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][lower-alpha 1]
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,
- [math]\displaystyle{ \| x \|_{Y} \leq \varepsilon \| x \|_{X} + C(\varepsilon) \| x \|_{Z} }[/math]
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:
- [math]\displaystyle{ \| \cdot \| : H^{k} (\Omega) \to \mathbf{R}: u \mapsto \| u \| := \sqrt{\sum_{| \alpha | \leq k} \| \mathrm{D}^{\alpha} u \|_{L^{2} (\Omega)}^{2}} }[/math]
and
- [math]\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}}. }[/math]
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
- ↑ Brezis, Haïm (2011). Functional analysis, Sobolev spaces and partial differential equations. New York: Springer-Verlag. ISBN 978-0-387-70913-0.
- ↑ 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.
- ↑ Fichera, Gaetano (1965). "The trace operator. Sobolev and Ehrling lemmas". Linear elliptic differential systems and eigenvalue problems. Lecture Notes in Mathematics. 8. pp. 24–29. doi:10.1007/BFb0079963. ISBN 978-3-540-03351-6. https://link.springer.com/chapter/10.1007/BFb0079963?noAccess=true. Retrieved 18 May 2022.
- ↑ Roubíček, Tomáš (2013). Nonlinear partial differential equations with applications. International Series of Numerical Mathematics. 153. Basel: Birkhäuser Verlag. p. 193. ISBN 9783034805131. https://books.google.com/books?id=peZHAAAAQBAJ&dq=nonlinear+partial+differential+equations+with+applications+roubicek&pg=PR3. Retrieved 18 May 2022.
Notes
- ↑ Fichera's statement of the lemma, which is identical to what we have here, is a generalization[4][lower-roman 1] 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).
- ↑ 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.
 |  |
