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 andthe 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.

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

Autorentext

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.

Inhalt
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.