View source for Sobolev space

{{expert-subject|1=needs a mathematician|date=March 2024}}

The '''Sobolev space''' is a [[function space]] in [[mathematics]]. These functions all have a [[Normed vector space|norm]].  This norm is a combination of [[Lp norm|''L<sup>p</sup>''-norms]] of the function together with its [[derivatives]] up to a given order. The derivatives are understood in a suitable [[Weak derivative|weak sense]] to make the space [[Complete metric space|complete]], i.e. a [[Banach space]]. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as [[Partial differential equation|partial differential equations]],. They are equipped with a norm that measures both the size and regularity of a function.  

Sobolev spaces are named after the Russian [[mathematician]] [[Sergei Lvovich Sobolev|Sergei Sobolev]]. Their importance comes from the fact that [[Weak solution|weak solutions]] of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of [[Continuous function|continuous functions]] with the [[Derivative|derivatives]] understood in the classical sense.  

The space is very useful to look at for [[partial differential equation]]. The spaces may be [[wikt:characterize|characterized]] by [[Smooth function|smooth functions]]. It is used by [[Mathematician|mathematicians]]. There is a strong relation between Soblev space and [[Besov space]].  

Sobolev spaces are named for the [[Russia|Russian]] mathematician [[Sergei Sobolev]] in the 1930s. His space has many uses. They are most used for understanding the solutions of ordinal [[Differential equation|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> 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 ℝ''<sup>n</sup>'' and 1&nbsp;≤&nbsp;''p''&nbsp;≤&nbsp;+∞.  The [[natural number]] ''m'' is called the order of the Sobolev space ''W<sup>k,p</sup>''(Ω). 

{{Functional analysis}}

[[Category:Calculus]]


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