Calculus of Variations

by Jürgen Jost , Xianqing Li-Jost

• ISBN13: 9780521642033
• ISBN10: 0521642035
• Format: Hardcover
• Copyright: 1999-01-28
• Publisher: Cambridge University Press

Summary

This textbook on the calculus of variations leads the reader from the basics to modern aspects of the theory. One-dimensional problems and the classical issues such as Euler-Lagrange equations are treated, as are Noether's theorem, Hamilton-Jacobi theory, and in particular geodesic lines, thereby developing some important geometric and topological aspects. The basic ideas of optimal control theory are also given. The second part of the book deals with multiple integrals. After a review of Lebesgue integration, Banach and Hilbert space theory and Sobolev spaces (with complete and detailed proofs), there is a treatment of the direct methods and the fundamental lower semicontinuity theorems. Subsequent chapters introduce the basic concepts of the modern calculus of variations, namely relaxation, Gamma convergence, bifurcation theory and minimax methods based on the Palais-Smale condition. The prerequisites are knowledge of the basic results from calculus of one and several variables. After having studied this book, the reader will be well equipped to read research papers in the calculus of variations.

Table of Contents

Part I. One-Dimensional Variational Problems: 1. The classical theory

2. Geodesic curves

3. Saddle point constructions

4. The theory of Hamilton and Jacobi

5. Dynamic optimization

Part II. Multiple Integrals in the Calculus of Variations: 6. Lebesgue integration theory

7. Banach spaces

8. Lp and Sobolev spaces

9. The direct methods

10. Nonconvex functionals: relaxation

11. G-convergence

12. BV-functionals and G-convergence: the example of Modica and Mortola

Appendix A. The coarea formula

Appendix B. The distance function from smooth hypersurfaces

13. Bifurcation theory

14. The Palais-Smale condition and unstable critical points of variational problems.