- Rellich-Kondrachov theorem
In
mathematics , the Rellich-Kondrachov theorem is a compact embeddingtheorem concerningSobolev space s. It is named after the Italian-Austria nmathematician Franz Rellich .tatement of the theorem
Let Ω ⊆ R"n" 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) mbox{ for } 1 leq q < p^{*}.
Consequences
Since an embedding is compact
if and only if the inclusion (identity) operator is acompact 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, the result is sometimes known as the Rellich-Kondrachov selection theorem (since one "selects" a convergent subsequence).The Rellich-Kondrachov theorem may be used to prove the
Poincaré inequality , 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 | abla u |_{L^{p} (Omega)}
for some constant "C" depending only on "p" and the geometry of the domain Ω, where
:u_{Omega} := frac{1}{mathrm{meas} (Omega)} int_{Omega} u(x) , mathrm{d} x
denotes the mean value of "u" over Ω.
References
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