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,

$\Vert x{\Vert }_Y\le \epsilon \Vert x{\Vert }_X+C(\epsilon )\Vert x{\Vert }_Z$

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:

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

and

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

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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