### Curvature properties of the slowness surface of the system of crystal acoustics for cubic crystals II

Otto Liess and Tetsuya Sonobe
Source: Osaka J. Math. Volume 49, Number 2 (2012), 357-391.

#### Abstract

In this paper we study geometric properties of the slowness surface of the system of crystal acoustics for cubic crystals in the special case when the stiffness constants satisfy the condition $a = -2b$. The paper is a natural continuation of the paper [9] in which related properties were studied for general constants $a$ and $b$, but assuming that we were in the nearly isotropic case, in which case $a - b$ has to be small. We also take this opportunity to correct a statement made in [9]: see Remark 1.3.

First Page:
Primary Subjects: 53A05, 35Q72
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.ojm/1340197931
Zentralblatt MATH identifier: 06060691
Mathematical Reviews number (MathSciNet): MR2945754

### References

A. Bannini and O. Liess: Estimates for Fourier transforms of surface carried densities on surfaces with singular points, II, Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 52 (2006), 211–232.
Mathematical Reviews (MathSciNet): MR2273095
Zentralblatt MATH: 1134.42309
Digital Object Identifier: doi:10.1007/s11565-006-0017-2
R. Courant and D. Hilbert: Methods of mathematical physics, II, Partial differential equations, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989.
Mathematical Reviews (MathSciNet): MR1013360
G.F.D. Duff: The Cauchy problem for elastic waves in an anisotropic medium, Phil. Transactions Royal Soc. London, Ser. A 252 (1960), 249–273.
Mathematical Reviews (MathSciNet): MR111293
Digital Object Identifier: doi:10.1098/rsta.1960.0006
B. Gross and J. Harris: Real algebraic curves, Ann. E.N.S., 4\textsuperscripte série t.14, (1981), 157–182.
Mathematical Reviews (MathSciNet): MR631748
Zentralblatt MATH: 0533.14011
L.D. Landau and E.M. Lifshits: Theoretical Physics; Elasticity Theory, first Russian edition, Mir, Moscow, 1953. (Available also in many other editions and translations in many languages.)
Mathematical Reviews (MathSciNet): MR1020299
O. Liess: Conical Refraction and Higher Microlocalization, Springer Lecture Notes in Mathematics 1555, 1993.
Mathematical Reviews (MathSciNet): MR1317813
Zentralblatt MATH: 0812.35002
O. Liess: Estimates for Fourier transforms of surface-carried densities on surfaces with singular points, Asymptotic analysis 37 (2004), 329–363.
Mathematical Reviews (MathSciNet): MR2047744
Zentralblatt MATH: 1075.35133
O. Liess: Decay estimates for the solutions of the system of crystal acoustics for cubic crystals, Kokyoroku series of the RIMS in Kyoto 1412 (2005), 1–13.
Mathematical Reviews (MathSciNet): MR2559826
Zentralblatt MATH: 1172.35505
O. Liess: Curvature properties of the slowness surface of the system of crystal acoustics for cubic crystals, Osaka J. Math. 45 (2008), 173–210.
Mathematical Reviews (MathSciNet): MR2416656
Zentralblatt MATH: 1140.53002
Project Euclid: euclid.ojm/1205503564
O. Liess: Decay estimates for the solutions of the system of crystal acoustics for cubic crystals, Asymptotic analysis 64 (2009), 1–27.
Mathematical Reviews (MathSciNet): MR2559826
Zentralblatt MATH: 1172.35505
G.F. Miller and M.J.P. Musgrave: On the propagation of elastic waves in aleotropic media, III, Media of cubic symmetry, Proc. Royal Soc. London 236 (1956), 352–383.
Mathematical Reviews (MathSciNet): MR79433
Digital Object Identifier: doi:10.1098/rspa.1956.0142
M.J.P. Musgrave: Crystal Acoustics, Holden and Day, San Francisco, 1979.
N. Ortner and P. Wagner: Fundamental matrices of homogeneous hyperbolic systems. Applications to crystal optics, elastodynamics, and piezoelectromagnetism, ZAMM 84 (2004), 314–346.
Mathematical Reviews (MathSciNet): MR2057145
H.G. Zeuthen: Sur les courbes du quatrième ordre, Mathematische Annalen 7, (1874), 410–432.