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The Pullback Equation for Differential Forms / by Gyula Csató, Bernard Dacorogna, Olivier Kneuss
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Progress in Nonlinear Differential Equations and Their Applications ; 83 | SpringerLink BücherPublisher: Boston : Springer Science+Business Media, LLC, 2012Description: Online-Ressource (XI, 436p, digital)ISBN:- 9780817683139
- 1283444445
- 9781283444446
- Differentialform
- Hodge-Zerlegung
- Hölder-Raum
- Mathematics
- Differential equations, partial
- Global differential geometry
- Algebra
- Differential geometry
- Partial differential equations
- Matrix theory
- Differential Equations
- Differential forms
- Differential equations, Nonlinear Numerical solutions
- Global differential geometry
- 515/.37 23
- 515.353
- 510
- QA370-380
- QA381
Contents:
Summary: Introduction -- Part I Exterior and Differential Forms -- Exterior Forms and the Notion of Divisibility -- Differential Forms -- Dimension Reduction -- Part II Hodge-Morrey Decomposition and Poincaré Lemma -- An Identity Involving Exterior Derivatives and Gaffney Inequality -- The Hodge-Morrey Decomposition -- First-Order Elliptic Systems of Cauchy-Riemann Type -- Poincaré Lemma -- The Equation div u = f -- Part III The Case k = n -- The Case f × g > 0 -- The Case Without Sign Hypothesis on f -- Part IV The Case 0 ≤ k ≤ n–1 -- General Considerations on the Flow Method -- The Cases k = 0 and k = 1 -- The Case k = 2 -- The Case 3 ≤ k ≤ n–1 -- Part V Hölder Spaces -- Hölder Continuous Functions -- Part VI Appendix -- Necessary Conditions -- An Abstract Fixed Point Theorem -- Degree Theory -- References -- Further Reading -- Notations -- Index. .Summary: An important question in geometry and analysis is to know when two k-forms f and g are equivalent through a change of variables. The problem is therefore to find a map φ so that it satisfies the pullback equation: φ*(g) = f. In more physical terms, the question under consideration can be seen as a problem of mass transportation. The problem has received considerable attention in the cases k = 2 and k = n, but much less when 3 ≤ k ≤ n–1. The present monograph provides the first comprehensive study of the equation. The work begins by recounting various properties of exterior forms and differential forms that prove useful throughout the book. From there it goes on to present the classical Hodge–Morrey decomposition and to give several versions of the Poincaré lemma. The core of the book discusses the case k = n, and then the case 1≤ k ≤ n–1 with special attention on the case k = 2, which is fundamental in symplectic geometry. Special emphasis is given to optimal regularity, global results and boundary data. The last part of the work discusses Hölder spaces in detail; all the results presented here are essentially classical, but cannot be found in a single book. This section may serve as a reference on Hölder spaces and therefore will be useful to mathematicians well beyond those who are only interested in the pullback equation. The Pullback Equation for Differential Forms is a self-contained and concise monograph intended for both geometers and analysts. The book may serve as a valuable reference for researchers or a supplemental text for graduate courses or seminars.PPN: PPN: 1651096767Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
The Pullback Equation for Differential Forms; Preface; Contents; Chapter 1 Introduction; 1.1 Statement of the Problem; 1.2 Exterior and Differential Forms; 1.2.1 Definitions and Basic Properties of Exterior Forms; 1.2.2 Divisibility; 1.2.3 Differential Forms; 1.3 Hodge-Morrey Decomposition and Poincaré Lemma; 1.3.1 A General Identity and Gaffney Inequality; 1.3.2 The Hodge-Morrey Decomposition; 1.3.3 First-Order Systems of Cauchy-Riemann Type; 1.3.4 Poincar´e Lemma; 1.4 The Case of Volume Forms; 1.4.1 Statement of the Problem; 1.4.2 The One-Dimensional Case; 1.4.3 The Case f·g > 0
1.4.4 The Case with No Sign Hypothesis on f1.5 The Case 0 = k = n-1; 1.5.1 The Flow Method; 1.5.2 The Cases k= 0 and k=1; 1.5.3 The Case k = 2; 1.5.4 The Case 3 = k = n-1; 1.6 Hölder Spaces; 1.6.1 Definition and Extension of Hölder Functions; 1.6.2 Interpolation, Product, Composition and Inverse; 1.6.3 Smoothing Operator; Part I Exterior and Differential Forms; Chapter 2 Exterior Forms and the Notion of Divisibility; 2.1 Definitions; 2.1.1 Exterior Forms and Exterior Product; 2.1.2 Scalar Product, Hodge Star Operator and Interior Product; 2.1.3 Pullback and Dimension Reduction
Chapter 5 An Identity Involving Exterior Derivatives and Gaffney Inequality5.1 Introduction; 5.2 An Identity Involving Exterior Derivatives; 5.2.1 Preliminary Formulas; 5.2.2 The Main Theorem; 5.3 Gaffney Inequality; 5.3.1 An Elementary Proof; 5.3.2 A Generalization of the Boundary Condition; 5.3.3 Gaffney-Type Inequalities in Lp and Hölder Spaces; Chapter 6 The Hodge-Morrey Decomposition; 6.1 Properties of Harmonic Fields; 6.2 Existence of Minimizers and Euler-Lagrange Equation; 6.3 The Hodge-Morrey Decomposition; 6.4 Higher Regularity
Chapter 7 First-Order Elliptic Systems of Cauchy-Riemann Type7.1 System with Prescribed Tangential Component; 7.2 System with Prescribed Normal Component; 7.3 Weak Formulation for Closed Forms; 7.4 Equivalence Between Hodge Decomposition and Cauchy-Riemann-Type Systems; Chapter 8 Poincaré Lemma; 8.1 The Classical Poincaré Lemma; 8.2 Global Poincaré Lemma with Optimal Regularity; 8.3 Some Preliminary Lemmas; 8.4 Poincaré Lemma with Dirichlet Boundary Data; 8.5 Poincaré Lemma with Constraints; 8.5.1 A First Result; 8.5.2 A Second Result; 8.5.3 Some Technical Lemmas
Chapter 9 The Equation div u = f
2.1.4 Canonical Forms for 1, 2, (n-2) and (n-1)-Forms2.2 Annihilators, Rank and Corank; 2.2.1 Exterior and Interior Annihilators; 2.2.2 Rank and Corank; 2.2.3 Properties of the Rank of Order 1; 2.3 Divisibility; 2.3.1 Definition and First Properties; 2.3.2 Main Result; 2.3.3 Some More Results; 2.3.4 Proof of the Main Theorem; Chapter 3 Differential Forms; 3.1 Notations; 3.2 Tangential and Normal Components; 3.3 Gauss-Green Theorem and Integration-by-Parts Formula; Chapter 4 Dimension Reduction; 4.1 Frobenius Theorem; 4.2 Reduction Theorem; Part II Hodge-Morrey Decomposition and Poincaré Lemma
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