In mathematics, the Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Italian-Austrian mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich proved the L2 theorem and Kondrachov the Lp theorem.
Statement of the theorem
Since an embedding is compact if and only if the inclusion (identity) operator is a compact operator, the Rellich–Kondrachov theorem implies that any uniformly bounded sequence in W1,p(Ω; R) has a subsequence that converges in Lq(Ω; R). Stated in this form, the result is sometimes known as the Rellich-Kondrachov selection theorem (since one "selects" a convergent subsequence).
for some constant C depending only on p and the geometry of the domain Ω, where
denotes the mean value of u over Ω.
- Evans, Lawrence C. (2010). "§5.8.1". Differential Equations, Partial (2nd ed.). p. 290. ISBN 0-8218-4974-3.
- Evans, Lawrence C. (2010). Differential Equations, Partial (2nd ed.). American Mathematical Society. ISBN 0-8218-4974-3.