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
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 > ℓ and 1 ≤ p < q < ∞ are two real numbers such that (k − ℓ)p < n and:
and the embedding is continuous. In the special case of k = 1 and ℓ = 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 (k − r − α)/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.
- M is a bounded open set in Rn with Lipschitz boundary (or whose boundary satisfies the cone condition; Adams 1975, Theorem 5.4)
- M is a compact Riemannian manifold
- M is a compact Riemannian manifold with boundary with Lipschitz boundary
- M is a complete Riemannian manifold with injectivity radius δ > 0 and bounded sectional curvature.
Kondrachov embedding theorem
On a compact manifold with C1 boundary, the Kondrachov embedding theorem states that if k > ℓ and k − n/p > ℓ − n/q then the Sobolev embedding
is completely continuous (compact).
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.
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.
Assume n < p ≤ ∞. Then there exists a constant C, depending only on p and n, such that
for all u ∈ C1(Rn) ∩ Lp(Rn), where
Thus if u ∈ W 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 u ∈ W k,p(U), then we consider two cases:
k < n/p
In this case u ∈ Lq(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:
We have in addition the estimate
the constant C depending only on k, p, n, γ, and U.
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.
The inequality follows from basic properties of the Fourier transform. Indeed, integrating over the complement of the ball of radius ρ,
by Parseval's theorem. On the other hand, one has
which, when integrated over the ball of radius ρ gives
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
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