# Math Help - prove these for this poisson equation

1. ## prove these for this poisson equation

2. Because by the maximum principle the function $u$ assumes it maximum and minimum on the boundary, $S$. Since it constatly assumes zero it must mean the harmonic function is identitically zero on $V$. As a corollary, we can show that the Dirichlet problem has at most one solution. Because if $\xi$ and $\mu$ are harmonic on an open set and solve the same Dirichlet problem then $\xi - \mu$ is also harmonic, and furthermore $\xi - \mu$ is identitcally zero on the boundary. Therefore, $\xi = \mu$ on all points in open set.