On orthogonal decomposition of a Sobolev space

Document Type: Original Article


Bryant & Stratton College


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.