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  "body": "\n## Definition\n\nLet $X$ be a set with a notion of convergence of sequences, and let $J \\colon X \\to \\R \\cup \\set{+\\infty}$ be a functional with $\\inf_X J \u003c +\\infty$.\nThe *direct method* is the following way to prove that $J$ has a minimiser.\n\n1. Take a minimising sequence $(u_k)$, that is, $J(u_k) \\to \\inf_X J$.\n2. Use *coercivity* to show that $(u_k)$ lies in a set that is sequentially compact, and extract a subsequence $u_{k_j} \\to u$ in $X$.\n3. Use sequential *lower semicontinuity*, $J(u) \\le \\liminf_j J(u_{k_j})$, to conclude that $J(u) = \\inf_X J$.\n\nThe standard abstract form is this.\nLet $X$ be a [[reflexive|NN8RWPC8WG]] Banach space and let $J$ be coercive, meaning $J(u) \\to +\\infty$ as $\\norm{u} \\to \\infty$.\nIf $J$ is sequentially [[lower semicontinuous|95BSKT0S6M]] for the [[weak topology|P6KHVSTPZV]], then $J$ attains its minimum on $X$.\n\n## Remarks\n\nIn the abstract form, coercivity makes a minimising sequence bounded, and bounded sequences in a reflexive Banach space have weakly convergent subsequences.\nFor integral functionals $J(u) = \\int_\\Omega L(x, u, \\nabla u) \\, dx$ of the [[calculus of variations|PQ25NHV0KS]] on $W^{1,q}(\\Omega)$ with $1 \u003c q \u003c \\infty$, weak lower semicontinuity holds when $L$ is continuous, bounded below and convex in $\\nabla u$.\n\nExample: let $\\Omega \\subseteq \\R^n$ be bounded and open, $f \\in L^2(\\Omega)$, and $J(u) = \\int_\\Omega \\big(\\frac{1}{2}\\abs{\\nabla u}^2 - f u\\big) \\, dx$ on $H^1_0(\\Omega)$.\nBy the Poincaré inequality $J(u) \\ge c\\norm{u}_{H^1}^2 - C\\norm{u}_{H^1}$, so $J$ is coercive; it is convex and continuous, hence weakly lower semicontinuous.\nSo $J$ has a minimiser, which is the weak solution of $-\\Delta u = f$ with $u = 0$ on $\\partial\\Omega$.\n\nNon-example: $J(u) = \\int_0^1 \\big((u'^2 - 1)^2 + u^2\\big) \\, dx$ on $W^{1,4}(0, 1)$ is coercive, but its integrand is not convex in $u'$, so it is not weakly lower semicontinuous.\nIndeed $\\inf J = 0$, approached by sawtooth functions with slopes $\\pm 1$ and small height, but no function has $J(u) = 0$, since this would need $u = 0$ and $\\abs{u'} = 1$ almost everywhere.\n\n## Explanations\n\n### L1 human\n\nThe direct method finds a minimiser by taking a minimising sequence, extracting a convergent subsequence, and passing to the limit.\nCoercivity gives the convergent subsequence and lower semicontinuity makes the limit a minimiser.\n",
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