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. It was proposed by Gunnar Ehrling.

Statement of the lemma

Let (X, ||·||X), (Y, ||·||Y) and (Z, ||·||Z) be three Banach spaces. Assume that:

Then, for every ε > 0, there exists a constant C(ε) such that, for all x ∈ X,

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

Corollary (equivalent norms for Sobolev spaces)

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

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

and

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

For the subspace of Hk(Ω) consisting of those Sobolev functions with zero trace (those that are "zero on the boundary" of Ω), the L1 norm of u can be left out to yield another equivalent norm.

References

• Renardy, Michael; Rogers, Robert C. (1992). An Introduction to Partial Differential Equations. Berlin: Springer-Verlag. ISBN 978-3-540-97952-4.