Reference Request: Simple Facts About Vector-Valued Sobolev Space – Functional Analysis

Question

Let $V,H$ be separable Hilbert spaces such that there are dense injections $V \hookrightarrow H \hookrightarrow V^*$. (For example, $H = L^2(\mathbb{R}^n)$, $V = H^1(\mathbb{R}^n)$, $V^* = H^{-1}(\mathbb{R}^n)$.) We can then define the vector-valued Sobolev space $W^{1,2}([0,1]; V, V^*)$ of functions $u \in L^2([0,1]; V)$ which have one weak derivative $u' \in L^2([0,1], V^*)$. Such spaces arise often in the study of PDE involving time.

I would like a reference for some simple facts about $W^{1,2}$. For example:

• Basic calculus, like integration by parts, etc.

• The "Sobolev embedding" result $W^{1,2} \subset C([0,1]; H)$;

• The "product rule" $\frac{d}{dt} \|u(t)\|_{H^2} = (u'(t), u(t))_{V^*, V}$

• $C^\infty([0,1]; V)$ is dense in $W^{1,2}$.

These are pretty easy to prove, but they should be standard and I don't want to waste space in a paper with proofs.

Some of these results, in the special case where $V$ is Sobolev space, are in L. C. Evans, Partial Differential Equations, section 5.9, but I'd rather not cite special cases. Also, in current editions of this book, there's a small but significant gap in one of the proofs (it is addressed briefly in the latest errata). So I'd prefer something else.

Thanks!

Answer 1

J. Wloka "Partial differential equations", § 25 (p. 390 on, in my 1992 CUP edition) has an account of the space $W(0,T)=W_2^1(0,T)$ which is essentially the space $W^{1,2}([0,T];V,V^*)$.