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Cartesian Currents in the Calculus of Variations II

Variational Integrals

  • Book
  • © 1998

Overview

Authors:
  1. Mariano Giaquinta

    1. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

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  2. Giuseppe Modica

    1. Dipartimento di Matematica Applicata, Università di Firenze, Firenze, Italy

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  3. Jiří Souček

    1. Faculty of Mathematics and Physics, Charles University, Praha 8, Čzech Republic

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  • Deals with non scalar variational problems arising in geometry
  • Selfcontained presentation
  • Accessible to non specialists
  • The two volumes are readable independently
  • Chapters and even sections readable independently of the general context
  • Elementary treatment
  • Illustrated with simple examples
  • Detailed Table of Contents and extensive Index

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xxiv
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  2. Regular Variational Integrals

    • Mariano Giaquinta, Giuseppe Modica, Jiří Souček
    Pages 1-135

  3. Finite Elasticity and Weak Diffeomorphisms

    • Mariano Giaquinta, Giuseppe Modica, Jiří Souček
    Pages 137-280

  4. The Dirichlet Integral in Sobolev Spaces

    • Mariano Giaquinta, Giuseppe Modica, Jiří Souček
    Pages 281-352

  5. The Dirichlet Energy for Maps into the Two Dimensional Sphere

    • Mariano Giaquinta, Giuseppe Modica, Jiří Souček
    Pages 353-465

  6. Some Regular and Non Regular Variational Problems

    • Mariano Giaquinta, Giuseppe Modica, Jiří Souček
    Pages 467-561

  7. The Non Parametric Area Functional

    • Mariano Giaquinta, Giuseppe Modica, Jiří Souček
    Pages 563-652

  8. Back Matter

    Pages 653-700
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Keywords

About this book

Non-scalar variational problems appear in different fields. In geometry, for in­ stance, we encounter the basic problems of harmonic maps between Riemannian manifolds and of minimal immersions; related questions appear in physics, for example in the classical theory of a-models. Non linear elasticity is another example in continuum mechanics, while Oseen-Frank theory of liquid crystals and Ginzburg-Landau theory of superconductivity require to treat variational problems in order to model quite complicated phenomena. Typically one is interested in finding energy minimizing representatives in homology or homotopy classes of maps, minimizers with prescribed topological singularities, topological charges, stable deformations i. e. minimizers in classes of diffeomorphisms or extremal fields. In the last two or three decades there has been growing interest, knowledge, and understanding of the general theory for this kind of problems, often referred to as geometric variational problems. Due to the lack of a regularity theory in the non scalar case, in contrast to the scalar one - or in other words to the occurrence of singularities in vector valued minimizers, often related with concentration phenomena for the energy density - and because of the particular relevance of those singularities for the problem being considered the question of singling out a weak formulation, or completely understanding the significance of various weak formulations becames non trivial.

Authors and Affiliations

  • Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Mariano Giaquinta

  • Dipartimento di Matematica Applicata, Università di Firenze, Firenze, Italy

    Giuseppe Modica

  • Faculty of Mathematics and Physics, Charles University, Praha 8, Čzech Republic

    Jiří Souček

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