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  "body": "\n## Summary\n\nA convex integrand is a Lagrangian $L(x, u, \\xi)$ of an integral functional $\\int_\\Omega L(x, u, \\nabla u) \\, dx$ that is a [[convex function|07GR3TNW4H]] of the gradient variable $\\xi$.\nConvexity in $\\xi$ is the hypothesis that makes such a functional weakly lower semicontinuous on Sobolev spaces, for scalar functions $u$.\nIt is therefore the basic structural assumption of existence theorems such as Tonelli's.\nFor vector-valued $u$ it can be weakened to quasiconvexity.\n\n## Explanations\n\n### L1 human\n\nThe integrand describes the cost of each local slope of a function.\nIf this cost is convex in the slope, mixing two slopes in fine oscillations never costs less than using their average slope.\nSo oscillating functions cannot undercut smooth ones, and a minimiser can be found as a limit.\nThe Dirichlet energy, with $L = \\frac{1}{2} \\abs{\\xi}^2$, has a convex integrand.\nThe double-well cost $(\\xi^2 - 1)^2$ is not convex, and indeed fine zigzags with slopes $\\pm 1$ beat the flat function.\n",
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