Lamé equations of elasticity

Remarks

The conditions $\mu > 0$ and $3\lambda + 2\mu > 0$ make the elastic energy positive definite on symmetric strains. They imply $\lambda + 2\mu > 0$, so the system is strongly elliptic.

Example: for $a, b \in \R^3$ and $f = 0$, the rigid displacement $u(x) = a + b \times x$ is a solution. Each component of $u$ is affine, so $\Delta u = 0$, and $\nabla \cdot u = 0$.

Non-example: for $f = 0$, $u(x) = (x_1^2, 0, 0)$ is not a solution. Here $\Delta u = (2, 0, 0)$ and $\nabla(\nabla \cdot u) = (2, 0, 0)$, so the left side is $(2(\lambda + 2\mu), 0, 0) \ne 0$.

The equations come from the balance of forces $\nabla \cdot \sigma + f = 0$ with Hooke's law $\sigma = \lambda (\nabla \cdot u) I + \mu (\nabla u + \nabla u^T)$ for the stress $\sigma$. In the dynamic equations, pressure waves travel at speed $\sqrt{(\lambda + 2\mu)/\rho}$ and shear waves at speed $\sqrt{\mu / \rho}$.