Solution
= Solution
If $\phi\in C^2(\overline U)^2$, take $v\in C_c^\infty(U)^2$ and integrate the gradient term by parts. The weak identity becomes
$$
\int_U(-\Delta\phi+A\phi-F)\cdot v=0.
$$
The <fundamental lemma of the calculus of variations> gives $-\Delta\phi+A\phi=F$ pointwise. Membership in $H_0^1(U)^2$, together with continuity up to the boundary, says that the boundary trace is zero, so the Dirichlet condition also holds classically.