On Orthogonal Decomposition of a Sobolev Space

Preprint, Other literature type English OPEN
Lakew, Dejenie A. (2016)
  • Publisher: Tusi Mathematical Research Group
  • Journal: (issn: 2538-225X)
  • Related identifiers: doi: 10.22034/aot.1703-1135
  • Subject: 46E35 | 46C15 | distance | Sobolev space | orthogonal decomposition | inner product | Mathematics - Functional Analysis | 46E35, 46C15

The theme of this short article is to investigate an orthogonal decomposition of the Sobolev space $W^{1,2}\left( \Omega \right) $ as $$ W^{1,2}\left( \Omega \right) =A^{2,2}\left( \Omega \right) \oplus D^{2}\left( W_{0}^{3,2}\left( \Omega \right) \right)$$ and look at some of the properties of the inner product therein and the distance defined from the inner product. We also determine the dimension of the orthogonal difference space $W^{1,2}\left( \Omega \right) \ominus \left(W_{0}^{1,2}\left( \Omega \right) \right) ^{\perp }$ and show the expansion of Sobolev spaces as their regularity increases.
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