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### The Pullback Equation for Differential Forms by Bernard Dacorogna, Gyula Csató, Olivier Kneuss

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In the present book we study the pullback equation for differential forms namely, given two differential k-forms f and g we want to discuss the equivalence of such forms. This turns out to be a system of nonlinear first-order partial differential equations in the unknown map ?. 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 f so that it satisfies the pullback equation: f*(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.

Book Detail :-
Title: The Pullback Equation for Differential Forms
Edition:
Author(s): Bernard Dacorogna, Gyula Csató, Olivier Kneuss
Publisher: Birkhäuser Basel
Series: Progress in Nonlinear Differential Equations and Their Applications
Year: 2012
Pages: 449
Type: PDF
Language: English
ISBN: 9780817683122,0817683127
Country: Switzerland
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The author Bernard Dacorogna is a PHD and Professor Emeritus of Statistics at the University of Florida. He is the author of Advanced Calculus with Applications in Statistics, Second Edition and co-author of Statistical Tests for Mixed Linear Models, all published by Wiley. Dr. Khuri is a member of numerous academic associations, among them the American Statistical Association and the Institute of Mathematical Statistics.

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Book Contents :- The Pullback Equation for Differential Forms written by Bernard Dacorogna, Gyula Csató, Olivier Kneuss cover the following topics.
1. Introduction
Part I Exterior and Differential Forms
2. Exterior Forms and the Notion of Divisibility
3. Differential Forms
4. Dimension Reduction
Part II Hodge–Morrey Decomposition and Poincar´e Lemma
5. An Identity Involving Exterior Derivatives and Gaffney Inequality
6. The Hodge–Morrey Decomposition
7. First-Order Elliptic Systems of Cauchy–Riemann Type
8. Poincar´e Lemma
9. The Equation divu = f
10. The Case f · g > 0
11. The Case Without Sign Hypothesis on f
Part IV The Case 0 = k = n-1
13. The Cases k = 0 and k = 1
14. The Case k = 2
15. The Case 3 = k = n-1
Part V H¨older Spaces
16. H¨older Continuous Functions
Part VI Appendix
17. Necessary Conditions
18. An Abstract Fixed Point Theorem
19. Degree Theory
References
Notations
Index

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