I'm trying to get an exact solution to a specific boundary value problem in linear elasticity. Standard techniques one would use to solve the heat or wave equations, such as separation of variables and Fourier series don't seem directly applicable here, and in my search I haven't been able to find a solution method for this problem.
I'm not looking for a complete solution, only tricks (if there are any) to simplify the equations so that they can be solved with standard techniques. I will state the full problem for the sake of completeness, although the specific boundary conditions may not be particularly relevant.
We have a box of uniform elastic material $0\le x\le a$, $0\le y\le b$, $0\le z\le c$. The solution variable is the displacement vector $\vec u = (u_x, u_y, u_z)$. The equations are the Navier-Cauchy equilibrium equations: $$ \mu \vec\nabla^2 \vec u + (\lambda + \mu)\vec\nabla\left(\vec\nabla \cdot \vec u\right) = 0 $$ or in scalar form, $$\mu \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right) u_x + (\lambda+\mu)\frac{\partial}{\partial x}\left(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}\right) = 0$$ $$\mu \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right) u_y + (\lambda+\mu)\frac{\partial}{\partial y}\left(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}\right) = 0$$ $$\mu \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right) u_z + (\lambda+\mu)\frac{\partial}{\partial z}\left(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}\right) = 0$$ where $\lambda$ and $\mu$ are material constants. All boundaries are kept fixed ($\vec u = 0$, Dirichlet boundary condition), except for $x = a$, where a uniform force per unit area of $\vec f$ is applied, corresponding to the equations $$\lambda \vec \nabla \cdot \vec u + 2\mu \frac{\partial u_x}{\partial x} = f_x$$ $$\mu \left(\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x}\right) = f_y$$ $$\mu \left(\frac{\partial u_x}{\partial z} + \frac{\partial u_z}{\partial x}\right) = f_z$$
My question is whether there is a standard way to approach this problem, something akin to separation of variables (which I can't apply directly because the equations are coupled), or a change of variables that will simplify the equations. Thank you.