Rellich–Kondrachov theorem

{{short description|Compact embedding theorem concerning Sobolev spaces}}

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 L^p theorem.

Statement of the theorem

Let Ω ⊆ Rⁿ be an open, bounded Lipschitz domain, and let 1 ≤ p < n. Set

: $p^{*} := \frac{n p}{n - p}.$

Then the Sobolev space W^1,p(Ω; R) 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 symbols,

: $W^{1, p} (\Omega) \hookrightarrow L^{p^{*}} (\Omega)$

and

: $W^{1, p} (\Omega) \subset \subset L^{q} (\Omega) \text{ for } 1 \leq q < p^{*}.$

=Kondrachov embedding theorem=

On a compact manifold with {{math|C¹}} boundary, the Kondrachov embedding theorem states that if {{math|k > ℓ}} and {{math|k − n/p > ℓ − n/q}} then the Sobolev embedding

: $W^{k,p}(M)\subset W^{\ell,q}(M)$

is completely continuous (compact).{{cite book | author=Taylor, Michael E. | title=Partial Differential Equations I - Basic Theory | year=1997 | edition=2nd | isbn=0-387-94653-5 | page=286}}

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 W^1,p(Ω; R) has a subsequence that converges in L^q(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,{{cite book | author=Evans, Lawrence C. | title=Partial Differential Equations | year=2010 | edition=2nd | isbn=978-0-8218-4974-3 | chapter=§5.8.1 | page=290}} which states that for u ∈ W^1,p(Ω; R) (where Ω satisfies the same hypotheses as above),

: $\| u - u_\Omega \|_{L^p (\Omega)} \leq C \| \nabla u \|_{L^p (\Omega)}$

for some constant C depending only on p and the geometry of the domain Ω, where

: $u_\Omega := \frac{1}{\operatorname{meas} (\Omega)} \int_\Omega u(x) \, \mathrm{d} x$

denotes the mean value of u over Ω.

References

Literature

{{cite book | author=Evans, Lawrence C. | title=Partial Differential Equations | year=2010 | edition=2nd | publisher=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 [https://www.ams.org/mathscinet-getitem?mr=2527916 2527916]. Zbl [https://zbmath.org/?format=complete&q=an:1180.46001 1180.46001]
{{cite journal

| last = Rellich

| first = Franz

| authorlink = Franz Rellich

| title = Ein Satz über mittlere Konvergenz

| journal = Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse

| volume = 1930

| pages = 30–35

| date = 24 January 1930

| language = German

| url = https://eudml.org/doc/59297

| jfm = 56.0224.02}}

Category:Theorems in mathematical analysis

Category:Sobolev spaces