Introduction to Sobolev Spaces and Partial Differential Equations

Modern analysis of partial differential equations (PDEs) requires a language that goes beyond classical derivatives. This course introduces you to Sobolev spaces, providing the essential mathematical toolkit to study weak solutions and their properties. You will start with the core foundational definitions, learning how to transition from classical calculus to weak derivatives and distribution theory. Through clear, written explanations and structured proofs, you will build a solid understanding of how these advanced spaces are constructed and applied. What you'll learn: Understand the limitations of classical derivatives and the necessity of weak derivatives; Define and analyze Sobolev spaces on various domains; Apply key embedding theorems to establish relations between different function spaces; Master the Lax-Milgram theorem to prove the existence and uniqueness of weak solutions; Practice formulating weak boundary value problems for elliptic equations. The course begins with essential prerequisite concepts, including L^p spaces and distributions, before guiding you through Sobolev space properties, trace theorems, and variational formulations of PDEs. This course is designed for advanced undergraduate students, beginning graduate students, and researchers in mathematics or theoretical physics who want a rigorous, beginner-friendly introduction to modern PDE theory. Start your journey into advanced mathematical analysis today.