Set $\mathcal H=H_0^1(U)^2$. A weak solution is a vector $\phi\in\mathcal H$ such that
$$
\boxed{
\int_U\big(D\phi:Dv+(A\phi)\cdot v\big)
=\langle F,v\rangle
\quad\text{for every }v\in\mathcal H,}
$$
where $D\phi:Dv=\sum_{j=1}^2D\phi_j\cdot Dv_j$ and the natural data space is $F\in H^{-1}(U)^2=(H_0^1(U)^2)^*$.


Solution

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

 Solution

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