The Rellich-Kondrachov Theorem is a compact embedding theorem for Sobolev spaces, named for the Austrian-German mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. It states that, on a sufficiently regular bounded domain, a Sobolev space of functions with weak derivatives embeds compactly into certain lower-order Sobolev or Lebesgue spaces, meaning any bounded sequence in the stronger space has a subsequence that converges in the weaker one. Rellich first proved the theorem for the L2 case and Kondrashov extended it to the general Lp case, and the result is a basic compactness tool throughout the modern theory of partial differential equations.
Facts
StatementOn a suitable bounded domain the Sobolev space W^{1,p} is continuously embedded in L^{p*} and compactly embedded in L^q for every 1 <= q < p*. 1 Classification
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Source Rellich-Kondrachov theorem, Wikipedia
Proved By
Source Rellich-Kondrachov theorem, Wikipedia
Sources
1. Rellich-Kondrachov theorem, Wikipedia
Statement of the Theorem
the Sobolev space W 1 , p ( Ω ) is continuously embedded in the L p space L p ∗ ( Ω ; R ) and is compactly embedded in L q ( Ω ; R ) for every 1 ≤ q < p ∗
In Branch: Functional Analysis, Lead section
theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician
Proved By: Franz Rellich, Lead
Rellich proved the L2 theorem and Kondrashov the Lp theorem.
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