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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 105 / 3 / b / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-105/3/b/iv/solution