Integration by parts in Bochner Lebesgue spaces.

Question

Does there exist an analogous of integration by parts for expressions such as: $$\int_0^T {\langle u(t),v(t) \rangle }\, \mathrm{d}t,$$ where $u,v\in L^2([0,T];H)$, for some Hilbert space $H$? If so, under what further assumptions on $u$ and $v$? Where can I possibly find some standard references in literature?

Answer 1

The standard integration by parts in Sobolev-Bochner spaces is as follows. If $u, v \in \{ w \in L^2(0,T;V) \mid w' \in L^2(0,T;V^*)\}$, then $$(u(T), v(T))_H - (u(0), v(0))_H = \int_0^T \langle u'(t), v(t) \rangle_{V^*, V} + \langle v'(t), u (t)\rangle_{V^*, V}$$ where $V \subset H \subset V^*$ is a Gelfand triple.

Evans probably has this. If not, you should read Nonlinear functional analysis and its applications II/A (subtitle Linear Monotone Operators) by Zeidler and Boron. Sections 23.2 are relevant.