Lecture Notes On Calculus Of Variations (Peking University Mathematics)

Preface
Contents
1. The theory and problems of calculus of variations
	1.1 Introduction
	1.2 Functionals
	1.3 Typical examples
	1.4 More examples
2. The Euler–Lagrange equation
	2.1 The necessary condition for the extremal values of functions —a review
	2.2 The derivation of the Euler–Lagrange equation
	2.3 Boundary conditions
	2.4 Examples of solving the Euler–Lagrange equations
	Exercises
3. The necessary condition and the sufficient condition on extremal values of functionals
	3.1 The extremal values of functions —a revisit
	3.2 Second order variations
	3.3 The Legendre–Hadamard condition
	3.4 The Jacobi field
	3.5 Conjugate points
	Exercises
4. Strong minima and extremal fields
	4.1 Strong minima and weak minima
	4.2 A necessary condition for strong minimal value and the Weierstrass excess function
	4.3 Extremal fields and strong minima
	4.4 Mayer field, Hilbert’s invariant integral
	4.5 A sufficient condition for strong minima
	4.6* The proof of Theorem 4.4 (for the case N > 1)
	Exercises
5. The Hamilton–Jacobi theory
	5.1 Eikonal and the Carathéodory system of equations
	5.2 The Legendre transformation
	5.3 The Hamilton system of equations
	5.4 The Hamilton–Jacobi equation
	5.5* Jacobi’s Theorem
	Exercises
6. Variational problems involving multivariate integrals
	6.1 Derivation of the Euler–Lagrange equation
	6.2 Boundary conditions
	6.3 Second order variations
	6.4 Jacobi fields
	Exercises
7. Constrained variational problems
	7.1 The isoperimetric problem
	7.2 Pointwise constraints
	7.3 Variational inequalities
	Exercises
8. The conservation law and Noether’s theorem
	8.1 One parameter diffeomorphisms and Noether’s theorem
		1. A special one parameter family of functions
		2. General local 1-parameter transformation group
	8.2 The energy–momentum tensor and Noether’s theorem
	8.3 Interior minima
	8.4* Applications
	Exercises
9. Direct methods
	9.1 The Dirichlet’s principle and minimization method
	9.2 Weak convergence and weak-* convergence
	9.3 Weak-* sequential compactness
	9.4* Reflexive spaces and the Eberlein–Šmulian theorem
	Exercises
10. Sobolev spaces
	10.1 Generalized derivatives
	10.2 The space W^{m,p} (\\omega)
	10.3 Representations of functionals
	10.4 Modifiers
	10.5 Some important properties of Sobolev spaces and embedding theorems
		Extension Theorem
		Approximation Theorem
		Poincaré’s Inequality
		Embedding Theorems
		Compact Embeddings
	10.6 The Euler–Lagrange equation
	Exercises
11. Weak lower semi-continuity
	11.1 Convex sets and convex functions
	11.2 Convexity and weak lower semi-continuity
	11.3 An existence theorem
	11.4* Quasi-convexity
	Exercises
12. Boundary value problems and eigenvalue problems of linear differential equations
	12.1 Linear boundary value problems and orthogonal projections
	12.2 The eigenvalue problems
	12.3 The eigenfunction expansions
	12.4 The minimax description of eigenvalues
		Courant’s Min-Max Theorem
	Exercises
13. Existence and regularity
	13.1 Regularity (n = 1)
	13.2 More on regularity (n > 1)
	13.3 The solutions of some variational problems
	13.4 The limitations of calculus of variations
	Exercises
14. The dual least action principle and the Ekeland variational principle
	14.1 The conjugate function of a convex function
	14.2 The dual least action principle
	14.3 The Ekeland variational principle
	14.4 The Fréchet derivative and the Palais–Smale condition
	14.5 The Nehari technique
	Exercises
15. The Mountain Pass Theorem, its generalizations, and applications
	15.1 The Mountain Pass Theorem
	15.2 Applications
16. Periodic solutions, homoclinic and heteroclinic orbits
	16.1 The simple pendulum
	16.2 Periodic solutions
	16.3 Heteroclinic orbits
	16.4 Homoclinic orbits
17. Geodesics and minimal surfaces
	17.1 Geodesics
	17.2 Minimal surfaces
18. Numerical methods for variational problems
	18.1 The Ritz method
	18.2 The finite element method
	18.3 Cea’s theorem
	18.4 An optimization method —the conjugate gradient method
19. Optimal control problems
	19.1 The formulation of problems
	19.2 The Pontryagin Maximal Principle
	19.3 The Bang-Bang principle
20. Functions of bounded variations and image processing
	20.1 Functions of bounded variations in one variable— a review
	20.2 Functions of bounded variations in several variables
	20.3 The relaxation function
	20.4 Image restoration and the Rudin–Osher–Fatemi model
Bibliography
Index
[Book Review]Lecture notes on calculus of variations by Kung Ching Chang; translated by Tan Zhang pp. 324, $56.00 (paper), ISBN 978-9-81314-623-5, World Scientific (2017).