Partial Differential Equations: Basics and Classification

This lecture covers the basics of partial differential equations (PDEs), including notation, definitions, and classification. It explains the concepts of open, connected subsets, smooth functions, partial derivatives, and classical solutions. The classification of PDEs into linear, semi-linear, quasi-linear, and fully nonlinear is discussed, with a focus on linear second-order PDEs. Examples of PDEs such as Laplace's equation, transport equation, heat equation, wave equation, and p-Laplacian are presented. The lecture also explores domains with smooth boundaries, Gauss-Green formula, integration by parts, and Green's identities.