Sobolev Spaces and Isoperimetric Inequalities

Modern mathematical analysis relies heavily on function spaces to study partial differential equations and geometry. This course provides a clear, conceptual introduction to Sobolev spaces and demonstrates how classical geometric concepts—such as isoperimetric and isocapacitary inequalities—provide essential analytical tools. You will build a solid theoretical foundation, moving from elementary measure and integration concepts to the equivalence between geometric profiles and Sobolev embeddings. Through written explanations and rigorous mathematical derivations, you will see how geometry controls the behavior of functions. What you'll learn: - Understand the core definitions, norms, and properties of Sobolev spaces - Explore classical isoperimetric inequalities and their analytical formulations - Master the fundamentals of capacity and isocapacitary inequalities on various domains - Analyze Sobolev-type embedding theorems and their geometric proofs - Apply capacity estimates to characterize admissible domain geometries and boundary regularity - Connect functional analysis techniques to modern partial differential equations The course begins with foundational terminology, weak derivatives, and basic measure-theoretic tools before systematically guiding you through geometric inequalities and their modern analytical applications. This course is designed for mathematics and physics students or practitioners with an introductory background in calculus and real analysis who want to understand function spaces from a geometric viewpoint. Start reading today to deepen your understanding of Sobolev spaces and geometric analysis.