Sobolev inequality

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 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 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 combines both cases.

Going deeper

For a student

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.