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.

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Sobolev embedding theorem

Let Wk,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-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  = 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 Cr(Rn). If (kr−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 Wk,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

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^{l,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)}

where

 p^*=\frac{pn}{n-p}>p

is the Sobolev conjugate of p.

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  < ∞. 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

\|I_\alpha f\|_q\le C\|f\|_p.

If p = 1, then the weak-type estimate holds:

m\{x : |I_\alpha f(x)| > \lambda\} \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

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) ∩ Lp(Rn), where

γ = 1 − n / p.

Thus if u ∈ W1,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 W1,p(U) to W1,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 ∈ Wk,p(U).

(i) If

k<\frac{n}{p}

then u\in 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.

(ii) If

k>\frac{n}{p}

then u belongs to the Hölder space  C^{k-[n/p]-1,\gamma}(U)\,, where

\gamma=\left[\frac{n}{p}\right]+1-\frac{n}{p} if n/p is not an integer, or
γ is any positive number < 1, if n/p is an integer

We have in addition the estimate

\|u\|_{C^{k-[n/p]-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

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

\|u\|_{BMO}<C\|Du\|_{L^n(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) ∩ W1,2(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} |\hat{u}(x)|^2\,dx \le \int_{|x|\ge\rho} \frac{x^2}{\rho^2}|\hat{u}(x)|^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 \scriptstyle{\|\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 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 ≤ 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 is defined by

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

References

  • Adams, Robert A. (1975), Sobolev spaces, Pure and Applied Mathematics,, 65., New York-London: Academic Press, pp. xviii+268, ISBN 978-0120441501, MR0450957 .
  • Aubin, Thierry (1976), "Espaces de Sobolev sur les variétés riemanniennes", Bulletin des Sciences Mathématiques. 2e Série 100 (2): 149–173, ISSN 0007-4497, MR0488125 
  • Aubin, Thierry (1982), Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 252, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90704-8, MR681859 .
  • Brezis, Haïm (1983), Analyse fonctionnelle : théorie et applications, Paris: Masson, ISBN 0-8218-0772-2 
  • Evans, Lawrence (1998), Partial Differential Equations, American Mathematical Society, Providence, ISBN 0-8218-0772-2 
  • Vladimir G., Maz'ja (1985), Sobolev spaces, Springer Series in Soviet Mathematics, Berlin: Springer-Verlag , Translated from the Russian by T. O. Shaposhnikova.
  • Nash, J. (1958), "Continuity of solutions of parabolic and elliptic equations", Amer. J. Math. (American Journal of Mathematics, Vol. 80, No. 4) 80 (4): 931–954, doi:10.2307/2372841, JSTOR 2372841 .
  • Nikol'skii, S.M. (2001), "Imbedding theorems", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/i/i050230.htm 
  • Stein, Elias (1970), Singular integrals and differentiability properties of functions, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8 

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