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October, 2018 Analysis of elastic symbols with the Cauchy integral and construction of asymptotic solutions
Hideo SOGA
J. Math. Soc. Japan 70(4): 1557-1580 (October, 2018). DOI: 10.2969/jmsj/75977597


This paper deals with the elastic wave equation $(D_t^2 - L(x, D_{x'}, D_{x_n})) u(t, x', x_n)=0$ in the half-space $x_n>0$. In the constant coefficient case, it is known that the solution is represented by using the Cauchy integral $\int_c e^{ix_n\zeta} (I-L(\xi', \zeta))^{-1} d\zeta$. In this paper this representation is extended to the variable coefficient case, and an asymptotic solution with the similar Cauchy integral is constructed. In this case, the terms $\partial_x^\alpha \int_c e^{ix_n\zeta} (I-L(x,\xi',\zeta))^{-1} d\zeta$ appear in the inductive process. These do not become lower terms necessarily, and therefore the principal part of asymptotic solution is a little different from the form in the constant coefficient case.


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Hideo SOGA. "Analysis of elastic symbols with the Cauchy integral and construction of asymptotic solutions." J. Math. Soc. Japan 70 (4) 1557 - 1580, October, 2018.


Received: 5 September 2016; Revised: 5 May 2017; Published: October, 2018
First available in Project Euclid: 3 October 2018

zbMATH: 07009712
MathSciNet: MR3868217
Digital Object Identifier: 10.2969/jmsj/75977597

Primary: 74B05
Secondary: 35C20 , 35L05 , 35L51 , 74J05

Keywords: asymptotic solutions , Elastic equations , singularities , wave equations

Rights: Copyright © 2018 Mathematical Society of Japan


Vol.70 • No. 4 • October, 2018
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