Sobolev space
The Sobolev space is a function space in mathematics. The space is very useful to look at for partial differential equation.
The spaces may be characterized by smooth functions. It is used by mathematicians. There is a strong relation between Soblev space and Besov space.
Sobolev spaces are named for the Russian mathematician Sergei Sobolev in the 1930s. His space has many uses, but usually are for understanding the solutions of ordinal differential equations and partial differential equations. According to the applications above, there were questions on how to characterize the space using classical functions. These questions were answered in 1964, by Meyers and Serrin.
Introduction and definition
Meyers and 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 Wk,p(Ω).