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Talk:Rellich–Kondrachov theorem

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In the book of Evans the theorem is only proofed for -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)[reply]

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)[reply]

An overview

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One way to prove the Rellich-Kondrachov theorem is the following: One establishes that the embedding into is compact and the embedding into is (well-defined and) continuous. Then, as a consequence (shown e.g. in [1] of Hölder's inequality, the embedding into is compact for any with . 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 (or more generally ) into with arbitrary boundary
  2. Embeddings of (or more generally ) into with a nice boundary
  3. Embeddings of (or more generally ) into with a nice boundary

Here are some sources:

  1. The spaces are compactly imbedded in the spaces for any , if [...] [2]
    • Assume is a bounded open subset of and is . Suppose . Then for each [3] [here, is the Sobolev conjugate of ]
    • Let be a bounded Lipschitz open subset of , where . If , then the embedding is compact for . [4]
      • Let be a bounded Lipschitz open set. We then have: If , then the embedding is compact for all exponents satisfying . [..] [5]
    • Let , , . The identity mapping is compact. [6]
    • Let be an open bounded subset of which has a boundary . Then, we have the following compact injections: If , for any , with . [..] [7]
    • Let , , . The mapping , which defines the traces, is compact. [8]
    • Let and let be the dimension of . We suppose that . The injection of into is then compact for all . [9]

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 , see [10].)
  • The Demengels' results fall into category (3) since is the trace space of .
  • 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

Notes

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  1. ^ Demengel/Demengel, Lemma 2.82, p97
  2. ^ Gilbarg/Trudinger, p167
  3. ^ Evans, p286
  4. ^ Demengel/Demengel, p96
  5. ^ Demengel/Demengel, p220
  6. ^ Necas, p102
  7. ^ Attouch; Buttazzo; Michaille
  8. ^ Necas, p103
  9. ^ Demengel/Demengel, p167
  10. ^ Necas, p49

References

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