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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^3$ be an open set, and let $\\lambda, \\mu$ be real constants, the Lamé parameters, with $\\mu \u003e 0$ and $3\\lambda + 2\\mu \u003e 0$.\nLet $f \\colon \\Omega \\to \\R^3$ be a given body force.\nThe *Lamé equations* of static linear elasticity are the system\n$$\\mu \\Delta u + (\\lambda + \\mu) \\nabla (\\nabla \\cdot u) + f = 0 \\quad \\text{in } \\Omega$$\nfor an unknown [[vector field|0299VCW1TH]] $u \\colon \\Omega \\to \\R^3$, the displacement.\nHere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] acting on each component, $\\nabla \\cdot u$ is the [[divergence|0MWV6ZTK6S]] and $\\nabla$ is the [[gradient|0ABK9V9Y64]].\nFor a body of constant density $\\rho \u003e 0$, the *dynamic Lamé equations* are $\\rho \\, \\partial_t^2 u = \\mu \\Delta u + (\\lambda + \\mu) \\nabla (\\nabla \\cdot u) + f$.\n\n## Remarks\n\nThe conditions $\\mu \u003e 0$ and $3\\lambda + 2\\mu \u003e 0$ make the elastic energy positive definite on symmetric strains.\nThey imply $\\lambda + 2\\mu \u003e 0$, so the system is strongly elliptic.\n\nExample: for $a, b \\in \\R^3$ and $f = 0$, the rigid displacement $u(x) = a + b \\times x$ is a solution.\nEach component of $u$ is affine, so $\\Delta u = 0$, and $\\nabla \\cdot u = 0$.\n\nNon-example: for $f = 0$, $u(x) = (x_1^2, 0, 0)$ is not a solution.\nHere $\\Delta u = (2, 0, 0)$ and $\\nabla(\\nabla \\cdot u) = (2, 0, 0)$, so the left side is $(2(\\lambda + 2\\mu), 0, 0) \\ne 0$.\n\nThe 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 $\\sigma$.\nIn the dynamic equations, pressure waves travel at speed $\\sqrt{(\\lambda + 2\\mu)/\\rho}$ and shear waves at speed $\\sqrt{\\mu / \\rho}$.\n",
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