Engine efficiency

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.

Statement 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 ⊂ Rⁿ 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:

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

and

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

For the subspace of H^k(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

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