Isoperimetric Inequalities and Their Applications

Payne, L. E.

doi:10.1137/1009070

Cited by 327 publications

(106 citation statements)

References 111 publications

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“…2 n = ) states that j is the first zero of the Bessel function of order zero, and A is the area of the membrane. Equality is attained in (8) only if the membrane is circular [9] [10] [11].…”

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confidence: 99%

The Uncertainty Principle in Terms of Isoperimetric Inequalities

Schürmann¹

2017

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confidence: 99%

The Uncertainty Principle in Terms of Isoperimetric Inequalities

Schürmann¹

2017

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“…where C stands for the capacity of S [10]. Therefore, an upper bound is furnished from (32): 4rip g0" ~s Tuo(r) a s ~< -7 -C.…”

Section: I't Fv [V F[2 DVmentioning

confidence: 99%

“…This was done in [4] where the exact solutions of the first two low-frequency approximations for scattering by a rigid triaxial ellipsoid was obtained. Ellipsoidal harmonics can be used to evaluate the capacity, the polarization tensor and the virtual mass tensor of a general ellipsoid [10,11].…”

Section: Z Fvmentioning

confidence: 99%

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Estimates for low-frequency elastic scattering by a rigid body

Dassios

Payne

1988

J Elasticity

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When a plane elastic wave is scattered by a rigid body the surface integral of the traction, projected along the direction of polarization of the incident wave, provides the leading low-frequency approximation for the scattering amplitudes. Two kinds of lower and upper bounds for the surface traction integral are given. One is based on the geometrical characteristics of the scatterer and is expressed in terms of corresponding values of the best fitting interior and exterior confocal triaxial ellipsoids. The case of best fitting interior and exterior spheres is examined as a special case. These bounds are sharp in the sense that they both become equalities when the scatterer degenerates to an ellipsoid. The other kind of lower and upper bounds involve the capacity of the scatterer. All estimates were obtained by using the generalized Dirichlet and Thomson Principles of Potential Theory in Elastostatics. Furthermore, all constants appearing in the bounds are given in terms of the ratio of the phase velocities for the transverse and the longitudinal wave. An upper bound for scattering by a cube at normal incidence is also included.

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“…Despite the large number of known results there are still many challenging conjectures in the area too numerous to list here. For references and more on the history of the problem the survey articles by Osserman [30] and Payne [31] are still a good source. A classic is also Bandle's book [4].…”

Section: Introductionmentioning

confidence: 99%

Krahn’s proof of the Rayleigh conjecture revisited

Daners

2011

Arch. Math.

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The paper is a discussion of Krahn's proof of the Rayleigh conjecture that amongst all membranes of the same area and the same physical properties, the circular one has the lowest ground frequency. We show how his approach coincides with the modern techniques of geometric measure theory using the co-area formula. We furthermore discuss some issues and generalisations of his proof.Mathematics Subject Classification (2010). 35P15; 01A60.

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Isoperimetric Inequalities and Their Applications

Cited by 327 publications

References 111 publications

The Uncertainty Principle in Terms of Isoperimetric Inequalities

The Uncertainty Principle in Terms of Isoperimetric Inequalities

Estimates for low-frequency elastic scattering by a rigid body

Krahn’s proof of the Rayleigh conjecture revisited

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