Calculus of Variations

This book originates from lecture notes developed and refined over several years for a course on the calculus of variations at the University of Cologne. The audience—primarily students of mathematics, and occasionally physics—typically brought with them little more than a solid background in elementary analysis. Addressing this audience shaped both the scope and the pedagogical approach of the book. The calculus of variations confronts students with a wide range of new ideas: abstract function spaces, functional-analytic tools, and subtle compactness and lower-semicontinuity arguments. These conceptual and technical challenges often make the subject difficult to access. This book is designed to lower that barrier, guiding readers step by step toward a sound understanding of the direct methods of the calculus of variations. The material is organized into 14 carefully structured chapters, corresponding to a typical semester’s schedule. The selection and sequencing of topics are deliberately focused: the goal is not encyclopedic coverage, but a clear path leading to the application of direct methods to semilinear elliptic partial differential equations. Throughout the book, the theory is consistently motivated and illustrated by numerous examples, and all arguments and computations are presented in full detail. By concentrating on what is truly essential and by carefully balancing intuition with precision, this book offers an ideal entry point for students of mathematics, physics, and related disciplines who wish to engage seriously with the calculus of variations and its analytical foundations. This book is a translation of the original German edition. The translation was done with the help of an artificial intelligence machine translation tool. A subsequent human revision was done primarily in terms of content, so that the book may read stylistically differently from a conventional translation.