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  "body": "\n## Summary\n\nThe relaxation of a variational problem replaces a functional $J$ that has no minimiser by its lower semicontinuous envelope $\\bar J$.\nThis is the largest lower semicontinuous functional below $J$.\nThe relaxed problem has the same infimum as the original one.\nEvery limit of a minimising sequence of the original problem, in the topology used, is a minimiser of the relaxed problem.\nFor integral functionals, relaxation typically replaces the integrand by its convex or quasiconvex envelope in the gradient variable.\n\n## Explanations\n\n### L1 human\n\nSometimes a minimisation problem has no solution because minimising sequences oscillate faster and faster, like a zigzag with ever finer teeth.\nThe limit of such a sequence is a smooth function, but the original energy undervalues what the zigzags achieved.\nRelaxation fixes the energy so that it assigns to each function the best value that nearby functions can approach.\nThen the limit becomes a true minimiser of the corrected energy.\nFor the double-well cost $(\\xi^2 - 1)^2$, relaxation replaces it by $0$ on $[-1, 1]$, because zigzags of slopes $\\pm 1$ can produce any average slope in between at no cost.\n",
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