Calculus of Variations (Cambridge Studies in Advanced Mathematics)

Jost, Jürgen/ Li-Jost, Xianqing

Cambridge University Press（2008/03発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 340 p.
• 言語 ENG
• 商品コード 9780521057127
• DDC分類 515.64

基本説明

New in paperback. Hardcover was published in 1999. Starting with classical issues like Euler-Lagrange equations, important geometric and topological aspects are developed and the basics of optimal control thoery are given.

Full Description

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.

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.