Ehrling's lemma

In mathematics, Ehrling's lemma, also known as Lions' lemma,[1] 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.[2][3][a]

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 ≤ ε ‖ x ‖ X + C ( ε ) ‖ x ‖ Z {\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:

‖ ⋅ ‖ : H k ( Ω ) → R : u ↦ ‖ u ‖ := ∑ | α | ≤ k ‖ D α u ‖ L 2 ( Ω ) 2 {\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

‖ ⋅ ‖ ′ : H k ( Ω ) → R : u ↦ ‖ u ‖ ′ := ‖ u ‖ L 2 ( Ω ) 2 + ∑ | α | = k ‖ D α u ‖ L 2 ( Ω ) 2 . {\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 Hk(Ω) consisting of those Sobolev functions with zero trace (those that are "zero on the boundary" of Ω), the L2 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[4][i] 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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