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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set and let $m \\ge 1$.\nLet $L \\colon \\overline{\\Omega} \\times \\R^m \\times \\R^{m \\times n} \\to \\R$, written $L(x, u, \\xi)$ with $\\xi = (\\xi^i_\\alpha)$, be of class $C^2$.\nLet $J(u) = \\int_\\Omega L(x, u, \\nabla u) \\, dx$ be the [[integral functional|JRQZ803Q5Y]] on $C^1(\\overline{\\Omega}; \\R^m)$.\nLet $u_0 \\in C^1(\\overline{\\Omega}; \\R^m)$ be a [[weak local minimiser|WMPCR723MJ]] of $J$ among functions with the same boundary values.\nThen for every $x \\in \\Omega$, every $a \\in \\R^m$ and every $b \\in \\R^n$,\n$$\\sum_{i, j = 1}^{m} \\sum_{\\alpha, \\beta = 1}^{n} \\frac{\\partial^2 L}{\\partial \\xi^i_\\alpha \\, \\partial \\xi^j_\\beta}\\big(x, u_0(x), \\nabla u_0(x)\\big) \\, a_i a_j b_\\alpha b_\\beta \\ge 0.$$\n\n## Remarks\n\nThe [[second variation|4C3BANJ8AH]] at $u_0$ is nonnegative for all test functions $\\varphi \\in C^1_c(\\Omega; \\R^m)$.\nOne tests it with $\\varphi(x) = \\varepsilon \\, \\eta(x) \\, a \\sin(b \\cdot x / \\varepsilon)$ for a fixed smooth cut-off function $\\eta$ supported near $x_0$.\nAs $\\varepsilon \\to 0$, only the term quadratic in $\\nabla \\varphi$ survives, and its limit is a positive multiple of the stated quadratic form at $x_0$ weighted by $\\eta^2$.\n\nThe condition asks for positivity only on rank-one matrices $a \\otimes b = (a_i b_\\alpha)$, not on all of $\\R^{m \\times n}$.\nFor $m = 1$ or $n = 1$ every matrix is of the form $a \\otimes b$.\nThen the condition says that the Hessian of $L$ in $\\xi$ is positive semidefinite, as in the one-dimensional Legendre condition.\n\nExample: for $m = n = 2$ and $L(\\xi) = \\det \\xi$, the quadratic form is $2 \\det(a \\otimes b) = 0$, so the condition holds although the Hessian of $\\det$ is indefinite.\nFor a $C^2$ integrand, the condition at every $\\xi$ is equivalent to rank-one convexity of $\\xi \\mapsto L(x, u, \\xi)$.\n\n## Explanations\n\n### L1 human\n\nFor maps with several components and several variables, a minimiser need not make the integrand convex in all gradient directions.\nIt only has to be convex along very special directions, the rank-one matrices $a \\otimes b$.\nThese are the directions in which a gradient can oscillate quickly, like a fine fold or a laminate.\n\n### L2 human\n\nThe Legendre–Hadamard condition is the necessary condition from the second variation for vector-valued problems in several variables.\nWhen it holds with a strict inequality and a uniform constant, the Euler–Lagrange system is called strongly elliptic, which is the basic hypothesis of linear elliptic theory for systems.\nIt is the infinitesimal form of rank-one convexity, which in turn is implied by Morrey's quasiconvexity.\n",
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