I'm trying to arrive at the weak form from the following strong form for the Anisotropic Poisson problem:
$$\begin{align*}-\nabla\cdot(\textbf{A}\nabla u) &= f \hspace{.5cm}\textrm{in }\Omega \\ u &= u_0 \hspace{.5cm}\textrm{on }\Gamma \end{align*}$$
After multiplying by a test function and integrating over the domain, I'm not sure what to do with the tensor $\textbf{A}$:
$$\int_{\Omega}-v\nabla\cdot(\textbf{A}\nabla u)=\int_{\Omega}fv$$
When it's isotropic, such as $\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}$, it acts simply as a constant; however, with anisotropy like $\begin{bmatrix}1 & 2 \\ 2 & 5 \end{bmatrix}$, it affects $\nabla u$ such that I don't believe Green's identity applies. So I don't think I can say
$$\int_{\Omega}-v\nabla\cdot(\textbf{A}\nabla u) = \int_{\Omega}\textbf{A}\nabla v\cdot\nabla u - \int_{\partial\Omega}u\textbf{A}\nabla v\cdot\textbf{n}$$
However, I don't know of any other identity which can help go from the strong form to a symmetric weak form. Any ideas?