 Solution

If $\varphi \in C^2(\overline{U}{)}^2$, take $v\in C_c^{\infty }(U{)}^2$ and integrate the gradient term by parts. The weak identity becomesThe fundamental lemma of the calculus of variations gives $-\Delta \varphi +A\varphi =F$ 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.