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 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.
Author(s): Filippo Santambrogio
Series: Universitext
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 338
City: Cham
Tags: Calculus of Variations, Euler-Lagrange Equation, Geodesics, Lower Semicontinuity, Hölder Regularity, Isoperimetric Problems, Gamma-Convergence
Preface
Acknowledgments
Notation
Contents
1 One-Dimensional Variational Problems
1.1 Optimal Trajectories
1.1.1 Three Classical 1D Examples
1.1.2 Geodesics on the Sphere
1.2 Examples of Existence and Non-existence
1.2.1 Existence
1.2.2 Non-existence
1.3 Optimality Conditions
1.3.1 The Euler–Lagrange Equation
1.3.2 Examples of Growth Conditions on L
1.3.3 Transversality Conditions
1.4 Curves and Geodesics in Metric Spaces
1.4.1 Absolutely Continuous Curves and Metric Derivative
1.4.2 Geodesics in Proper Metric Spaces
1.4.3 Geodesics and Connectedness in Thin Spaces
1.4.4 The Minimal Length Problem in Weighted Metric Spaces
1.4.5 An Application to Heteroclinic Connections
1.5 The Ramsey Model for Optimal Growth
1.6 Non-autonomous Optimization in BV
1.7 Discussion: Optimal Control and the Value Function
1.8 Exercises
Hints
2 Multi-Dimensional Variational Problems
2.1 Existence in Sobolev Spaces
2.2 The Multi-Dimensional Euler–Lagrange Equation
2.3 Harmonic Functions
2.4 Discussion: p-Harmonic Functions for 1leqpleqinfty
2.5 Exercises
Hints
3 Lower Semicontinuity
3.1 Convexity and Lower Semicontinuity
3.2 Integral Functionals
3.3 Functionals on Measures
3.4 Existence of Minimizers
3.5 Discussion: Lower Semicontinuity for Vector-Valued Maps
3.6 Exercises
Hints
4 Convexity and its Applications
4.1 Uniqueness and Sufficient Conditions
4.2 Some Elements of Convex Analysis
4.2.1 The Fenchel–Legendre Transform
4.2.2 Subdifferentials
4.2.3 Recession Functions
4.2.4 Formal Duality for Constrained and PenalizedOptimization
4.3 An Example of Convex Duality: Minimal Flows and Optimal Compliance
4.3.1 Rigourous Duality with No-Flux Boundary Conditions
4.4 Regularity via Duality
4.4.1 Pointwise Vector Inequalities
4.4.2 Applications to (Degenerate) Elliptic PDEs
4.4.3 Variants: Local Regularity and Dependence on x
4.5 A Proof of the Fenchel–Rockafellar Duality Theorem
4.6 Discussion: From Optimal Transport to Congested Traffic and Mean Field Games
4.7 Exercises
Hints
5 Hölder Regularity
5.1 Morrey and Campanato Spaces
5.2 Regularity for Elliptic Equations in Divergence Form and Smooth Coefficients
5.2.1 Preliminaries
5.2.2 Schauder Regularity
5.3 Hilbert's 19th Problem and the De Giorgi Theorem
5.4 Moser's Proof
5.4.1 Linfty Bounds
5.4.2 Continuity
5.4.3 The Non-Homogeneous Case
5.5 Discussion: Lp Elliptic Regularity and Boundary Data
5.6 Exercises
Hints
6 Variational Problems for Sets
6.1 The Isoperimetric Problem
6.1.1 BV Functions and Finite Perimeter Sets
6.1.2 The Isoperimetric Property of the Ball
6.1.3 Steiner Symmetrization of Sets With Respect to a Hyperplane
6.2 Optimization of the First Dirichlet Eigenvalue
6.3 Optimal 1D Networks and the Gołąb Theorem
6.3.1 Steiner Trees, Irrigation and Transportation Networks, and Membrane Reinforcement
6.3.2 Proof of the Gołąb Semicontinuity Theorem
6.4 Discussion: Direct Proofs, Quantitative Versions, and Variants for the Isoperimetric Inequality
6.5 Exercises
Hints
7 Gamma-Convergence: Theory and Examples
7.1 General Metric Theory and Main Properties
7.2 Gamma-Convergence of Classical Integral Functionals
7.3 Optimal Location and Optimal Quantization
7.4 The Modica–Mortola Functional and the Perimeter
7.5 Discussion: Phase-Field Approximation of Free Discontinuity and Optimal Network Problems
7.6 Exercises
Hints
References
Index
