divergence form of the determinant

Question

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}

The hint is to write $\partial_1u_1^n\partial_2u_2^n-\partial_1u_2^n\partial_2u_1^n$ as a divergence.

How would you see that as a divergence ?

Answer 1

Notice that

\begin{equation} \begin{split} \partial_1 u_{1}^{n} \partial_2 u_{2}^{n} - \partial_1 u_{2}^{n} \partial_2 u_{1}^{n}& = \partial_1 (u_{1}^{n} \partial_2 u_{2}^{n}) - \partial_2(u_{1}^{n} \partial_1 u_{2}^{n}) \\ & = \text{div}(u_{1}^{n} \partial_2 u_{2}^{n}, -u_{1}^{n} \partial_1 u_{2}^{n}) \end{split} \end{equation}

which is what you're after.