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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.
A long list of problems will guide the student through the study of the subject. Almost all the problems come with their fully detailed solutions. The book is complemented by four appendices, which contribute to making it self-contained, as well as to deepening the study of certain parts.
Despite being designed for students, even the researchers in the field could find a reading of the book profitable, at least for certain parts concerning the properties of Sobolev spaces, functional inequalities of the Sobolev-Poincaré type, tricks to handle nonlinear elliptic PDEs, and a gentle introduction to some techniques of modern regularity theory for elliptic PDEs.
List of contents
- 1. Tools.- 2. Some One-Dimensional Variational Problems.- 3. Sobolev Spaces.- 4. The Direct Method in Sobolev Spaces.- 5. Lipschitz Functions.- 6. The Direct Method in Lipschitz Spaces.- 7. Excerpts from Regularity Theory.- 8. Solutions to Problems.
About the author
Lorenzo Brasco was born in Firenze in 1981. He received his Ph.D. in mathematics in 2010 from the University of Pisa and University of Paris-Dauphine in a joint Ph.D. program. From 2010–2011, he held a postdoc position at the University of Naples, funded by an ERC project. From 2011 to 2015, he was Maître de Conférences at Aix-Marseille Université. From 2015 to 2023, he was an Associate Professor of Mathematical Analysis at the University of Ferrara. He is now a Full Professor of Mathematical Analysis at the same university. Author of more than 60 research papers published in international peer-reviewed journals. Invited speaker in several international workshops and research institutes, including the Banff International Research Station, the Mathematisches Forschunginstitut Oberwolfach and the Mittag-Leffler Institute. His research areas are the calculus of variations and partial differential equations.
