= Solution
Use the bilinear form
$$
B(\phi,v)=\int_U\big(D\phi:Dv+(A\phi)\cdot v\big)
$$
on $H_0^1(U)^2$. Smoothness on the compact set $\overline U$ makes $A$ bounded, so $B$ is bounded. Positive semidefiniteness gives
$$
B(\phi,\phi)\geq\lVert D\phi\rVert_2^2.
$$
The <Poincare 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 every
$$
\boxed{F\in H^{-1}(U)^2.}
$$
In particular, every $F\in L^2(U)^2$ is admissible.
Back to article page
