A course in the calculus of variations
optimization, regularity, and modeling
Seiten
2023
|
1. Auflage
Springer International Publishing (Verlag)
978-3-031-45035-8 (ISBN)
Springer International Publishing (Verlag)
978-3-031-45035-8 (ISBN)
This book provides an introduction to the broad topic of the calculus of variations. It addresses the most natural questions on variational problems and the mathematical complexities they present.
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 and the Steiner tree problem. The book concludes with a chapter on the limits of sequences of variational problems, expressed in terms of -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.
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 and the Steiner tree problem. The book concludes with a chapter on the limits of sequences of variational problems, expressed in terms of -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.
Filippo Santambrogio is currently professor at Université Claude Bernard Lyon 1, after having completed his studies in Pisa and starting his career in Paris. He is a specialist of the calculus of variations and in particular optimal transport, mean field games, and gradient flows.
1 One-dimensional variational problems.- 2 Multi-dimensional variational problems.- 3 Lower semicontinuity.- 4 Convexity and its applications.- 5 Hölder regularity.- 6 Variational problems for sets.- 7 -convergence: theory and examples.
| Erscheinungsdatum | 20.12.2023 |
|---|---|
| Reihe/Serie | Universitext |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 551 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Schlagworte | Convex duality • elliptic regularity • Euler-Lagrange equations • Geodesic and optimal curves • Geometric variational problems for sets • Lower semicontinuity of integral functionals • Partial differential equations • textbook on calculus of variations • Variational convergence |
| ISBN-10 | 3-031-45035-3 / 3031450353 |
| ISBN-13 | 978-3-031-45035-8 / 9783031450358 |
| Zustand | Neuware |
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