Estimates for Fourier Transforms of Surface-Carried Densities on Surfaces with Singular Points

Liess, Otto

doi:10.1007/978-3-0348-9092-2_26

Cited by 4 publications

(15 citation statements)

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“…It is with these examples in mind that we decided to restrict attention (mostly) to surfaces in R 3 . In a paper in preparation it will be shown that the decay estimates for the solutions of the system of crystal optics in cubic crystals mentioned in [8] can be slightly improved if the arguments are based on proposition 3.2 below rather than on theorem 1.1 in [9]. There is also a nontrivial intersection of the present paper with [1], but most of the result here are new.…”

Section: Introductionmentioning

confidence: 58%

“…In this section we shall consider surfaces related to the uniplanar surfaces in [9]. Our main point is that one can not discard the assumptions (we refer to assumption (1.12) there; for brevity we do not state it here) in theorem 1.1 in that paper completely if one wants to obtain decay of the quality described there, and in fact that if one weakens the assumptions there in a controlled way, then the decay one obtains is strictly weaker.…”

Section: Decay For Weakly Singular Surfacesmentioning

confidence: 99%

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Estimates for Fourier transforms of surface carried densities on surfaces with singular points, II

Bannini¹,

Liess

2006

Ann. Univ. Ferrara

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Section: Introductionmentioning

confidence: 58%

Section: Decay For Weakly Singular Surfacesmentioning

confidence: 99%

Estimates for Fourier transforms of surface carried densities on surfaces with singular points, II

Bannini¹,

Liess

2006

Ann. Univ. Ferrara

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“…We have written this out with details, to show the relation with Theorem 3.1 in the form in which this theorem has been stated in [16], but when we apply the results from [16], we shall practically return to the form which the inner integral has in (6.10). (Also see Section 7.)…”

Section: Reduction To the Slowness Surfacementioning

confidence: 99%

“…If it holds, then this part of the argument will be applicable for the crystal at hand, even though we are not assuming it to be nearly isotropic.) Given the special form which the curves Γ ± have, we may use here a result from [16], which we shall now recall for the convenience of the reader. This result gives a criterium to check wether or not quartics in the plane of a special form have inflection points.…”

Section: The Case Of Uniplanar Pointsmentioning

confidence: 99%

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Decay estimates for the solutions of the system of crystal acoustics for cubic crystals

Liess

2009

Asymptotic Analysis

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The aim of this paper is to study decay estimates for global solutions of the system of linear crystal elasticity in three space dimensions for cubic crystals in the generic nearly isotropic case. Our main result is that for large time, the solutions to the system decay with a decay rate of order t −1/2−1/κ for some κ ∈ N which is strictly positive (cf. Remark 1.2). We should observe that in the isotropic case, the optimal decay rate is t −1 and that the decay rate for the wave equation in n space variables is t −(n−1)/2 . On the other hand, the decay rate for solutions of the Maxwell system for optically biaxial crystals is of order t −1/2 . (See O. Liess, Asymptot. Anal. 4 (1991), 61-95.)where the a, b, c, are three real constants which can be calculated in terms of the stiffness constants of the crystal (cf., e.g., [4,23]). Equation (1.1) describes the evolution of elastic waves in the given crystal. Several restrictions on the constants a, b, c come from physical considerations and give, among others, that the system is hyperbolic. Of these restrictions we mention the following ones (see, e.g., [15]): c > 0, 3c − b + a > 0, a = 0, a + c > 0. As in [4] we shall also assume that b > 0 and we shall justify this assumption later on. The main reference in mathematical crystal elasticity is perhaps still Musgrave [22], but most of the results on crystals used later on are from the paper of Duff [4]. 0921-7134/09/$17.00 © 2009 -IOS Press and the authors. All rights reserved 2 O. Liess / Solutions of the system of crystal acoustics for cubic crystalsWith the system (1.1) we now associate the initial conditions:Here we shall assume that the functions f i , g i , i = 1, 2, 3, are C ∞ -functions on R 3 and have compact support. Since the system (1.1) is hyperbolic (and with constant coefficients) it clearly admits global solutions on R 4 . It is, moreover, standard to observe that the functions x → u i (t, x) are compactly supported in x for any fixed t, so it will in particular make sense to consider partial Fourier transforms in x, with t considered a parameter. In the case a = b the system reduces to Lamé's system of isotropic elasticity with the Lamé constants λ, μ given by the conditions λ + 2μ = c + a, μ = c. Here √ μ is the velocity of the two shear waves and √ c + 2a the velocity of the pressure wave. Since the velocity of the shear wave is in the isotropic case bigger than that of the pressure wave, we must have b > 0 (cf. [12]). Since we shall assume in most of the paper that b − a is small compared with c, we take this as a justification to assume that b 0 whenever this simplifies the situation. We denote b − a henceforth by d, so d becomes a measure of the anisotropy of the crystal. We shall say that some statement holds in the nearly isotropic case if for every fixedWe shall moreover say, that a property holds generically, if for every (b 0 , c 0 , d 0 ) under consideration there is a neighborhood U ⊂ R 3 of (b 0 , c 0 , d 0 ), a constantc and an algebraic function ϕ : U → R, such that the property holds wh...

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Remarks on the Lax conjecture for hyperbolic polynomials

Liess

2009

Linear Algebra and its Applications

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Estimates for Fourier Transforms of Surface-Carried Densities on Surfaces with Singular Points

Cited by 4 publications

References 9 publications

Estimates for Fourier transforms of surface carried densities on surfaces with singular points, II

Estimates for Fourier transforms of surface carried densities on surfaces with singular points, II

Decay estimates for the solutions of the system of crystal acoustics for cubic crystals

Remarks on the Lax conjecture for hyperbolic polynomials

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