---
tag: B03CA692K3
kind: statement
role: theorem
name: Sobolev inequality
msc: ["46E35", 26D10]
uses: [M4CEZ5K9HP, SASP14MA1Z, C3EC148VZD, JQ9WXC729D, GZTT6FT18G]
about: [M4CEZ5K9HP]
sources:
  - url: https://en.wikipedia.org/wiki/Sobolev_inequality
---

## Statement

Let $n \ge 2$, let $1 \le p < n$, and let $p^* = \frac{np}{n - p}$.

1. 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.
2. 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 $u \in W^{1,p}(U)$.
3. Let $U$ be as in part 2, let $k \ge 1$ be an integer with $kp < n$, 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 $C = C(n, k, p, U)$.

Here $W^{k,p}(U)$ is the [[Sobolev space|M4CEZ5K9HP]] and $L^q(U)$ is the [[Lp space|SASP14MA1Z]].

## Remarks

Part 1 is the Gagliardo–Nirenberg–Sobolev inequality.
For $p = 1$, 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.
The case $p > 1$ follows by applying the case $p = 1$ to $\abs{u}^\gamma$ for a suitable $\gamma > 1$.
Part 1 extends to all $u \in W^{1,p}(\R^n)$ by density, with the [[weak derivative|C3EC148VZD]] in place of $\nabla u$.
Part 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 $k$.

The exponent $p^*$ is forced by scaling.
Replacing $u(x)$ 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 $\lambda > 0$ only when $p^* = np/(n - p)$.
The 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.
For $p > n$ the corresponding result is Morrey's inequality.
The [[Sobolev embedding theorem|GZTT6FT18G]] combines both cases.

## Explanations

### L1 human

If the slope of a function is small in an average sense, then the function itself cannot be too large on a big set.
The Sobolev inequality makes this precise: control of $\nabla u$ in $L^p$ gives control of $u$ in the better space $L^{p^*}$.
In dimension $3$ with $p = 2$, finite energy $\int \abs{\nabla u}^2$ forces $u$ to lie in $L^6$.

### L2 human

The inequality is the basic tool to pass from energy estimates, which control derivatives, to estimates of the solution itself.
It 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 $q \le p^*$.
Both sides of part 1 scale in the same way under dilations $u(x) \mapsto u(\lambda x)$, which makes the critical exponent $p^*$ special.
The embedding into $L^{p^*}$ is not compact, while the embeddings into $L^q$ for $q < p^*$ on bounded sets are compact by the Rellich–Kondrachov theorem.