Analysis of elastic symbols with the Cauchy integral and construction of asymptotic solutions (Q1712761)

The paper deals with the Dirichlet boundary value problem for the elastic wave equation in the halfspace \(\mathbb{R}_+^n:\) \[ (D_t^2-L(x,D_x)) u(t,x)=0\;\text{in}\;\mathbb{R} \times \mathbb{R}_+^n,\;u|_{x_n=0} =f(t,x') \;\text{on}\;\mathbb{R}\times \mathbb{R}_+^n \] with \[ L(x,D_x)=\sum_{j,\ell=1}^n a_{j,\ell}(x)(-i\partial_{x_j})(-i\partial_{x_\ell})+\sum_{j=1}^n b_j(x)(-i\partial_{x_j})+b_0(x). \] The author generalizes the case of constant coefficients with the solution represented by using the Cauchy integral \(\;\int_C e^{ix_n}(I-L(\xi',\xi))^{-1}\,d\zeta\;\) to variable coefficients case and an asymptotic solution with the similar Cauchy integral is constructed.