Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 4 / c / i / Solution

The zero-Dirichlet Green function of the Laplacian $G_D(x,y)$ is symmetric, vanishes at the boundary in the appropriate sense, is harmonic in $x$ away from $y$, and satisfiesas a distribution. Equivalently, for suitable $\varphi$,