Viscosity Solutions for Nonlinear and Degenerate Elliptic PDEs

Nonlinear and degenerate elliptic equations often lack classical smooth solutions, making modern weak solution frameworks essential for rigorous analysis. This text-based course guides you step-by-step through the theory and application of viscosity solutions, providing intuitive explanations for mathematical tools used in modern partial differential equations. You will build a solid theoretical foundation by exploring how regularization techniques, comparison principles, and maximum principles work together to handle equations like the p-Laplacian and Hamilton-Jacobi equations. What you'll learn: - Understand foundational definitions of sub-solutions, super-solutions, and viscosity solutions - Apply the vanishing viscosity method to analyze singular perturbations and regularized equations - Analyze degenerate elliptic problems, including the eikonal equation and p-Laplace operators - Master comparison principles and uniqueness proofs for nonlinear differential equations - Explore modern connections between viscosity solutions, optimal control, and geometric analysis - Practice step-by-step analytical reasoning through structured written exercises and mathematical proofs The curriculum begins with essential terminology, differential inequalities, and motivation before guiding you through core proofs, maximum principles, and detailed case studies. This course is designed for learners, students, and practitioners in applied mathematics and engineering who want an accessible, self-paced introduction to nonlinear PDEs without requiring advanced prior expertise in viscosity theory. Begin reading today to strengthen your analytical mastery of modern partial differential equations.