跳转到内容

User:Lihuapeng/沙盒

维基百科,自由的百科全书

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蓋·劉維奇·索伯列夫的名字命名。

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 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

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

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

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

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

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

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

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

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

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

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[來源請求] 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