I have a problem coming from linear elasticity in $(x,y,z)\in\mathbb{R}^2\times \mathbb{R}^+$, $t\in \mathbb R$:

$$\left\{\begin{aligned}\partial_{tt} \sigma&=A(\partial_x,\partial_y,\partial_z) \sigma\\
 \sigma\big|_{t=0}&=0\\
\partial_t\sigma\big|_{t=0}&=0\\
\sigma\big|_{z=0}&=f(x,y,t),
\end{aligned}\right. $$

where $\sigma=(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{xy},\sigma_{xz},\sigma_{yz})$ is the vector of stresses. $A$ is assumend to be a constant, second order differential Matrix, symmetric and positive semidefinite.

This equation describes the equation of motion for the stresses, if one knows the stresses on the boundary.

Since I would like to apply Fourier Transform, I need to transform this into a whole space problem.

Can anyone help me here? Is it possible to write this somehow like this:


$$\left\{\begin{aligned}\partial_{tt} \widetilde{\sigma}&=A(\partial_x,\partial_y,\partial_z) \widetilde{\sigma} + \text{something}\\
 \widetilde{\sigma}\big|_{t=0}&=0\\
\partial_t\widetilde{\sigma}\big|_{t=0}&=0,
\end{aligned}\right. $$

where $\widetilde\sigma$ is definded on the whole space and $\widetilde\sigma\big|_{z>0}\equiv \sigma$?

If anyone has some literature for me, that would be great too!

thanks for the help