"An Introduction to Partial Differential Equations" is a comprehensive textbook that addresses the fundamentals and applications of partial differential equations (PDEs). These equations are crucial for modeling natural phenomena and are utilized in nearly all scientific disciplines. The book is aimed at aspiring doctoral students and provides the necessary theoretical foundation to work in this complex field. It assumes knowledge of advanced calculus and basic complex variables, while the required tools from functional analysis are developed throughout the book. The new edition includes numerous new exercises, arranged from easy to difficult, as well as additional examples and expanded content on Sobolev spaces. A new section on nonlinear variational problems with Young measures has also been added, making the book a valuable resource for students and educators.