Lions–Magenes lemma

In mathematics, the Lions–Magenes lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a criterion for moving a time derivative of a function out of its action (as a functional) on the function itself.

Statement of the lemma
Let X0, X and X1 be three Hilbert spaces with X0 ⊆ X ⊆ X1. Suppose that X0 is continuously embedded in X and that X is continuously embedded in X1, and that X1 is the dual space of X0. Denote the norm on X by || &sdot; ||X, and denote the action of X1 on X0 by $$\langle\cdot,\cdot\rangle$$. Suppose for some $$T>0$$ that $$u \in L^2 ([0, T]; X_0)$$ is such that its time derivative $$\dot{u} \in L^2 ([0, T]; X_1)$$. Then $$u$$ is almost everywhere equal to a function continuous from $$[0,T]$$ into $$X$$, and moreover the following equality holds in the sense of scalar distributions on $$(0,T)$$:


 * $$\frac{1}{2}\frac{d}{dt} \|u\|_X^2 = \langle\dot{u},u\rangle$$

The above equality is meaningful, since the functions


 * $$t\rightarrow \|u\|_X^2, \quad t\rightarrow \langle \dot{u}(t),u(t)\rangle$$

are both integrable on $$[0,T]$$.