Advanced Partial Differential Equations: Distributions and Sobolev Spaces

To truly understand modern partial differential equations, standard calculus is not enough; you must master the language of weak derivatives and generalized functions. This text-based course bridges the gap between classical calculus and the advanced functional analysis required for modern PDE theory. You will transition from classical solutions to weak solutions, establishing a rigorous mathematical foundation that is essential for research, applied mathematics, and advanced engineering analysis. By reading this course, you will learn to confidently apply distribution theory and work within Sobolev spaces to solve complex differential problems. What you'll learn: Understand the foundational concepts of test functions, weak derivatives, and the space of distributions; Master the mathematical definitions and properties of Sobolev spaces; Apply embedding theorems and trace operators to boundary value problems; Learn how to formulate weak solutions for elliptic partial differential equations; Practice verifying the existence and uniqueness of solutions using Lax-Milgram theorem. The course begins with essential definitions of topological vector spaces and test functions, gradually building up to distribution theory, Sobolev spaces, and their direct application to solving differential equations. This course is designed for advanced undergraduate or beginning graduate students in mathematics, physics, or engineering who have a basic background in real analysis and linear algebra, with no prior knowledge of distributions required. Begin reading today to elevate your mathematical analysis skills to the next level.