Maximum Principle in Harmonic Functions

This lecture delves into the properties of harmonic functions and solutions of the Laplace equation, focusing on the maximum principle and its implications for establishing uniqueness and bounds on solutions. The instructor also introduces the fundamental solution of the Laplace equation and discusses the mean value property, real analytic functions, and sub-harmonic and super-harmonic functions. The strong and weak versions of the maximum principle are explained, showcasing how they determine the behavior of sub-harmonic and super-harmonic functions in relation to the boundaries of the domain. The lecture concludes with a detailed proof of the weak maximum principle and its application to elliptic partial differential equations.