Ehrling's lemma

In mathematics, Ehrling's lemma, also known as Lions' lemma,{{cite book | last1 = Brezis | first1 = Haïm | title = Functional analysis, Sobolev spaces and partial differential equations | publisher = Springer-Verlag | location = New York | year= 2011 | isbn= 978-0-387-70913-0}} 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.{{cite journal |last1=Ehrling |first1=Gunnar |title=On a type of eigenvalue problem for certain elliptic differential operators |journal=Mathematica Scandinavica |date=1954 |volume= 2|issue=2 |pages=267–285 |doi=10.7146/math.scand.a-10414 |jstor=24489040 |doi-access=free }}{{cite book |last1=Fichera |first1=Gaetano |author-link=Gaetano Fichera|title=Linear elliptic differential systems and eigenvalue problems |date=1965 |pages=24–29 |chapter-url=https://link.springer.com/chapter/10.1007/BFb0079963?noAccess=true |access-date=18 May 2022 |chapter=The trace operator. Sobolev and Ehrling lemmas|series=Lecture Notes in Mathematics |volume=8 |doi=10.1007/BFb0079963 |isbn=978-3-540-03351-6 }}{{efn|Fichera's statement of the lemma, which is identical to what we have here, is a generalization{{cite book |last1=Roubíček |first1=Tomáš |title=Nonlinear partial differential equations with applications |date=2013 |publisher=Birkhäuser Verlag|location=Basel |page=193|volume=153|series=International Series of Numerical Mathematics|isbn=9783034805131 |url=https://books.google.com/books?id=peZHAAAAQBAJ&dq=nonlinear+partial+differential+equations+with+applications+roubicek&pg=PR3 |access-date=18 May 2022}}{{efn-lr|In subchapter 7.3 "Aubin-Lions lemma", footnote 9, Roubíček says: "In the original paper, Ehrling formulated this sort of assertion in less generality."}} 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).}}

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

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

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 L² norm of u can be left out to yield another equivalent norm.

References

=Notes=

Bibliography

{{cite book

| last1 = Renardy

| first1 = Michael

| last2 = Rogers

| first2 = Robert C.

| title = An Introduction to Partial Differential Equations

| publisher = Springer-Verlag

| location = Berlin

| year=1992

| isbn=978-3-540-97952-4

}}

Category:Banach spaces

Category:Sobolev spaces

Category:Lemmas in mathematical analysis