General Existence Results for PDEs

This lecture covers the general existence result for the Laplace-Dirichlet problem, properties of the Newtonian potential, and the existence of solutions for the Poisson-Dirichlet problem. The instructor discusses the construction of candidate solutions using Green's representation formula and the regularity of boundary points. The lecture progresses to the verification of solutions and the conditions for solvability. The proof involves extending functions by zero and applying integral representations. The lecture concludes with the verification of solutions and the conditions for the solvability of the Poisson equation.