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 Kondrakov 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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[edit] Sobolev embedding theorem

Write Wk,p for the Sobolev space of some compact Riemannian manifold of dimension n, consisting (roughly) of functions whose first k derivatives are in Lp. Here k can be any real number, and 1≤p≤∞. (For p=∞ the Sobolev space W^{k,\infty} is defined to be the Hölder space Cm,α where k=m+α and 0<α≤1 and m is integer.) The Sobolev embedding theorem states that if k≥ l and k−n/p ≥ l−n/q then

W^{k,p}\subseteq W^{l,q}

and the embedding is continuous. Moreover if k> l and k−n/p > l−n/q then the embedding is completely continuous (this is sometimes called Kondrakov's theorem). Functions in W^{l,\infty} have all derivatives of order less than l continuous, so in particular this gives conditions on Sobolev spaces for various derivatives to be continuous. Informally these embeddings say that to convert an Lp estimate to a boundedness estimate costs 1/p derivatives per dimension.

There are similar variations of the embedding theorem for non-compact manifolds such as Rn (Stein 1970).

These embedding theorems follow from various inequalities, given in the sections below.

[edit] Gagliardo-Nirenberg-Sobolev inequality

Assume that u(x) is continuously differentiable function with compact support from \mathbb{R}^n to \mathbb{R}. Then for 1\leq p <n there is a constant Cn(p) such that

\|u\|_{L^{p^*}(\mathbb{R}^n)}\leq C_n(p) \|Du\|_{L^{p}(\mathbb{R}^n)}

where

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

is the Sobolev conjugate of p.

[edit] Nash inequality

There exists a constant C>0, such that for all u\in L^1(R^n)\cap H^1(R^n),

\|u\|_{L^2(R^n)}^{1+2/n}\leq C\|u\|_{L^1(R^n)}^{2/n} \| Du\|_{L^2(R^n)}

[edit] Morrey's inequality

Assume n<p\leq \infty. Then there exists a constant C, depending only on p and n, such that

\|u\|_{C^{0,\gamma}(R^n)}\leq C \|u\|_{W^{1,p}(R^n)}

for all u\in C^1 (R^n), where

γ: = 1 − n / p

In other words, if u\in W^{1,p}(U), then u is in fact Hölder continuous (with parameter γ), after possibly being redefined on a set of measure 0.

[edit] General Sobolev inequalities

Let U be a bounded open subset of Rn, with a C1 boundary. Assume u\in W^{k,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 Ck − [n / p] − 1,γ(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.

[edit] Case p = n

If u\in W^{1,n}(R^n)\cap L^1_{loc}(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.

[edit] References

  • Adams, Robert A. (1975), Sobolev spaces, vol. 65., Pure and Applied Mathematics,, New York-London: Academic Press, pp. xviii+268, MR0450957, ISBN 978-0120441501 
  • Lawrence C. Evans. Partial differential equations. Graduate studies in Mathematics, Vol 19. American Mathematical Society. 1998. ISBN 0-8218-0772-2
  • Maz'ja, Vladimir G., Sobolev spaces, Translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985. xix+486 pp
  • Nikol'skii, S.M. (2001), “Imbedding theorems”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
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