Poisson’s equation and eigenfunctions of the Laplacian

Abstract

This work aims to explore the foundations of partial differential equations (PDEs) by focusing specifically on Poisson’s equation with Dirichlet boundary conditions and the eigenvalue problem for the Laplacian. These equations are of special interest in both mathematics and physics. Although they are among the simplest cases of PDEs, they introduce techniques and results that are key to solving more complex equations. In particular, we will introduce the weak formulation of both equations and prove the existence of weak solutions in two different ways. The first method uses Hilbert space techniques, such as the Lax-Milgram theorem and the Spectral theorem, while the second method involves the minimization of functionals. Ultimately, we will study the regularity of weak solutions and examine a practical case in which the previous theory is very useful.