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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be open and let $m \\ge 1$.\nLet $L \\colon \\Omega \\times \\R^m \\times \\R^{m \\times n} \\to \\R$, written $L(x, u, \\xi)$, be the integrand of an [[integral functional|JRQZ803Q5Y]] $J(u) = \\int_\\Omega L(x, u, \\nabla u) \\, dx$.\nThe integrand $L$ is *convex* (in the gradient variable) if for every $x \\in \\Omega$ and $u \\in \\R^m$ the map $\\xi \\mapsto L(x, u, \\xi)$ is a [[convex function|07GR3TNW4H]] on $\\R^{m \\times n}$.\n\n## Remarks\n\nExample: $L(x, u, \\xi) = \\frac{1}{2} \\abs{\\xi}^2 - f(x) u$ is convex in $\\xi$ for any function $f$.\nIndeed $\\xi \\mapsto \\frac{1}{2} \\abs{\\xi}^2$ is convex, and the term $-f(x) u$ does not depend on $\\xi$.\n\nNon-example: $L(\\xi) = (\\xi^2 - 1)^2$ with $n = m = 1$ is not convex.\nHere $L(-1) = L(1) = 0$, but $L(0) = 1 \u003e \\frac{1}{2} L(-1) + \\frac{1}{2} L(1)$.\n\nA $C^2$ integrand is convex in $\\xi$ exactly when its Hessian $\\big(\\partial^2 L / \\partial \\xi^i_\\alpha \\partial \\xi^j_\\beta\\big)$ is positive semidefinite everywhere.\nSome authors use the term for integrands that are jointly convex in $(u, \\xi)$, which is a stronger condition.\nFor $m \\ge 2$ and $n \\ge 2$, convexity is stronger than polyconvexity, quasiconvexity and rank-one convexity.\n",
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