What is 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 characteri