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I'm having problems with the following question:

Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that $u^n\rightharpoonup u$ in $H_0^1(\Omega:\mathbb{R}^2)$. Show that for all $\phi\in C_0^\infty(\Omega)$ \begin{equation} \int_\Omega(\partial_1u_1^n\partial_2u_2^n-\partial_1u_2^n\partial_2u_1^n)\phi dx\to\int_\Omega(\partial_1u_1\partial_2u_2-\partial_1u_2\partial_2u_1)\phi dx \end{equation}

Anyone got any good hints?

Davide Giraudo
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Thom Kostova
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  • So actually you are trying to prove that $\partial_1 u_1^2\partial_2u_2^n$ is convergence weakly in $L^1$ right? Because if $u_n\in H_0^1$ then $\partial_1 u_1^n\partial_2u_2^n$ will belong to $L^1$. Hence, should we test against $\phi\in L^\infty$ but not just $\phi\in C_c^\infty$? after all, $C_c^\infty$ is not dense in $L^\infty$. – spatially Nov 10 '14 at 00:43
  • Maybe this is not true. Take a look in this answer. Maybe with some adaptation, it is possible to give a conter example for your problem. I was thinking in defining $u_n=(\cos(nx)/n,\cos(nx)/n)$. – Tomás Nov 18 '14 at 10:53

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