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

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Graphical representation of the embedding conditions. The space W 3,p, represented by a blue dot at the point (1/p, 3), embeds into the spaces indicated by red dots, all lying on a line with slope n. The white circle at (0,0) indicates the impossibility of optimal embeddings into L ∞.

Let W k,p(Rn) denote the Sobolev space consisting of all real-valued functions on Rn whose weak derivatives up to order k 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 > , p < n and 1 ≤ p < q < ∞ are two real numbers such that

 

(given  ,  ,   and   this is satisfied for some   provided  ), then

 

and the embedding is continuous: for every  , one has  , and

 

In the special case of k = 1 and = 0, Sobolev embedding gives

 

where p is the Sobolev conjugate of p, given by

 

and for every  , one has   and

 

This special case of the Sobolev embedding is a direct consequence of the Gagliardo–Nirenberg–Sobolev inequality. The result should be interpreted as saying that if a function   in   has one derivative in  , then   itself has improved local behavior, meaning that it belongs to the space   where  . (Note that  , so that  .) Thus, any local singularities in   must be more mild than for a typical function in  .

 
If the line from the picture above intersects the y-axis at s = r + α, the embedding into a Hölder space C r, α (red) holds. White circles indicate intersection points at which optimal embeddings are not valid.

The second part of the Sobolev embedding theorem applies to embeddings in Hölder spaces C r,α(Rn). If n < pk and

 

with α ∈ (0, 1) then one has the embedding

 

In other words, for every   and  , one has  

 

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. If   then   for every  .

In particular, as long as  , the embedding criterion will hold with   and some positive value of  . That is, for a function   on  , if   has   derivatives in   and  , then   will be continuous (and actually Hölder continuous with some positive exponent  ).

Generalizations

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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

If M is a bounded open set in Rn with continuous boundary, then W 1,2(M) is compactly embedded in L2(M) (Nečas 2012, Section 1.1.5, Theorem 1.4).

Kondrachov embedding theorem

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On a compact manifold M with C1 boundary, the Kondrachov embedding theorem states that if k > and then the Sobolev embedding

 

is completely continuous (compact).[1] Note that the condition is just as in the first part of the Sobolev embedding theorem, with the equality replaced by an inequality, thus requiring a more regular space W k,p(M).

Gagliardo–Nirenberg–Sobolev inequality

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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

 

with  . The case   is due to Sobolev[2] and the case   to Gagliardo and Nirenberg independently.[3][4] The Gagliardo–Nirenberg–Sobolev inequality implies directly the Sobolev embedding

 

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

Hardy–Littlewood–Sobolev lemma

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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 1970, 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

 

there exists a constant C depending only on p such that

 

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

 

where 1/q = 1 − α/n. Alternatively one has the estimate where   is the vector-valued Riesz transform, c.f. (Schikorra, Spector & Van Schaftingen 2017). The boundedness of the Riesz transforms implies that the latter inequality gives a unified way to write the family of inequalities for the Riesz potential.

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

Morrey's inequality

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

 

for all uC1(Rn) ∩ Lp(Rn), where

 

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 Lipschitz boundary. In this case,

 

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). The inequality is named after Charles B. Morrey Jr.

General Sobolev inequalities

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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 or k = n, p = 1

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In this case we conclude that uLq(U), where

 

We have in addition the estimate

 ,

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

k > n/p

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Here, we conclude that u belongs to a Hölder space, more precisely:

 

where

 

We have in addition the estimate

 

the constant C depending only on k, p, n, γ, and U. In particular, the condition   guarantees that   is continuous (and actually Hölder continuous with some positive exponent).

Case p=n, k=1

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If  , then u is a function of bounded mean oscillation and

 

for some constant C depending only on n.[5]: §I.2  This estimate is a corollary of the Poincaré inequality.

Nash inequality

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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),

 

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

  (1)

because  . On the other hand, one has

 

which, when integrated over the ball of radius ρ gives

  (2)

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

 

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 2011, Comments on Chapter 8). In fact, if I is a bounded interval, then for all 1 ≤ r < ∞ and all 1 ≤ qp < ∞ the following inequality holds

 

where:

 

Logarithmic Sobolev inequality

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The simplest of the Sobolev embedding theorems, described above, states that if a function   in   has one derivative in  , then   itself is in  , where

 

We can see that as   tends to infinity,   approaches  . Thus, if the dimension   of the space on which   is defined is large, the improvement in the local behavior of   from having a derivative in   is small (  is only slightly larger than  ). In particular, for functions on an infinite-dimensional space, we cannot expect any direct analog of the classical Sobolev embedding theorems.

There is, however, a type of Sobolev inequality, established by Leonard Gross (Gross 1975) and known as a logarithmic Sobolev inequality, that has dimension-independent constants and therefore continues to hold in the infinite-dimensional setting. The logarithmic Sobolev inequality says, roughly, that if a function is in   with respect to a Gaussian measure and has one derivative that is also in  , then   is in " -log", meaning that the integral of   is finite. The inequality expressing this fact has constants that do not involve the dimension of the space and, thus, the inequality holds in the setting of a Gaussian measure on an infinite-dimensional space. It is now known that logarithmic Sobolev inequalities hold for many different types of measures, not just Gaussian measures.

Although it might seem as if the  -log condition is a very small improvement over being in  , this improvement is sufficient to derive an important result, namely hypercontractivity for the associated Dirichlet form operator. This result means that if a function is in the range of the exponential of the Dirichlet form operator—which means that the function has, in some sense, infinitely many derivatives in  —then the function does belong to   for some   (Gross 1975 Theorem 6).

References

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  1. ^ Taylor, Michael E. (1997). Partial Differential Equations I - Basic Theory (2nd ed.). p. 286. ISBN 0-387-94653-5.
  2. ^ Sobolev, Sergeĭ L’vovich (1938). "Sur un théorème de l'analyse fonctionnelle". Comptes Rendus (Doklady) de l'Académie des Sciences de l'URSS, Nouvelle Série. 20: 5–9.
  3. ^ Gagliardo, Emilio (1958). "Proprietà di alcune classi di funzioni in più variabili". Ricerche di Matematica. 7: 102–137.
  4. ^ Nirenberg, Louis (1959). "On elliptic partial differential equations". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie III. 13: 115–162.
  5. ^ Brezis, H.; Nirenberg, L. (September 1995). "Degree theory and BMO; part I: Compact manifolds without boundaries". Selecta Mathematica. 1 (2): 197–263. doi:10.1007/BF01671566. S2CID 195270732.