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, u\in C^1_c(R^n)). There is a constant C = C(n,p) for 1\leq p <n such that

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

  • 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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