Sobolev inequality

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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[edit]

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 Lp. Here k is a non-negative integer and 1 ≤ p ≤ ∞. The first part of the Sobolev embedding theorem states that if k > and 1 ≤ p < q ≤ ∞ are two extended real numbers such that (k)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 = 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 (krα)/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[edit]

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

Kondrachov embedding theorem[edit]

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

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

is completely continuous (compact).

Gagliardo–Nirenberg–Sobolev inequality[edit]

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 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[edit]

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 Iα = (−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 the weak-type estimate holds:

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.

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

Morrey's inequality[edit]

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 uC1(Rn) ∩ Lp(Rn), where

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

Thus if uW 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[edit]

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 uW k,p(U), then we consider two cases:

k < n/p[edit]

In this case uLq(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[edit]

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[edit]

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[edit]

The Nash inequality, introduced by John Nash (1958), states that there exists a constant C > 0, such that for all uL1(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 ωn 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[citation needed] to the Lp 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 ≤ qp < ∞ 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[edit]