Calculus of Variations

This textbook on the calculus of variations covers from the basics to the modern aspects of the theory.

Jürgen Jost (Author), Xianqing Li-Jost (Author)

9780521057127, Cambridge University Press

Paperback / softback, published 27 March 2008

340 pages
22.9 x 15.2 x 1.9 cm, 0.51 kg

"This modern self-contained exposition...is an excellent textbook for graduate students and a good source of information in the calculus of variations." Mathematical Reviews

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 like 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 only prerequisites are 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.

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.

Subject Areas: Calculus of variations [PBKQ]