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El. knyga: Pullback Equation for Differential Forms

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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 so that it satisfies the pullback equation: *(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 n1. 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 HodgeMorrey 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 n1 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 serveas a valuable reference for researchers or a supplemental text for graduate courses or seminars.

Recenzijos

From the reviews:

This monograph provides a systematic study of the pullback equation, presenting results on local and global existence of solutions and regularity. It is very likely that this book will become an indispensable reference and source of inspiration for everybody interested in this subject. The book starts with an introductory chapter which serves as a users guide for the rest of the book . The book is completed by an index and a list of references consisting of over 100 entries. (Pietro Celada, Mathematical Reviews, April, 2013)

This book studies the pullback equation for differential forms . The principal emphasis of this book is put upon regularity and boundary conditions. Special attention has been paid upon getting optimal regularity, which requires estimates for elliptic equations and fine properties of Hölder spaces. The book will presumably appeal to both geometers and analysts. (Hirokazu Nishimura, Zentralblatt MATH, Vol. 1247, 2012)

1 Introduction
1(32)
1.1 Statement of the Problem
1(2)
1.2 Exterior and Differential Forms
3(7)
1.2.1 Definitions and Basic Properties of Exterior Forms
3(3)
1.2.2 Divisibility
6(1)
1.2.3 Differential Forms
7(3)
1.3 Hodge-Morrey Decomposition and Poincare Lemma
10(5)
1.3.1 A General Identity and Gaffney Inequality
10(1)
1.3.2 The Hodge-Morrey Decomposition
11(1)
1.3.3 First-Order Systems of Cauchy-Riemann Type
12(1)
1.3.4 Poincare Lemma
13(2)
1.4 The Case of Volume Forms
15(5)
1.4.1 Statement of the Problem
15(2)
1.4.2 The One-Dimensional Case
17(1)
1.4.3 The Case f.g > 0
18(1)
1.4.4 The Case with No Sign Hypothesis on f
19(1)
1.5 The Case 0 ≤ k ≤ n - 1
20(5)
1.5.1 The Flow Method
20(1)
1.5.2 The Cases k = 0 and k = 1
21(1)
1.5.3 The Case k = 2
22(2)
1.5.4 The Case 3 ≤ k ≤ n - 1
24(1)
1.6 Holder Spaces
25(8)
1.6.1 Definition and Extension of Holder Functions
25(2)
1.6.2 Interpolation, Product, Composition and Inverse
27(1)
1.6.3 Smoothing Operator
28(5)
Part I Exterior and Differential Forms
2 Exterior Forms and the Notion of Divisibility
33(42)
2.1 Definitions
34(12)
2.1.1 Exterior Forms and Exterior Product
34(2)
2.1.2 Scalar Product, Hodge Star Operator and Interior Product
36(3)
2.1.3 Pullback and Dimension Reduction
39(3)
2.1.4 Canonical Forms for 1, 2, (n - 2) and (n - 1)-Forms
42(4)
2.2 Annihilators, Rank and Corank
46(11)
2.2.1 Exterior and Interior Annihilators
46(2)
2.2.2 Rank and Corank
48(5)
2.2.3 Properties of the Rank of Order 1
53(4)
2.3 Divisibility
57(18)
2.3.1 Definition and First Properties
57(6)
2.3.2 Main Result
63(4)
2.3.3 Some More Results
67(4)
2.3.4 Proof of the Main Theorem
71(4)
3 Differential Forms
75(16)
3.1 Notations
75(4)
3.2 Tangential and Normal Components
79(8)
3.3 Gauss-Green Theorem and Integration-by-Parts Formula
87(4)
4 Dimension Reduction
91(10)
4.1 Frobenius Theorem
91(2)
4.2 Reduction Theorem
93(8)
Part II Hodge-Morrey Decomposition and Poincare Lemma
5 An Identity Involving Exterior Derivatives and Gaffney Inequality
101(20)
5.1 Introduction
101(2)
5.2 An Identity Involving Exterior Derivatives
103(10)
5.2.1 Preliminary Formulas
103(4)
5.2.2 The Main Theorem
107(6)
5.3 Gaffney Inequality
113(8)
5.3.1 An Elementary Proof
113(2)
5.3.2 A Generalization of the Boundary Condition
115(3)
5.3.3 Gaffney-Type Inequalities in Lp and Holder Spaces
118(3)
6 The Hodge-Morrey Decomposition
121(14)
6.1 Properties of Harmonic Fields
121(3)
6.2 Existence of Minimizers and Euler-Lagrange Equation
124(3)
6.3 The Hodge-Morrey Decomposition
127(3)
6.4 Higher Regularity
130(5)
7 First-Order Elliptic Systems of Cauchy-Riemann Type
135(12)
7.1 System with Prescribed Tangential Component
135(5)
7.2 System with Prescribed Normal Component
140(2)
7.3 Weak Formulation for Closed Forms
142(3)
7.4 Equivalence Between Hodge Decomposition and Cauchy-Riemann-Type Systems
145(2)
8 Poincare Lemma
147(32)
8.1 The Classical Poincare Lemma
147(1)
8.2 Global Poincare Lemma with Optimal Regularity
148(2)
8.3 Some Preliminary Lemmas
150(7)
8.4 Poincare Lemma with Dirichlet Boundary Data
157(4)
8.5 Poincare Lemma with Constraints
161(18)
8.5.1 A First Result
161(1)
8.5.2 A Second Result
161(5)
8.5.3 Some Technical Lemmas
166(13)
9 The Equation div u = f
179(12)
9.1 The Main Theorem
179(2)
9.2 Regularity of Divergence-Free Vector Fields
181(1)
9.3 Some More Results
182(9)
9.3.1 A First Result
182(2)
9.3.2 A Second Result
184(7)
Part III The Case k = n
10 The Case f.g > 0
191(20)
10.1 The Main Theorem
191(2)
10.2 The Flow Method
193(5)
10.3 The Fixed Point Method
198(3)
10.4 Two Proofs of the Main Theorem
201(8)
10.4.1 First Proof
201(5)
10.4.2 Second Proof
206(3)
10.5 A Constructive Method
209(2)
11 The Case Without Sign Hypothesis on f
211(44)
11.1 Main Result
211(2)
11.2 Remarks and Related Results
213(4)
11.3 Proof of the Main Result
217(5)
11.4 Radial Solution
222(7)
11.5 Concentration of Mass
229(6)
11.6 Positive Radial Integration
235(20)
Part IV The Case 0 ≤ k ≤ n - 1
12 General Considerations on the Flow Method
255(12)
12.1 Basic Properties of the Flow
255(3)
12.2 A Regularity Result
258(3)
12.3 The Flow Method
261(6)
13 The Cases k = 0 and k = 1
267(18)
13.1 The Case of 0-Forms and of Closed 1-Forms
267(4)
13.1.1 The Case of 0-Forms
267(2)
13.1.2 The Case of Closed 1-Forms
269(2)
13.2 Darboux Theorem for 1-Forms
271(14)
13.2.1 Main Results
271(5)
13.2.2 A Technical Result
276(9)
14 The Case k = 2
285(34)
14.1 Notations
285(1)
14.2 Local Result for Forms with Maximal Rank
286(4)
14.3 Local Result for Forms of Nonmaximal Rank
290(2)
14.3.1 The Theorem and a First Proof
290(1)
14.3.2 A Second Proof
291(1)
14.4 Global Result with Dirichlet Data
292(27)
14.4.1 The Main Result
292(2)
14.4.2 The Flow Method
294(1)
14.4.3 The Key Estimate for Regularity
295(7)
14.4.4 The Fixed Point Method
302(6)
14.4.5 A First Proof of the Main Theorem
308(6)
14.4.6 A Second Proof of the Main Theorem
314(5)
15 The Case 3 ≤ k ≤ n - 1
319(16)
15.1 A General Theorem for Forms of Rank = k
319(2)
15.2 The Case of (n - 1)-Forms
321(3)
15.2.1 The Case of Closed (n - 1)-Forms
321(1)
15.2.2 The Case of Nonclosed (n - 1)-Forms
322(2)
15.3 Simultaneous Resolutions and Applications
324(11)
15.3.1 Simultaneous Resolution for 1-Forms
324(2)
15.3.2 Applications to k-Forms
326(9)
Part V Holder Spaces
16 Holder Continuous Functions
335(72)
16.1 Definitions of Continuous and Holder Continuous Functions
335(6)
16.1.1 Definitions
335(3)
16.1.2 Regularity of Boundaries
338(1)
16.1.3 Some Elementary Properties
339(2)
16.2 Extension of Continuous and Holder Continuous Functions
341(15)
16.2.1 The Main Result and Some Corollaries
341(4)
16.2.2 Preliminary Results
345(8)
16.2.3 Proof of the Main Theorem
353(3)
16.3 Compact Imbeddings
356(2)
16.4 A Lower Semicontinuity Result
358(1)
16.5 Interpolation and Product
359(10)
16.5.1 Interpolation
359(7)
16.5.2 Product and Quotient
366(3)
16.6 Composition and Inverse
369(6)
16.6.1 Composition
369(1)
16.6.2 Inverse
370(2)
16.6.3 A Further Result
372(3)
16.7 Difference of Composition
375(11)
16.7.1 A First Result
376(1)
16.7.2 A Second Result
377(7)
16.7.3 A Third Result
384(2)
16.8 The Smoothing Operator
386(10)
16.8.1 The Main Theorem
386(4)
16.8.2 A First Application
390(3)
16.8.3 A Second Application
393(3)
16.9 Smoothing Operator for Differential Forms
396(11)
Part VI Appendix
17 Necessary Conditions
407(6)
18 An Abstract Fixed Point Theorem
413(4)
19 Degree Theory
417(8)
19.1 Definition and Main Properties
417(1)
19.2 General Change of Variables Formula
418(2)
19.3 Local and Global Invertibility
420(5)
References 425(4)
Further Reading 429(2)
Notations 431(4)
Index 435
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