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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set with Lipschitz boundary, and let $1 \u003c p \u003c \\infty$.\nLet $L \\colon \\overline{\\Omega} \\times \\R \\times \\R^n \\to \\R$, written $L(x, u, \\xi)$, be continuous.\nAssume:\n\n1. $L$ is a [[convex integrand|X0Q69ZS09T]]: $\\xi \\mapsto L(x, u, \\xi)$ is convex for all $(x, u)$;\n2. there are constants $a \u003e 0$ and $b \\in \\R$ with $L(x, u, \\xi) \\ge a \\abs{\\xi}^p + b$ for all $(x, u, \\xi)$.\n\nLet $u_0$ be in the [[Sobolev space|M4CEZ5K9HP]] $W^{1,p}(\\Omega)$ and let $J(u) = \\int_\\Omega L(x, u, \\nabla u) \\, dx$, an [[integral functional|JRQZ803Q5Y]].\nIf $J(u_0) \u003c +\\infty$, then $J$ attains its minimum on $u_0 + W^{1,p}_0(\\Omega)$.\n\n## Remarks\n\nThe proof is the [[direct method|TE379HBTY0]].\nThe growth bound and the Poincaré inequality make $J$ [[coercive|NGT5YCVAMR]] on $u_0 + W^{1,p}_0(\\Omega)$, so a minimising sequence is bounded in $W^{1,p}(\\Omega)$.\nSince $1 \u003c p \u003c \\infty$, a subsequence converges weakly in $W^{1,p}(\\Omega)$, and strongly in $L^p(\\Omega)$ by the Rellich–Kondrachov theorem.\nConvexity in $\\xi$ makes $J$ [[weakly lower semicontinuous|H6GTJRG26G]], so the weak limit is a minimiser.\n\nExample: for $L = \\frac{1}{2} \\abs{\\xi}^2 + \\frac{1}{4} u^4$ and $p = 2$, the theorem gives a minimiser with boundary values $u_0$ whenever $J(u_0) \u003c +\\infty$; it is a weak solution of $-\\Delta u + u^3 = 0$.\n\nEach hypothesis matters.\nFor $L = (\\xi^2 - 1)^2 + u^2$ with $n = 1$ and $u_0 = 0$, convexity fails and there is no minimiser.\nFor $L = \\sqrt{1 + \\abs{\\xi}^2}$, which has growth $p = 1$, minimisers of the area of a graph need not exist in $W^{1,1}$ for some boundary data.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that an energy that is convex in the slope and grows fast enough has a lowest state with given boundary values.\nFast growth keeps minimising sequences from running away.\nConvexity prevents them from lowering the energy by oscillating.\nTogether they force a minimising sequence to settle on a true minimiser.\n\n### L2 human\n\nThis is the model existence theorem of the calculus of variations, and it is the template for the direct method.\nThe minimiser is a priori only a Sobolev function; whether it is smooth is the separate question of regularity, raised in Hilbert's nineteenth problem.\nFor vector-valued $u$, convexity can be weakened to Morrey quasiconvexity, and in nonlinear elasticity to polyconvexity, as in Ball's existence theorem.\nWithout convexity one replaces the integrand by its convex envelope; this is relaxation, and it computes the limits of minimising sequences.\n",
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