Sobolev space

Template:Complex The Sobolev space is a function space in mathematics. The space is very useful to analyze for partial differential equation. The spaces can be characterized by smooth functions. It is a well known work from some mathematicians.There is a strong relation between Soblev space and Besov space. Sobolev spaces was introduced by Russian mathematician Sergei Sobolev in 1930s. His creating space has great application for analyzing of solution of ordinal differential equations and partial differential equations. According to above the applications, we had had natural question of how to characterize of the space by space of classical functions. However, in 1964 year, we had a partial answer of that from working of Meyers and Serrin.

Introduction and definition

Meyers-Serrin's theorem is well known for how to characterize to Sobolev spaces by collection of functions can classical derivative up to given oder same as order of distributional derivative of the Sobolev space. In this section, we shall first recall that definition of Sobolev spaces and set to several preliminaries as follows.

[math]\displaystyle{ W^{m,p}(\Omega) = \left \{ u \in L^p(\Omega) : D^{\alpha}u \in L^p(\Omega) \,\, \forall |\alpha| \leq m \right \}. }[/math]

Here, Ω is an open set in ℝⁿ and 1 ≤ p ≤ +∞. The natural number m is called the order of the Sobolev space W^k,p(Ω).