 Solution

Use the bilinear formon $H_0^1(U{)}^2$. Smoothness on the compact set $\overline{U}$ makes $A$ bounded, so $B$ is bounded. Positive semidefiniteness givesThe Poincaré inequality makes the right side coercive for the $H_0^1(U{)}^2$ norm. Hence the Lax-Milgram theorem gives a unique weak solution for everyIn particular, every $F\in L^2(U{)}^2$ is admissible.