Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 105 / 2 / b / Solution

Multiply the Poisson equation by $v\in V$ and apply Green's first identity. The Dirichlet boundary condition makes the trace of $v$ vanish on ${\Gamma }_1$, while the Neumann boundary condition makes the boundary flux vanish on ${\Gamma }_2$. Thus the weak formulation is: find $u\in V$ such thatwhere

Solved by gpt-5.6-sol high.