Lecture Notes on Calculus of Variations

This is based on the course "Calculus of Variations" taught at Peking University from 2006 to 2010 for advanced undergraduate to graduate students majoring in mathematics. The book contains 20 lectures covering both the theoretical background material as well as an abundant collection of applications. Lectures 1–8 focus on the classical theory of calculus of variations. Lectures 9–14 introduce direct methods along with their theoretical foundations. Lectures 15–20 showcase a broad collection of applications. The book offers a panoramic view of the very important topic on calculus of variations. This is a valuable resource not only to mathematicians, but also to those students in engineering, economics, and management, etc.

Contents:

• The Theory and Problems of Calculus of Variations
• The Euler–Lagrange Equation
• The Necessary Condition and the Sufficient Condition on Extremal Values of Functionals
• Strong Minima and Extremal Fields
• The Hamilton–Jacobi Theory
• Variational Problems Involving Multivariate Integrals
• Constrained Variational Problems
• The Conservation Law and Noether's Theorem
• Direct Methods
• Sobolev Spaces
• Weak Lower Semi-Continuity
• Boundary Value Problems and Eigenvalue Problems of Linear Differential Equations
• Existence and Regularity
• The Dual Least Action Principle and the Ekeland Variational Principle
• The Mountain Pass Theorem, Its Generalizations, and Applications
• Periodic Solutions, Homoclinic and Heteroclinic Orbits
• Geodesics and Minimal Surfaces
• Numerical Methods for Variational Problems
• Optimal Control Problems
• Functions of Bounded Variations and Image Processing

Readership: Undergraduates in calculus of variations.
Key Features:

• It is a unique set of lecture notes for a unique course only taught at the Peking University from 2006 to 2010
• It gives a nice blend of both theoretical and applicable treatment on calculus of variations