The Poisson Equation

Abstract

In Section 3.1 the Poisson equation –Δu=f is introduced, and the uniqueness of the solution is proved. The Green function is defined in Section 3.2. It allows the representation (3.6) of the solution, provided it is existing. Concerning the existence, Theorem 3.13 contains a negative statement (cf. Section 3.3): The Poisson equation with a continuous right-hand side f may possess no classical solution. A sufficient condition for a classical solution is the Hölder continuity of f as stated in Theorem 3.18. Section 3.4 introduces Green’s function for the ball. In the two-dimensional case, Riemann’s mapping theorem allows the construction of the Green function for a large class of domains. In Section 3.5 we replace the Dirichlet boundary condition by the Neumann condition. The final Section 3.6 is a short introduction into the integral equation method. The solution of the boundary-value problem can indirectly be obtained by solving an integral equation.