Viscosity Solutions for Non-Linear Partial Differential Equations

Understanding weak solutions to non-linear partial differential equations is essential for modern applied mathematics, optimal control, and mathematical physics. This text-based guide introduces the theory of viscosity solutions from the ground up, making advanced analytical concepts approachable. By working through clear explanations and structured mathematical steps, you will learn how to analyze equations where classical derivatives fail to exist, including Hamilton-Jacobi and p-Laplacian formulations. What you will learn: - Understand core definitions and foundational terminology of viscosity subsolutions and supersolutions - Apply maximum principles and comparison theorems to establish uniqueness - Analyze degenerate elliptic equations and non-linear differential operators - Master vanishing viscosity approximations and regularization methods - Explore modern connections to optimal control theory and numerical schemes Starting with key definitions and essential analytical tools, the course systematically builds toward regularizing degenerate equations and studying their limiting behavior. Written exercises and fully worked derivations reinforce each concept at every step. This course is designed for students, researchers, and engineers with basic familiarity in calculus who want a clear, accessible introduction to non-linear PDE theory. Begin reading today to expand your mathematical toolkit with modern PDE methods.