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I've read the following in a few papers:

Given: Let $\Omega \subset R^d$ be a Lipschitz domain. A sequence $f_n$ converges strongly to $f$ in $L^2(0,T;L^2(\Omega))$ and weakly in $L^2(0,T;H^2(\Omega))$.

Then by a standard interpolation argument it follows that $f_n$ converges strongly to $f$ in $L^2(0,T;H^1(\Omega))$.

Now my question is what is meant by standard interpolation argument?

Thanks for your answers!

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  • $\begingroup$ A comment about the title "Standard interpolation between..." -- the word "standard" in your papers refers to "argument", not to "interpolation". They are not talking about some "standard interpolation". $\endgroup$ – 40 votes Jul 17 '13 at 21:47
  • $\begingroup$ You are right, I was to quick writing the title. Any ideas what is mathematically meant by this standard argument? $\endgroup$ – Chuck Jul 18 '13 at 7:46
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You may want to have a look for a book on interpolation theory, there is, e.g., a great introduction by Tartar.

There are, typically, two interpolation results: $$\|u\|_{L^p} \le \|u\|_{L^r}^\mu \, \|u\|_{L^s}^{1-\mu},$$ (Lebesgue scale), if $1/p = \mu/r + (1-\mu)/s$ and $$\|u\|_{H^1} \le C \, \|u\|_{L^2}^{1/2} \, \|u\|_{H^2}^{1/2}$$ (Sobolev scale).

Your proof requires the second one (use boundedness in $H^2$).

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  • $\begingroup$ Ok thanks. So taking the square in the second interpolation inequality, integration over (0,T) and applying Hölder's inequality does the job... $\endgroup$ – Chuck Jul 20 '13 at 20:34

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