Ehrling's lemma

In mathematics, Ehrling's lemma is a result concerning Banach spaces. It is often used in functional analysis to demonstrate the equivalence of certain norms on Sobolev spaces.

tatement of the lemma

Let ("X", ||·||_"X"), ("Y", ||·||_"Y") and ("Z", ||·||_"Z") be three Banach spaces. Assume that:
* "X" is compactly embedded in "Y": i.e. "X" ⊆ "Y" and every ||·||_"X"-bounded sequence in "X" has a subsequence that is ||·||_"Y"-convergent; and
* "Y" is continuously embedded in "Z": i.e. "Y" ⊆ "Z" and there is a constant "k" so that ||"y"||_"Z" ≤ "k"||"y"||_"Y" for every "y" ∈ "Y".Then, for every "ε" > 0, there exists a constant "C"("ε") such that, for all "x" ∈ "X",

:$|\; x\; |\_\{Y\}\; leq\; varepsilon\; |\; x\; |\_\{X\}\; +\; C(varepsilon)\; |\; x\; |\_\{Z\}$

Corollary (equivalent norms for Sobolev spaces)

Let Ω ⊂ R^"n" be open and bounded, and let "k" ∈ N. Suppose that the Sobolev space "H"^"k"(Ω) is compactly embedded in "H"^"k"−1(Ω). Then the following two norms on "H"^"k"(Ω) are equivalent:

:$|\; cdot\; |\; :\; H^\{k\}\; (Omega)\; o\; mathbf\{R\}:\; u\; mapsto\; |\; u\; |\; :=\; sqrt\{sum\_\{|\; alpha\; |\; leq\; k\}\; |\; mathrm\{D\}^\{alpha\}\; u\; |\_\{L^\{2\}\; (Omega)\}^\{2$

and

:$|\; cdot\; |\text{'}\; :\; H^\{k\}\; (Omega)\; o\; mathbf\{R\}:\; u\; mapsto\; |\; u\; |\text{'}\; :=\; sqrt\{|\; u\; |\_\{L^\{1\}\; (Omega)\}^\{2\}\; +\; sum\_\{|\; alpha\; |\; =\; k\}\; |\; mathrm\{D\}^\{alpha\}\; u\; |\_\{L^\{2\}\; (Omega)\}^\{2.$

For the subspace of "H"^"k"(Ω) consisting of those Sobolev functions with zero trace (those that are "zero on the boundary" of Ω), the "L"¹ norm of "u" can be left out to yield another equivalent norm.

References

* cite book
last = Rennardy
first = Michael
coauthors = Rogers, Robert C.
title = An Introduction to Partial Differential Equations
publisher = Springer-Verlag
location = Berlin
year=1992
id=ISBN 978-3-540-97952-4