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 include results showing spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev.
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[edit] Gagliardo-Nirenberg-Sobolev inequality
Assume that u(x) is continuously differentiable function from Rn to R with compact support (that is, 
). There is a constant C = C(n,p) for 
such that
where
is the Sobolev conjugate of p.
[edit] Nash Inequality
There exists a constant C>0, such that for all 
,
[edit] Morrey's inequality
Assume 
. Then there exists a constant C, depending only on p and n, such that
for all 
, where
- γ: = 1 − n / p
In other words, if 
, 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 
.
- (i) If
then 
, where
We have in addition the estimate
-
,
the constant C depending only on k, p, n, and U.
- (ii) If
then u belongs to the Hölder space Ck − [n / p] − 1,γ(U), where
-
![\gamma=\left[\frac{n}{p}\right]+1-\frac{n}{p}](../../../math/f/5/b/f5b26c90d96c4564a1badfa322b13ed7.png)
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)}](../../../math/1/8/1/18113e38a5bd64fa5a45f1dd00e24252.png)
,
the constant C depending only on k, p, n, γ, and U.
[edit] Case p = n
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.
[edit] 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
