用户:Lihuapeng/沙盒

数学中,数学分析领域有一类索伯列夫不等式,与索伯列夫空间的范数有关。这些不等式用于证明给出特定索伯列夫空间之间的包含关系的 索伯列夫嵌入定理,而Rellich–Kondrachov定理的条件稍强,表明一些索伯列夫空间是紧嵌入英语Compact embedding在其他空间中的。这些索伯列夫空间以舍盖·刘维奇·索伯列夫的名字命名。

Sobolev embedding theorem

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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 real numbers such that (k)p < n and:

 

then

 

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

 

where p is the Sobolev conjugate of p, given by

 

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

 

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

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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

Kondrachov embedding theorem

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

 

is completely continuous (compact).

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

 

The case   is due to Sobolev,   to Gagliardo and Nirenberg independently. 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,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). 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

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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 C1 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).

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

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In this case 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, 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.

Case

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

 

for some constant C depending only on n. 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

by Parseval's theorem. 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 again applying Parseval's theorem:

 

gives the inequality.

In the special case of n = 1, the Nash inequality can be extended[来源请求] 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

 

where:

 

References

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