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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $J \\colon X \\to (-\\infty, +\\infty]$ be a [[functional|1EQM0DJ4NC]].\nThe *relaxed functional*, or lower semicontinuous envelope, of $J$ is\n$$\\bar J(u) = \\sup \\set{ G(u) : G \\colon X \\to [-\\infty, +\\infty] \\text{ is lower semicontinuous and } G \\le J }.$$\nThe *relaxed problem* of the problem $\\inf_X J$ is the problem $\\inf_X \\bar J$.\n\n## Remarks\n\nThe function $\\bar J$ is itself [[lower semicontinuous|RN4G8SGP6P]], and $\\bar J \\le J$.\nIf $X$ is a metric space, then $\\bar J(u) = \\inf \\set{ \\liminf_k J(u_k) : u_k \\to u }$.\nIn the calculus of variations one uses the sequential form of this formula for weak convergence in a Sobolev space.\n\nThe relaxed problem has the same infimum: $\\inf_X \\bar J = \\inf_X J$.\nIf $X$ is a metric space, every limit of a subsequence of a minimising sequence of $J$ is a minimiser of $\\bar J$.\n\nExample: on $X = \\R$, let $J(u) = u^2$ for $u \\ne 0$ and $J(0) = 1$.\nThen $\\bar J(u) = u^2$ for all $u$, which attains the common infimum $0$ at $u = 0$, while $J$ does not attain it.\n\nExample: let $J(u) = \\int_0^1 \\big( (u'^2 - 1)^2 + u^2 \\big) \\, dx$ on $W^{1,4}_0(0, 1)$ with weak convergence.\nIts relaxation is $\\bar J(u) = \\int_0^1 \\big( f^{**}(u') + u^2 \\big) \\, dx$, where $f^{**}(\\xi) = \\big(\\max(\\xi^2 - 1, 0)\\big)^2$ is the convex envelope of $f(\\xi) = (\\xi^2 - 1)^2$.\nThe function $u = 0$ minimises $\\bar J$, with $\\bar J(0) = 0 = \\inf J$, although $J$ has no minimiser.\n\nNon-example: for maps $u \\colon \\Omega \\to \\R^m$ with $m, n \\ge 2$, replacing the integrand by its convex envelope does not give $\\bar J$ in general.\nFor $2 \\times 2$ matrices, $\\det$ is weakly continuous on $W^{1,p}$ for $p \u003e 2$, so it needs no relaxation, but its convex envelope is $-\\infty$.\nThe correct integrand is the quasiconvex envelope, under growth conditions.\n",
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