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]
Let (X, ||⋅||X), (Y, ||⋅||Y) and (Z, ||⋅||Z) be three Banach spaces. Assume that:
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
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:
\| ⋅ \|:Hk(\Omega)\toR:u\mapsto\|u\|:=\sqrt{\sum|\|D\alphau
2 | |
\| | |
L2(\Omega) |
and
\| ⋅ \|':Hk(\Omega)\toR:u\mapsto\|u\|':=\sqrt{\|u
2 | |
\| | |
L2(\Omega) |
+\sum|\|D\alphau
2 | |
\| | |
L2(\Omega) |
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.