 1 (1 self)
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ABSTRACT. For a class of onedimensional linear elliptic fourthorder equations with homogeneous Dirichlet boundary conditions it is shown that a nonpositive and nonvanishing righthand side gives rise to a negative solution. A similar result is obtained for the same class of equations for radially symmetric solutions in a ball or in an annulus. Several applications are given, including applications to nonlinear equations and eigenvalue problems. hal00798584, version 1 8 Mar 2013 1.