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# Sobolev inequality

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.

## Sobolev embedding theorem

Write $W^\left\{k,p\right\}$ 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 $W^\left\{k,infty\right\}$ is defined to be the Hölder space Cm where k=m+α and 0<α≤1 and m is integer.) The Sobolev embedding theorem states that if kl and kn/pln/q then
$W^\left\{k,p\right\}subseteq W^\left\{l,q\right\}$
and the embedding is continuous. Moreover if k> l and kn/p > ln/q then the embedding is completely continuous (this is sometimes called Kondrakov's theorem). Functions in $W^\left\{l,infty\right\}$ 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 .

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

## Gagliardo-Nirenberg-Sobolev inequality

Assume that u(x) is continuously differentiable function with compact support from $mathbb\left\{R\right\}^n$ to $mathbb\left\{R\right\}$. Then for

$|u|_\left\{L^\left\{p^*\right\}\left(mathbb\left\{R\right\}^n\right)\right\}leq C_n\left(p\right) |Du|_\left\{L^\left\{p\right\}\left(mathbb\left\{R\right\}^n\right)\right\}$
where
$p^*=frac\left\{pn\right\}\left\{n-p\right\}>p$
is the Sobolev conjugate of p.

## Nash inequality

There exists a constant C>0, such that for all $uin L^1\left(R^n\right)cap H^1\left(R^n\right)$,
$|u|_\left\{L^2\left(R^n\right)\right\}^\left\{1+2/n\right\}leq C|u|_\left\{L^1\left(R^n\right)\right\}^\left\{2/n\right\} | Du|_\left\{L^2\left(R^n\right)\right\}$

## Morrey's inequality

Assume C, depending only on p and n, such that
$|u|_\left\{C^\left\{0,gamma\right\}\left(R^n\right)\right\}leq C |u|_\left\{W^\left\{1,p\right\}\left(R^n\right)\right\}$
for all $uin C^1 \left(R^n\right)$, where
$gamma:=1-n/p$
In other words, if $uin W^\left\{1,p\right\}\left(U\right)$, then u is in fact Hölder continuous (with parameter $gamma$), after possibly being redefined on a set of measure 0.

## General Sobolev inequalities

Let U be a bounded open subset of $R^n$, with a $C^1$ boundary. (U may also be unbounded, but in this case its boundary, if it exists, must be sufficiently well-behaved.) Assume $uin W^\left\{k,p\right\}\left(U\right)$.
(i) If
then $uin L^q\left(U\right)$, where
$frac\left\{1\right\}\left\{q\right\}=frac\left\{1\right\}\left\{p\right\}-frac\left\{k\right\}\left\{n\right\}$
We have in addition the estimate
$|u|_\left\{L^q\left(U\right)\right\}leq C |u|_\left\{W^\left\{k,p\right\}\left(U\right)\right\}$,
the constant C depending only on k, p, n, and U.
(ii) If
$k>frac\left\{n\right\}\left\{p\right\}$
then $u$ belongs to the Hölder space $C^\left\{k-\left[n/p\right]-1,gamma\right\}\left(U\right)$, where
$gamma=left\left[frac\left\{n\right\}\left\{p\right\}right\right]+1-frac\left\{n\right\}\left\{p\right\}$ if n/p is not an integer, or
$gamma$ is any positive number <1, if n/p is an integer
We have in addition the estimate
$|u|_\left\{C^\left\{k-\left[n/p\right]-1,gamma\right\}\left(U\right)\right\}leq C |u|_\left\{W^\left\{k,p\right\}\left(U\right)\right\}$,
the constant C depending only on k, p, n, $gamma$, and U.

## Case $p=n$

If $uin W^\left\{1,n\right\}\left(R^n\right)cap L^1_\left\{loc\right\}\left(R^n\right)$, then $u$ is a function of bounded mean oscillation and
C depending only on n.
This estimate is a corollary of the Poincaré inequality.

## References

• 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

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