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I have got a problem regarding the weak differentiability of Bochner-integrable functions. Let $(V,H,V^*)$ be a Gelfand-triple and \begin{align*} w \in W(0,T) := \{w\in L^2(0,T;V) ~\vert~ \exists w' \in L^2(0,T;V^*) \}. \end{align*} We know $w$ is equal to a continuous function in $H$. Does the differential quotient converge in $L^2(0,T;V^*)$, namely \begin{align*} \frac{w(t+h) - w(t)}{h} \to w'(t) \text{ in } L^2(0,T;V^*)? \end{align*} I need this result to prove the convergence of an implicit Euler scheme.

Thanks you, FFoDWindow

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I suppose that $w'$ for you indicates $dw/dt$. From your formula it seems that you want strong convergence of the difference quotient, correct?

At the moment I don't see that you have the necessary time regularity to prove such a statement. You know at the moment only that $w$ and $w'$ are square-integrable in time, and a simple continuity in time is not enough. Can you prove absolute continuity?

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  • $\begingroup$ Hey Kira, thank you for your reply. It should be absoluly continuous in $L^2(0,T;V^*)$, since $V,H,V^*$ form a Gelfand-Triple. I'll verify it tomorrow. $\endgroup$
    – malwin
    Commented Feb 9, 2017 at 23:58
  • $\begingroup$ Our $w$ is absolutely continuous with values in $V^*$. $\endgroup$
    – malwin
    Commented Feb 10, 2017 at 13:22
  • $\begingroup$ @FFoDWindow Did you take a look at math.stackexchange.com/a/980049/10311 (which in turn refers to Evans, section 5.8.2) $\endgroup$
    – anonymous
    Commented Feb 12, 2017 at 12:36
  • $\begingroup$ @anonymous: What you are referring to is the situation for real-valued functions. The question of FFoDWindow was about Banach-space-valued functions. The book of Evans has here some limited material in Chapter 7, however not for the question asked. $\endgroup$
    – Kira G.
    Commented Feb 13, 2017 at 14:10
  • $\begingroup$ @KiraG. Fair enough. The equivalence between $p$-integrability of the gradient and difference quotients of a vector-valued $L^p(0,T,X)$ function can e.g. be found as Theorem 3.20 in M. Kreuter's master thesis (assuming that $X$ has the Radon-Nikodým property). $\endgroup$
    – anonymous
    Commented Feb 13, 2017 at 20:25

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