Overview
- Inspired by scientific modeling in physical as well as decision sciences
- Applies convex duality to the calculus of variations, including to regularity theory
- Includes over a hundred exercises, with hints
Part of the book series: Universitext (UTX)
Access this book
Tax calculation will be finalised at checkout
Other ways to access
About this book
Beginning with the scientific modeling that motivates the subject, the book then tackles mathematical questions such as the existence and uniqueness of solutions, their characterization in terms of partial differential equations, and their regularity. It includes both classical and recent results on one-dimensional variational problems, as well as the adaptation to the multi-dimensional case. Here, convexity plays an important role in establishing semi-continuity results and connections with techniques from optimization, and convex duality is even used to produce regularity results. This is then followed by the more classical Hölder regularity theory for elliptic PDEs and some geometric variational problems on sets, including the isoperimetric inequality andthe Steiner tree problem. The book concludes with a chapter on the limits of sequences of variational problems, expressed in terms of Γ-convergence.
While primarily designed for master's-level and advanced courses, this textbook, based on its author's instructional experience, also offers original insights that may be of interest to PhD students and researchers. A foundational understanding of measure theory and functional analysis is required, but all the essential concepts are reiterated throughout the book using special memo-boxes.
Keywords
Table of contents (7 chapters)
Authors and Affiliations
About the author
Bibliographic Information
Book Title: A Course in the Calculus of Variations
Book Subtitle: Optimization, Regularity, and Modeling
Authors: Filippo Santambrogio
Series Title: Universitext
DOI: https://doi.org/10.1007/978-3-031-45036-5
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
Softcover ISBN: 978-3-031-45035-8Published: 18 December 2023
eBook ISBN: 978-3-031-45036-5Published: 17 December 2023
Series ISSN: 0172-5939
Series E-ISSN: 2191-6675
Edition Number: 1
Number of Pages: XXI, 338
Number of Illustrations: 1 b/w illustrations
Topics: Optimization, Functional Analysis, Analysis