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Calculus of Variations

Calculus of Variations

Part of Cambridge Studies in Advanced Mathematics

  • Authors:
  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig
  • Xianqing Li-Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig
  • Date Published: April 2008
  • availability: Available
  • format: Paperback
  • isbn: 9780521057127

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  • 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.

    • Plenty of new material, much of it basic
    • Relevant for many applications in physics and engineering
    • Many key examples treated in detail
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    Reviews & endorsements

    "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

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    Product details

    • Date Published: April 2008
    • format: Paperback
    • isbn: 9780521057127
    • length: 340 pages
    • dimensions: 229 x 152 x 19 mm
    • weight: 0.51kg
    • availability: Available
  • 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.

  • Authors

    Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig

    Xianqing Li-Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig

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