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In mathematics, Ehrling's lemma, also known as Lions' 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 named after Gunnar Ehrling.

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,

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

Corollary (equivalent norms for Sobolev spaces)

Let Ω ⊂ R be open and bounded, and let k ∈ N. Suppose that the Sobolev space H(Ω) is compactly embedded in H(Ω). Then the following two norms on H(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

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

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

References

1. Brezis, Haïm (2011). Functional analysis, Sobolev spaces and partial differential equations. New York: Springer-Verlag. ISBN 978-0-387-70913-0.
2. Ehrling, Gunnar (1954). "On a type of eigenvalue problem for certain elliptic differential operators". Mathematica Scandinavica. 2 (2): 267–285. doi:10.7146/math.scand.a-10414. JSTOR 24489040.
3. Fichera, Gaetano (1965). "The trace operator. Sobolev and Ehrling lemmas". Linear elliptic differential systems and eigenvalue problems. Lecture Notes in Mathematics. Vol. 8. pp. 24–29. doi:10.1007/BFb0079963. ISBN 978-3-540-03351-6. Retrieved 18 May 2022.
4. Roubíček, Tomáš (2013). Nonlinear partial differential equations with applications. International Series of Numerical Mathematics. Vol. 153. Basel: Birkhäuser Verlag. p. 193. ISBN 9783034805131. Retrieved 18 May 2022.

Notes

1. Fichera's statement of the lemma, which is identical to what we have here, is a generalization of a result in the Ehrling article that Fichera and others cite, although the lemma as stated does not appear in Ehrling's article (and he did not number his results).

1. In subchapter 7.3 "Aubin-Lions lemma", footnote 9, Roubíček says: "In the original paper, Ehrling formulated this sort of assertion in less generality."

Bibliography

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

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