Cartesian currents in the calculus of variations

著者

• Giaquinta, Mariano
• Modica, Giuseppe
• Souček, Jiří

書誌事項

Cartesian currents in the calculus of variations

Mariano Giaquinta, Giuseppe Modica, Jiří Souček

（Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 37-38）

Springer, c1998

• v. 1
• v. 2

注記

Vol.1. Cartesian currents. -- v. 2. Variational integrals.

Includes bibliographical references and index

内容説明・目次

巻冊次

v. 1 ISBN 9783540640097

内容説明

This monograph (in two volumes) deals with non scalar variational problems arising in geometry, as harmonic mappings between Riemannian manifolds and minimal graphs, and in physics, as stable equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and accessible to non specialists. Topics are treated as far as possible in an elementary way, illustrating results with simple examples; in principle, chapters and even sections are readable independently of the general context, so that parts can be easily used for graduate courses. Open questions are often mentioned and the final section of each chapter discusses references to the literature and sometimes supplementary results. Finally, a detailed Table of Contents and an extensive Index are of help to consult this monograph

目次

Part I: General Measure Theory.- Integer Rectifiable Currents.- Cartesian Maps.- Cartesian Currents in Euclidean Spaces.- Cartesian Currents in Riemannian Manifolds.- Part II: Regular Variational Integrals.- Finite Elasticity and Weak Diffeomorphisms.- The Dirichlet Integral in Sobolev Spaces.- The Dirichlet Energy for Maps into S2.- Regular and Non Regular Integrals.- The Non Parametric Area Functional.

巻冊次

v. 2 ISBN 9783540640103

内容説明

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.

目次

1. Regular Variational Integrals.- 2. Finite Elasticity and Weak Diffeomorphisms.- 3. The Dirichlet Integral in Sobolev Spaces.- 4. The Dirichlet Energy for Maps into S2.- 5. Some Regular and Non Regular Variational Problems.- 6. The Non Parametric Area Functional.- Symbols.

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詳細情報

• NII書誌ID(NCID): BA37405138
• ISBN:
  • 3540640096
  • 354064010X
• LCCN: 98018195
• 出版国コード: gw
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Berlin
• ページ数/冊数: 2 v.
• 大きさ: 25 cm
• 分類:
  • LCC : QA316
  • DC21 : 515/.64
  • NDC8 : 413.3
• 件名:
  • LCSH : Calculus of variations
• 親書誌ID:
  • BA00035552