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  "body": "\n## Definition\n\nLet $a \u003c b$, let $\\mathcal{A} \\subseteq C^1([a, b])$ be a class of [[admissible functions|G7T008BEWY]], and let $J \\colon \\mathcal{A} \\to \\R$ be a [[functional|1EQM0DJ4NC]].\nFor $y \\in C^1([a, b])$ use the [[norms|0V2WP9B1MF]]\n$$\\norm{y}_0 = \\max_{[a, b]} \\abs{y}, \\qquad \\norm{y}_1 = \\max_{[a, b]} \\abs{y} + \\max_{[a, b]} \\abs{y'}.$$\nLet $y_0 \\in \\mathcal{A}$.\n\n1. $y_0$ is a *strong local minimum* of $J$ if there is $\\varepsilon \u003e 0$ with $J(y_0) \\le J(y)$ for all $y \\in \\mathcal{A}$ with $\\norm{y - y_0}_0 \u003c \\varepsilon$.\n2. $y_0$ is a *weak local minimum* of $J$ if there is $\\varepsilon \u003e 0$ with $J(y_0) \\le J(y)$ for all $y \\in \\mathcal{A}$ with $\\norm{y - y_0}_1 \u003c \\varepsilon$.\n\nStrong and weak local maxima are defined with $J(y_0) \\ge J(y)$.\nA strong (weak) local extremum is a strong (weak) local minimum or maximum.\n\n## Remarks\n\nSince $\\norm{y}_0 \\le \\norm{y}_1$, every strong local minimum is a weak local minimum.\n\nExample: let $J(y) = \\int_0^1 y'^2 \\, dx$ on $\\mathcal{A} = \\set{y \\in C^1([0, 1]) : y(0) = 0, \\ y(1) = 1}$, and $y_0(x) = x$.\nFor $y \\in \\mathcal{A}$ write $y' = 1 + \\eta$; then $\\int_0^1 \\eta \\, dx = 0$, so $J(y) = 1 + \\int_0^1 \\eta^2 \\, dx \\ge J(y_0)$.\nSo $y_0$ is a global, hence strong, minimum.\n\nNon-example: let $J(y) = \\int_0^1 y'^3 \\, dx$ on the same class, and $y_0(x) = x$.\nWith $y' = 1 + \\eta$ one gets $J(y) = 1 + \\int_0^1 \\eta^2 (3 + \\eta) \\, dx$, which is at least $1$ when $\\abs{\\eta} \u003c 3$; so $y_0$ is a weak local minimum.\nIt is not a strong local minimum.\nFix $h \u003e 0$, let $N$ be large and $\\delta = h/N$, and let $y$ have slope $-N$ on $[0, \\delta]$ and slope $(1 + h)/(1 - \\delta)$ on $[\\delta, 1]$, with corners rounded off.\nThen $\\norm{y - y_0}_0 \\le h + \\delta$, while $J(y)$ is close to $-N^2 h + (1 + h)^3/(1 - \\delta)^2$, which tends to $-\\infty$ as $N \\to \\infty$.\n",
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