Let $\Omega \subset \mathbb{R}^n$ be bounded, open, Lipschitz. Let $A_n,A^{-1}_n \in L^{\infty}(\Omega,\mathbb{R}^n \times \mathbb{R}^n )$ such that:
$$A_n,A^{-1}_n \to I_d \quad \text{uniformly}$$
We can then define a "divergence type operator" $div_{An} :H^1(\Omega)^n\to L^2(\Omega)$ as:
$$div_{An}(u) = Tr(A_n\nabla u)$$
Cleary $div_{An}\to div$ in the operator norm since:
$$\|div_{An}(u)-div(u)\|_{L^{2}(\Omega)}\leq\|A_n-I\|_{L^\infty}\|u\|_{H^{1}(\Omega)}$$
I want to show that if $v \in Kernel(div)$ then there is sequence $v_n\in Kernel(div_{A_n})$ such that:
$$v_n \to v $$
In other words, that the corresponding orthogonal projections converges. I have try to use the Leray projection, but this approach involves a Neumann problem that it's not unique up to a constant.
Any ideas or thought are very welcome. Thanks!