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Rellich–Kondrachov theorem

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(Redirected from Rellich's lemma) 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 L theorem and Kondrashov the L theorem.

Statement of the theorem

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

p ∗ := n p n − p . {\displaystyle p^{*}:={\frac {np}{n-p}}.}

Then the Sobolev space W(Ω; R) is continuously embedded in the L space L(Ω; R) and is compactly embedded in L(Ω; R) for every 1 ≤ q < p. In symbols,

W 1 , p ( Ω ) ↪ L p ∗ ( Ω ) {\displaystyle W^{1,p}(\Omega )\hookrightarrow L^{p^{*}}(\Omega )}

and

W 1 , p ( Ω ) ⊂ ⊂ L q ( Ω )  for  1 ≤ q < p ∗ . {\displaystyle W^{1,p}(\Omega )\subset \subset L^{q}(\Omega ){\text{ for }}1\leq q<p^{*}.}

Kondrachov embedding theorem

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

W k , p ( M ) ⊂ W ℓ , q ( M ) {\displaystyle W^{k,p}(M)\subset W^{\ell ,q}(M)}

is completely continuous (compact).

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(Ω; R) has a subsequence that converges in L(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, which states that for u ∈ W(Ω; R) (where Ω satisfies the same hypotheses as above),

‖ u − u Ω ‖ L p ( Ω ) ≤ C ‖ ∇ u ‖ L p ( Ω ) {\displaystyle \|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 Ω := 1 meas ⁡ ( Ω ) ∫ Ω u ( x ) d x {\displaystyle u_{\Omega }:={\frac {1}{\operatorname {meas} (\Omega )}}\int _{\Omega }u(x)\,\mathrm {d} x}

denotes the mean value of u over Ω.

References

  1. Taylor, Michael E. (1997). Partial Differential Equations I - Basic Theory (2nd ed.). p. 286. ISBN 0-387-94653-5.
  2. Evans, Lawrence C. (2010). "§5.8.1". Partial Differential Equations (2nd ed.). p. 290. ISBN 978-0-8218-4974-3.

Literature

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