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 satisfies
$$
-\Delta_xG_D(x,y)=2\pi\delta_y
$$
as a <distribution>. Equivalently, for suitable $\phi$,
$$
(-2\pi\Delta^{-1}\phi)(x)
=\int_DG_D(x,y)\phi(y)\,dy.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-203/4/c/i/solution