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