Rellich–Kondrachov theorem
In mathematics, the Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich proved the L2 theorem and Kondrashov the Lp theorem.
Statement of the theorem
Let Ω ⊆ Rn be an open, bounded Lipschitz domain, and let 1 ≤ p < n. Set
- [math]\displaystyle{ p^{*} := \frac{n p}{n - p}. }[/math]
Then the Sobolev space W1,p(Ω; R) is continuously embedded in the Lp space Lp∗(Ω; R) and is compactly embedded in Lq(Ω; R) for every 1 ≤ q < p∗. In symbols,
- [math]\displaystyle{ W^{1, p} (\Omega) \hookrightarrow L^{p^{*}} (\Omega) }[/math]
and
- [math]\displaystyle{ W^{1, p} (\Omega) \subset \subset L^{q} (\Omega) \text{ for } 1 \leq q \lt p^{*}. }[/math]
Kondrachov embedding theorem
On a compact manifold with C1 boundary, the Kondrachov embedding theorem states that if k > ℓ and k − n/p > ℓ − n/q then the Sobolev embedding
- [math]\displaystyle{ W^{k,p}(M)\subset W^{\ell,q}(M) }[/math]
is completely continuous (compact).[1]
Consequences
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, in the past the result was sometimes referred to as the Rellich–Kondrachov selection theorem, since one "selects" a convergent subsequence. (However, today the customary name is "compactness theorem", whereas "selection theorem" has a precise and quite different meaning, referring to set-valued functions).
The Rellich–Kondrachov theorem may be used to prove the Poincaré inequality,[2] which states that for u ∈ W1,p(Ω; R) (where Ω satisfies the same hypotheses as above),
- [math]\displaystyle{ \| u - u_\Omega \|_{L^p (\Omega)} \leq C \| \nabla u \|_{L^p (\Omega)} }[/math]
for some constant C depending only on p and the geometry of the domain Ω, where
- [math]\displaystyle{ u_\Omega := \frac{1}{\operatorname{meas} (\Omega)} \int_\Omega u(x) \, \mathrm{d} x }[/math]
denotes the mean value of u over Ω.
References
- ↑ Taylor, Michael E. (1997). Partial Differential Equations I - Basic Theory (2nd ed.). p. 286. ISBN 0-387-94653-5.
- ↑ Evans, Lawrence C. (2010). "§5.8.1". Partial Differential Equations (2nd ed.). p. 290. ISBN 978-0-8218-4974-3.
Literature
- Evans, Lawrence C. (2010). Partial Differential Equations (2nd ed.). American Mathematical Society. ISBN 978-0-8218-4974-3.
- Kondrachov, V. I., On certain properties of functions in the space L p .Dokl. Akad. Nauk SSSR 48, 563–566 (1945).
- Leoni, Giovanni (2009). A First Course in Sobolev Spaces. Graduate Studies in Mathematics. 105. American Mathematical Society. pp. xvi+607. ISBN:978-0-8218-4768-8. MR 2527916. Zbl 1180.46001
- Rellich, Franz (24 January 1930). "Ein Satz über mittlere Konvergenz" (in German). Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1930: 30–35. https://eudml.org/doc/59297.
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