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Let $f_n$ be bounded uniformly in the $H^1$ norm, so we have (weak convergence) $f_n \rightharpoonup f \qquad \text{in} \qquad H^1(\Omega\times [0,T]).$ Then by compact embedding, we have strong convergence $f_n \to f \qquad \text{in}\qquad L^2(\Omega \times [0,T]).$

How can I show that $\frac{d}{dt}\int_{\Omega(t)}f_n\phi \to \frac{d}{dt}\int_{\Omega(t)}f\phi$ for $\phi \in H^1$?

How to approach this problem? What conditions do I need? Obviously the statement is true if the derivatives weren't there but I'm not sure what to do. I can't take the absolute value of the difference and put the absolute value into the integral either.

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    Thanks @NateEldredge. Showing what you wrote above is fine I think (take the difference and use Cauchy-Schwarz etc.) I take it: $\int_0^T \chi g_n \to \int_0^T \chi g \qquad \text{for all $\chi$}$ implies $g_n \to g \qquad \text{a.e.}$ is what you were hinting at?2012-06-14

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