Solution
= Solution
Set $\mathcal H=H_0^1(U)^2$. A weak solution is a vector $\phi\in\mathcal H$ such that
$$
\boxed{
\int_U\big(D\phi:Dv+(A\phi)\cdot v\big)
=\langle F,v\rangle
\quad\text{for every }v\in\mathcal H,}
$$
where $D\phi:Dv=\sum_{j=1}^2D\phi_j\cdot Dv_j$ and the natural data space is $F\in H^{-1}(U)^2=(H_0^1(U)^2)^*$.