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Suppose $p\geq 2$ and $\Omega\subset\mathbb{R}^n$ is a bounded domain. Suppose that $u_n\in L^p (\Omega)$ and $u_n\rightharpoonup u$. is true that $u_n^2\rightharpoonup u^2$ in $L^\frac{p}{2}(\Omega)$?


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No. Let $\Omega=[0,2\pi]$ and $f_n(x)=\sin(n\pi x)$, $n\ge1$. Then $f_n$ converges weakly to $0$ in $L^2$, but $f_n^2$ does not converge weakly in $L^1$ to $0$, since if $\phi(x)=1$, then $\phi\in L^\infty$ and $\int_0^{2\pi}f_n(x)^2\phi(x)dx=\pi\ne0$.

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Thank you Julián. –  Tomás Nov 12 '12 at 23:49

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