Sobolev inequality

In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev.

Sobolev embedding theorem

Let W^k,p(Rⁿ) denote the Sobolev space consisting of all real-valued functions on Rⁿ whose first k weak derivatives are functions in L^p. Here k is a non-negative integer and 1 ≤ p < ∞. The first part of the Sobolev embedding theorem states that if k > ℓ and 1 ≤ p < q < ∞ are two real numbers such that (k − ℓ)p < n and:

$\frac{1}{q} = \frac{1}{p}-\frac{k-\ell}{n},$

then

$W^{k,p}(\mathbf{R}^n)\subseteq W^{\ell,q}(\mathbf{R}^n)$

and the embedding is continuous. In the special case of k = 1 and ℓ = 0, Sobolev embedding gives

$W^{1,p}(\mathbf{R}^n) \subseteq L^{p^*}(\mathbf{R}^n)$

where p^∗ is the Sobolev conjugate of p, given by

$\frac{1}{p^*} = \frac{1}{p} - \frac{1}{n}.$

This special case of the Sobolev embedding is a direct consequence of the Gagliardo–Nirenberg–Sobolev inequality.

The second part of the Sobolev embedding theorem applies to embeddings in Hölder spaces C^r,α(Rⁿ). If (k − r − α)/n = 1/p with α ∈ (0, 1), then one has the embedding

$W^{k,p}(\mathbf{R}^n)\subset C^{r,\alpha}(\mathbf{R}^n).$

This part of the Sobolev embedding is a direct consequence of Morrey's inequality. Intuitively, this inclusion expresses the fact that the existence of sufficiently many weak derivatives implies some continuity of the classical derivatives.

Generalizations

The Sobolev embedding theorem holds for Sobolev spaces W^k,p(M) on other suitable domains M. In particular (Aubin 1982, Chapter 2; Aubin 1976), both parts of the Sobolev embedding hold when

• M is a bounded open set in Rⁿ with Lipschitz boundary (or whose boundary satisfies the cone condition; Adams 1975, Theorem 5.4)
• M is a compact Riemannian manifold
• M is a compact Riemannian manifold with boundary with Lipschitz boundary
• M is a complete Riemannian manifold with injectivity radius δ > 0 and bounded sectional curvature.

Kondrachov embedding theorem

Main article: Rellich–Kondrachov theorem

On a compact manifold with C¹ boundary, the Kondrachov embedding theorem states that if k > ℓ and k − n/p > ℓ − n/q then the Sobolev embedding

$W^{k,p}(M)\subset W^{\ell,q}(M)$

is completely continuous (compact).

Gagliardo–Nirenberg–Sobolev inequality

Assume that u is a continuously differentiable real-valued function on Rⁿ with compact support. Then for 1 ≤ p < n there is a constant C depending only on n and p such that

$\|u\|_{L^{p^*}(\mathbf{R}^n)}\leq C \|Du\|_{L^{p}(\mathbf{R}^n)}.$

The case $1< p < n$ is due to Sobolev, $p =1$ to Gagliardo and Nirenberg independently. The Gagliardo–Nirenberg–Sobolev inequality implies directly the Sobolev embedding

$W^{1,p}(\mathbf{R}^n) \sub L^{p^*}(\mathbf{R}^n).$

The embeddings in other orders on Rⁿ are then obtained by suitable iteration.

Hardy–Littlewood–Sobolev lemma

Sobolev's original proof of the Sobolev embedding theorem relied on the following, sometimes known as the Hardy–Littlewood–Sobolev fractional integration theorem. An equivalent statement is known as the Sobolev lemma in (Aubin 1982, Chapter 2). A proof is in (Stein, Chapter V, §1.3).

Let 0 < α < n and 1 < p < q < ∞. Let I_α = (−Δ)^−α/2 be the Riesz potential on Rⁿ. Then, for q defined by

$q = \frac{pn}{n-\alpha p}$

there exists a constant C depending only on p such that

$\left \|I_\alpha f \right \|_q \le C \|f\|_p.$

If p = 1, then one has two possible replacement estimates. The first is the more classical weak-type estimate:

$m \left \{x : \left |I_\alpha f(x) \right | > \lambda \right \} \le C \left( \frac{\|f\|_1}{\lambda} \right )^q,$

where 1/q = 1 − α/n. Alternatively one has the estimate

$\left \|I_\alpha f \right \|_q \le C \|Rf\|_1,$

where $Rf$ is the vector-valued Riesz transform, c.f. (Schikorra & Spector Van Schaftingen). Interestingly the boundedness of the Riesz transforms implies that the latter inequality gives a unified way to write the family.

The Hardy–Littlewood–Sobolev lemma implies the Sobolev embedding essentially by the relationship between the Riesz transforms and the Riesz potentials.

Morrey's inequality

Assume n < p ≤ ∞. Then there exists a constant C, depending only on p and n, such that

$\|u\|_{C^{0,\gamma}(\mathbf{R}^n)}\leq C \|u\|_{W^{1,p}(\mathbf{R}^n)}$

for all u ∈ C¹(Rⁿ) ∩ L^p(Rⁿ), where

$\gamma=1-\frac{n}{p}.$

Thus if u ∈ W^1,p(Rⁿ), then u is in fact Hölder continuous of exponent γ, after possibly being redefined on a set of measure 0.

A similar result holds in a bounded domain U with C¹ boundary. In this case,

$\|u\|_{C^{0,\gamma}(U)}\leq C \|u\|_{W^{1,p}(U)}$

where the constant C depends now on n, p and U. This version of the inequality follows from the previous one by applying the norm-preserving extension of W^1,p(U) to W^1,p(Rⁿ).

General Sobolev inequalities

Let U be a bounded open subset of Rⁿ, with a C¹ boundary. (U may also be unbounded, but in this case its boundary, if it exists, must be sufficiently well-behaved.) Assume u ∈ W^k,p(U), then we consider two cases:

k < n/p

In this case u ∈ L^q(U), where

$\frac{1}{q}=\frac{1}{p}-\frac{k}{n}.$

We have in addition the estimate

$\|u\|_{L^q(U)}\leq C \|u\|_{W^{k,p}(U)}$,

the constant C depending only on k, p, n, and U.

k > n/p

Here, u belongs to a Hölder space, more precisely:

$u \in C^{k-\left[\frac{n}{p}\right]-1,\gamma}(U),$

where

$\gamma = \begin{cases} \left[\frac{n}{p}\right]+1-\frac{n}{p} & \frac{n}{p} \notin \mathbf{Z} \\ \text{any element in } (0, 1) & \frac{n}{p} \in \mathbf{Z} \end{cases}$

We have in addition the estimate

$\|u\|_{C^{k-\left[\frac{n}{p}\right]-1,\gamma}(U)}\leq C \|u\|_{W^{k,p}(U)},$

the constant C depending only on k, p, n, γ, and U.

Case $p=n, k=1$

If $u\in W^{1,n}(\mathbf{R}^n)$, then u is a function of bounded mean oscillation and

$\|u\|_{BMO} \leq C \|Du\|_{L^n(\mathbf{R}^n)},$

for some constant C depending only on n. This estimate is a corollary of the Poincaré inequality.

Nash inequality

The Nash inequality, introduced by John Nash (1958), states that there exists a constant C > 0, such that for all u ∈ L¹(Rⁿ) ∩ W^1,2(Rⁿ),

$\|u\|_{L^2(\mathbf{R}^n)}^{1+2/n} \leq C\|u\|_{L^1(\mathbf{R}^n)}^{2/n} \| Du\|_{L^2(\mathbf{R}^n)}.$

The inequality follows from basic properties of the Fourier transform. Indeed, integrating over the complement of the ball of radius ρ,

$\int_{|x|\ge\rho} \left |\hat{u}(x) \right |^2\,dx \le \int_{|x|\ge\rho} \frac{|x|^2}{\rho^2} \left |\hat{u}(x) \right |^2\,dx\le \rho^{-2}\int_{\mathbf{R}^n}|D u|^2\,dx$

(1)

by Parseval's theorem. On the other hand, one has

$|\hat{u}| \le \|u\|_{L^1}$

which, when integrated over the ball of radius ρ gives

$\int_{|x|\le\rho} |\hat{u}(x)|^2\,dx \le \rho^n\omega_n \|u\|_{L^1}^2$

(2)

where ω_n is the volume of the n-ball. Choosing ρ to minimize the sum of (1) and (2) and again applying Parseval's theorem:

$\|\hat{u}\|_{L^2} = \|u\|_{L^2}$

gives the inequality.

In the special case of n = 1, the Nash inequality can be extended to the L^p case, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality (Brezis 1999). In fact, if I is a bounded interval, then for all 1 ≤ r < ∞ and all 1 ≤ q ≤ p < ∞ the following inequality holds

$\| u\|_{L^p(I)}\le C\| u\|^{1-a}_{L^q(I)} \|u\|^a_{W^{1,r}(I)},$

where:

$a\left(\frac{1}{q}-\frac{1}{r}+1\right)=\frac{1}{q}-\frac{1}{p}.$

References

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