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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $1 \\le p \u003c n$, and let $p^* = \\frac{np}{n - p}$.\n\n1. There is a constant $C = C(n, p)$ such that $\\norm{u}_{L^{p^*}(\\R^n)} \\le C \\norm{\\nabla u}_{L^p(\\R^n)}$ for every continuously differentiable $u \\colon \\R^n \\to \\R$ with compact support.\n2. Let $U \\subset \\R^n$ be a bounded open set with $C^1$ boundary. Then $W^{1,p}(U) \\subseteq L^{p^*}(U)$, and there is a constant $C = C(n, p, U)$ with $\\norm{u}_{L^{p^*}(U)} \\le C \\norm{u}_{W^{1,p}(U)}$ for all $u \\in W^{1,p}(U)$.\n3. Let $U$ be as in part 2, let $k \\ge 1$ be an integer with $kp \u003c n$, and let $\\frac{1}{q} = \\frac{1}{p} - \\frac{k}{n}$. Then $W^{k,p}(U) \\subseteq L^{q}(U)$, and $\\norm{u}_{L^{q}(U)} \\le C \\norm{u}_{W^{k,p}(U)}$ for all $u \\in W^{k,p}(U)$, with $C = C(n, k, p, U)$.\n\nHere $W^{k,p}(U)$ is the [[Sobolev space|M4CEZ5K9HP]] and $L^q(U)$ is the [[Lp space|SASP14MA1Z]].\n\n## Remarks\n\nPart 1 is the Gagliardo–Nirenberg–Sobolev inequality.\nFor $p = 1$, the proof writes $\\abs{u(x)} \\le \\int_{-\\infty}^{x_i} \\abs{\\partial_i u} \\, dt$ in each coordinate direction, multiplies the $n$ bounds, and applies a generalized [[Hölder's inequality|JQ9WXC729D]] one variable at a time.\nThe case $p \u003e 1$ follows by applying the case $p = 1$ to $\\abs{u}^\\gamma$ for a suitable $\\gamma \u003e 1$.\nPart 1 extends to all $u \\in W^{1,p}(\\R^n)$ by density, with the [[weak derivative|C3EC148VZD]] in place of $\\nabla u$.\nPart 2 follows from part 1 by extending $u$ to a compactly supported function on $\\R^n$, and part 3 follows from part 2 by induction on $k$.\n\nThe exponent $p^*$ is forced by scaling.\nReplacing $u(x)$ by $u(\\lambda x)$ multiplies the two sides of part 1 by $\\lambda^{-n/p^*}$ and $\\lambda^{1 - n/p}$, and these agree for all $\\lambda \u003e 0$ only when $p^* = np/(n - p)$.\nThe inequality fails for $p = n$ with $p^* = \\infty$: for $n \\ge 2$ there are unbounded functions in $W^{1,n}$ of a ball, such as $\\log \\log (1 / \\abs{x})$ near $0$ in the plane.\nFor $p \u003e n$ the corresponding result is Morrey's inequality.\nThe [[Sobolev embedding theorem|GZTT6FT18G]] combines both cases.\n\n## Explanations\n\n### L1 human\n\nIf the slope of a function is small in an average sense, then the function itself cannot be too large on a big set.\nThe Sobolev inequality makes this precise: control of $\\nabla u$ in $L^p$ gives control of $u$ in the better space $L^{p^*}$.\nIn dimension $3$ with $p = 2$, finite energy $\\int \\abs{\\nabla u}^2$ forces $u$ to lie in $L^6$.\n\n### L2 human\n\nThe inequality is the basic tool to pass from energy estimates, which control derivatives, to estimates of the solution itself.\nIt is used to show that the energy of a nonlinear elliptic problem is well defined, for example for terms like $\\int \\abs{u}^q$ with $q \\le p^*$.\nBoth sides of part 1 scale in the same way under dilations $u(x) \\mapsto u(\\lambda x)$, which makes the critical exponent $p^*$ special.\nThe embedding into $L^{p^*}$ is not compact, while the embeddings into $L^q$ for $q \u003c p^*$ on bounded sets are compact by the Rellich–Kondrachov theorem.\n",
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