Talk:Rellich–Kondrachov theorem

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In the book of Evans the theorem is only proofed for $C^{1}$-domains. Could somebody give a reference, where the theorem is proved in this general setting? However the book of Evans and Gariepy on Meassure Theory and Fine Properties of Functions shows how similar estimates can be treated for Lipschitz domains. --78.55.145.142 (talk) 21:34, 18 December 2009 (UTC)

I just found [[1]], which is also a generalization of a statement in Wells' "Differential Analysis on Complex Manifolds". --Konrad (talk) 09:17, 15 June 2010 (UTC)

An overview

One way to prove the Rellich-Kondrachov theorem is the following: One establishes that the embedding into $L^{1}$ is compact and the embedding into $L^{q}$ is (well-defined and) continuous. Then, as a consequence (shown e.g. in of Hölder's inequality, the embedding into $L^{r}$ is compact for any $r$ with $1\leq r\leq q$. Since the following works proceed in this manner, each contains two proofs: One of continuity and one of compactness. I only cite the latter for brevity.

There are three types of theorems that fall into this category:

1. Embeddings of $W_{0}^{1,p}(\Omega )$ (or more generally $W_{0}^{k,p}(\Omega )$) into $L^{r}(\Omega )$ with arbitrary boundary
2. Embeddings of $W^{1,p}(\Omega )$ (or more generally $W^{k,p}(\Omega )$) into $L^{r}(\Omega )$ with a nice boundary
3. Embeddings of $W^{1,p}(\Omega )$ (or more generally $W^{k,p}(\Omega )$) into $L^{r}(\partial \Omega )$ with a nice boundary

Here are some sources:

1. The spaces $W_{0}^{1,p}(\Omega )$ are compactly imbedded in the spaces $L^{q}(\Omega )$ for any $q<np/(n-p)$, if $p<n$ [...]
2. • Assume $U$ is a bounded open subset of $\mathbb {R} ^{n}$ and $\partial U$ is $C^{1}$. Suppose $1\leq p<n$. Then $W^{1,p}(U)\subset \subset L^{q}(U)$ for each $1\leq q<p^{*}$ [here, $p^{*}$ is the Sobolev conjugate of $p$]
   • Let $\Omega$ be a bounded Lipschitz open subset of $\mathbb {R} ^{N}$, where $N>1$. If $N>mp$, then the embedding $W^{m,p}(\Omega )\to L^{q}(\Omega )$ is compact for $q<Np/(N-mp)$.
     • Let $\Omega$ be a bounded Lipschitz open set. We then have: If $sp<N$, then the embedding $W^{s,p}(\Omega )\to L^{q}(\Omega )$ is compact for all exponents $q$ satisfying $q<Np/(N-sp)$. [..]
   • Let $\Omega \in {\mathfrak {N}}^{0,1}$, $1\leq p<N$, $1\geq 1/q>1/p-1/N$. The identity mapping $I\colon W^{1,p}(\Omega )\to L^{q}(\Omega )$ is compact.
   • Let $\Omega$ be an open bounded subset of $\mathbb {R} ^{n}$ which has a $C^{1}$ boundary $\partial \Omega$. Then, we have the following compact injections: If $p<N$, $W^{1,p}(\Omega )\to L^{q}(\Omega )$ for any $q<p^{*}$, with $1/p^{*}=1/p-1/N$. [..]
3. • Let $\Omega \in {\mathfrak {N}}^{0,1}$, $1<p<N$, $1\geq 1/q>1/p-[1/(N-1)](p-1)/p$. The mapping $Z\in [W^{1,p}(\Omega )\to L^{q}(\partial \Omega )]$, which defines the traces, is compact.
   • Let $p>1$ and let $N-1$ be the dimension of $\partial \Omega$. We suppose that $kp<N$. The injection of $W^{k-1/p,p}(\partial \Omega )$ into $L^{q}(\partial \Omega )$ is then compact for all $q<(N-1)p/(N-kp)$.

Some remarks are in order:

• Necas' book is the only source I know for a results of type (3) with a Lipschitz boundary (denoted by ${\mathfrak {N}}^{0,1}$, see .)
• The Demengels' results fall into category (3) since $W^{k-1/p,p}(\partial \Omega )$ is the trace space of $W^{k,p}(\Omega )$.
• The Demengels have the only result for fractional Sobolev spaces that I'm aware of.

Question: While Necas' results are very general, they are not as accessible as others. Are results of type (3) with a Lipschitz boundary presented anywhere else?

Answer: This was answered here: http://math.stackexchange.com/a/261788/10311

References

• Demengel, Françoise; Demengel, Gilbert - Functional spaces for the theory of elliptic partial differential equations; http://www.ams.org/mathscinet-getitem?mr=2895178
• Gilbarg, David; Trudinger, Neil S. - Elliptic partial differential equations of second order. http://www.ams.org/mathscinet-getitem?mr=1814364
• Evans, Lawrence C. - Partial differential equations. Second edition. http://www.ams.org/mathscinet-getitem?mr=2597943
• Necas, Jindrich - Direct Methods in the Theory of Elliptic Equations. http://www.springer.com/mathematics/dynamical+systems/book/978-3-642-10454-1
• Attouch, Hedy; Buttazzo, Giuseppe; Michaille, Gérard - Variational analysis in Sobolev and BV spaces. http://www.ams.org/mathscinet-getitem?mr=2192832

160.45.109.44 (talk) 12:42, 24 October 2012 (UTC)