Analysis of elastic symbols with the Cauchy integral and construction of asymptotic solutions

Abstract

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.