### On orthogonal decomposition of a Sobolev space

Dejenie Lakew

#### Abstract

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.

#### Article information

Source
Adv. Oper. Theory, Volume 2, Number 4 (2017), 419-427.

Dates
Accepted: 8 June 2017
First available in Project Euclid: 4 December 2017

https://projecteuclid.org/euclid.aot/1512431718

Digital Object Identifier
doi:10.22034/aot.1703-1135

Mathematical Reviews number (MathSciNet)
MR3730037

Zentralblatt MATH identifier
1385.46023

#### Citation

Lakew, Dejenie. On orthogonal decomposition of a Sobolev space. Adv. Oper. Theory 2 (2017), no. 4, 419--427. doi:10.22034/aot.1703-1135. https://projecteuclid.org/euclid.aot/1512431718

#### References

• D. A. Lakew, New proofs on properties of an orthogonal decomposition of a Hilbert space, $arXiv:1510.07944v1$.
• D. A. Lakew, On Orthogonal decomposition of $L^2(\Omega)$, J. Math. Comput. Sci. 10 (2015), no. 1, 27–37.