Existence of Solutions for Poisson-Dirichlet Problem

This lecture covers the existence of solutions for the Poisson-Dirichlet problem, focusing on showing that certain conditions hold for locally bounded and Hölder continuous functions. The lecture progresses by defining Holder continuity and demonstrating properties of the fundamental solution. It then delves into the concept of the Newtonian potential and its properties, emphasizing the locally integrable nature of the solutions. The lecture concludes by discussing the uniform continuity and differentiability of the solutions, showcasing the smoothness and boundedness of the functions.