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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.


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I think this is a rather common situation in PDE's and analysis, because there are so many slight variants possible but it is often difficult to identify the right general formulation that covers them all. Whenever I encountered this, my solution was to write out careful statements and proofs of what I needed and put them all into an appendix of my paper. – Deane Yang Feb 4 '12 at 11:37
@Deane: That's what I have in my current draft, and it was a good exercise, but the appendix is half as long as the paper itself. – Nate Eldredge Feb 4 '12 at 15:43
Keep it unless a referee or editor makes you take it out. You won't regret it. – Deane Yang Feb 5 '12 at 19:08
up vote 4 down vote accepted

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 $W^{1,2}([0,T];V,V^*)$.

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This looks like just what I want. Thanks, and thanks to all for the other suggestions, which also look good. – Nate Eldredge Feb 6 '12 at 17:43

Herbert Ammann's book on parabolic problems contains an excellent introduction.

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If you read French then this book is the place you are looking for

Brézis, H. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. (French) North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. vi+183 pp.Inc.,

Another source

Barbu, Viorel(R-IASIM) Nonlinear differential equations of monotone types in Banach spaces. Springer Monographs in Mathematics. Springer, New York, 2010. x+272 pp. ISBN: 978-1-4419-5541-8978-1-4419-5541-8

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