Brasco L. Handbook of Calculus of Variations for Absolute Beginners 2025
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Brasco L. Handbook of Calculus of Variations for Absolute Beginners 2025
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Textbook in PDF format
The book aims at endowing any student with a survival toolkit to start safely diving into the realm of Calculus of Variations. In summary, the latter is a part of mathematical analysis devoted to minimization/maximization problems. A great effort has been made to present the themes and methods considered in the book in the simplest possible way: the reader will not find here general statements or proofs based on general abstract theories. In contrast, the main focus of the book is on introducing some key concepts "from scratch", by means of simple and meaningful explicit examples (including for instance, the classical isoperimetric and brachistocrone problems, as well as the boundary value problem for harmonic functions). In particular, the book is mainly (but not exclusively) designed to smoothly introduce the reader to the so-called Direct Method of the Calculus of Variations, which is a central concept in the field. Accordingly, a good part of the book is devoted to discussing spaces of weakly differentiable functions (i.e., Sobolev and Lipschitz functions), which are essential tools of the Direct Method