The Sobolev Embedding Theorem gives conditions under which a function with a fixed number of weak derivatives, themselves integrable to some power, must also belong to a better-behaved function space, such as a space of continuous or Holder-continuous functions, depending on the relation between the number of derivatives, the integrability power, and the dimension of the domain. Named for Sergei Sobolev, it is a foundational result of the modern theory of partial differential equations, converting rough integrability information into concrete regularity.
Facts
StatementGives inclusions between certain Sobolev spaces; it is proved using the class of Sobolev inequalities, which relate norms including those of Sobolev spaces. 1 Classification
Statement FormCharacterization Theorem 1 Connections
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In Branch
Source Sobolev inequality, Wikipedia
Sources
1. Sobolev inequality, Wikipedia
Lead paragraph
These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces,
In Branch: Analysis, Lead sentence
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev space
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