Viscosity Solutions for Degenerate Elliptic Equations

Non-linear and degenerate elliptic partial differential equations appear throughout modern applied mathematics, optimal control, and geometric analysis, yet classical solutions often fail to exist. This text-based course introduces you to the theory of viscosity solutions, providing the analytical framework needed to understand generalized solutions with precision and rigor. Starting from foundational calculus and differential equation concepts, you will explore why classical derivatives break down and how weak formulations bridge the gap. You will read structured explanations of regularized systems, vanishing viscosity techniques, and key comparison principles, building intuition for complex mathematical structures step by step. What you'll learn: - Understand the fundamental definition and motivation behind viscosity sub-solutions and super-solutions - Apply the vanishing viscosity method to analyze singular and degenerate equations - Examine the properties of the p-Laplacian operator and the eikonal equation - Formulate maximum principles and comparison theorems for non-linear equations - Analyze modern regularization techniques and asymptotic limits - Practice step-by-step mathematical reasoning on standard benchmark problems The course begins with foundational definitions, terminology, and test-function mechanics before progressing through comparison principles, existence proofs, and modern analytic applications. Written for students and researchers with a basic background in multivariable calculus and differential equations who are new to non-smooth analysis. Start reading today to deepen your understanding of modern non-linear partial differential equations.