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
Statement Form Sources
1. Rellich-Kondrachov theorem, Wikipedia
Statement of the TheoremQuote, 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 ∗
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.