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E-raamat: Sobolev Maps to the Circle: From the Perspective of Analysis, Geometry, and Topology

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Sobolev Maps to the Circle: From the Perspective of Analysis, Geometry, and TopologySuurem pilt
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The theory of real-valued Sobolev functions is a classical part of analysis and has a wide range of applications in pure and applied mathematics. By contrast, the study of manifold-valued Sobolev maps is relatively new. The incentive to explore these spaces arose in the last forty years from geometry and physics.  This monograph is the first to provide a unified, comprehensive treatment of Sobolev maps to the circle, presenting numerous results obtained by the authors and others. Many surprising connections to other areas of mathematics are explored, including the Monge-Kantorovich theory in optimal transport, items in geometric measure theory, Fourier series, and non-local functionals occurring, for example, as denoising filters in image processing. Numerous digressions provide a glimpse of the theory of sphere-valued Sobolev maps. 

Each chapter focuses on a single topic and starts with a detailed overview, followed by the most significant results, and rather complete proofs. The “Complements and Open Problems” sections provide short introductions to various subsequent developments or related topics, and suggest new
directions of research.  Historical perspectives and a comprehensive list of references close out each chapter.  Topics covered include lifting, point and line singularities, minimal connections and minimal surfaces, uniqueness spaces, factorization, density, Dirichlet problems, trace theory, and gap phenomena.

Sobolev Maps to the Circle will appeal to mathematicians working in various areas, such as nonlinear analysis, PDEs, geometric analysis, minimal surfaces, optimal transport, and topology.  It will also be of interest to physicists working on liquid crystals and the Ginzburg-Landau theory of superconductors.

Arvustused

A nice and unique feature of the book that every chapter contains a section on Complements and open problems. All in all, I find this text very well written and it would definitely be recommended to graduate students and researchers who would like to learn about the fine properties of Sobolev functions valued in the circle. (Alpár R. Mészáros, zbMATH 1501.46001, 2023)

This monograph offers a rigorous discussion to a fascinating topic of mathematics with multiple relevant applications to various fields. Also for this reason it is highly recommended both for mathematicians and physicists working in the various fields involved in the theory of Sobolev maps to the circle . the book may also be useful for Ph.D students who are interested in the topic, and the content of the individual chapters could be used in advanced courses and seminars. (Antonia Chinnì, Mathematical Reviews, February, 2023)

Preface vii
Overview xi
Notation xxiii
Contents xxvii
1 Lifting in W1,p
1(64)
1.0 Introduction
1(4)
1.1 Warm up. Lifting of continuous maps: local and global aspects
5(6)
1.2 Lifting of §1-valued maps in W1,p Definition of Ju
11(11)
1.3 Square root
22(1)
1.4 Lifting of §1-valued maps in W1,1 and BV. Definition of Σ(u)
23(14)
1.5 Σ(u) computed via duality: the basic relation between Σ{u) and Ju
37(7)
1.6 An excursion into Monge--Kantorovich (=MK) territory
44(5)
1.7 Relaxed energy
49(2)
1.8 Least energy with prescribed Jacobian
51(1)
1.9 Complements and open problems
52(4)
1.10 Comments
56(9)
2 The geometry of Ju and Σ(u) in 2D; point singularities and minimal connections
65(68)
2.0 Introduction
65(6)
2.1 Ju as a sum of Dirac masses
71(5)
2.2 Where optimal transport (=OT) enters: the quantity L(a, d) and minimal configurations
76(12)
2.3 Returning to u: Σ(h) = 2πL(a, d)
88(3)
2.4 Σ(u) = S1(u) ≥ 2π L(a, d) via the coarea formula
91(3)
2.5 Σ(u) = S1(u) ≤ 2π L(a, d) via the dipole construction
94(2)
2.6 Σ(u) ≤ ∫Ω|Vu| + 2πL(a, d) via minimal configurations
96(5)
2.7 Connections associated with (a, d). Minimal connections
101(4)
2.8 Returning to u: a bijective correspondence between BV liftings and connections
105(2)
2.9 Describing Ju and Σ(u) for a general u W1,1
107(12)
2.10 An integral representation of the distribution Ju
119(4)
2.11 Complements and open problems
123(5)
2.12 Comments
128(5)
3 The geometry of Ju and Σ(u) in 3D (and higher); line singularities and minimal surfaces
133(40)
3.0 Introduction
133(2)
3.1 Examples. Ju as path integration
135(7)
3.2 Σ(u) as a least area: E(u) = 2πA0(Γ)
142(3)
3.3 Σ(u) ≥ 2πA0(Γ) via the coarea formula
145(3)
3.4 Σ(u) ≤ 2πA0(Γ) via the dipole construction
148(1)
3.5 Least area spanned by a contour from the perspective of MK
149(1)
3.6 The structure of Ju for a general u W1,1. Where Federer encounters Kantorovich
150(7)
3.7 Further properties when Ju is a measure
157(3)
3.8 Complements and open problems
160(10)
3.9 Comments
170(3)
4 A digression: sphere-valued maps
173(32)
4.0 Introduction
173(4)
4.1 The "historical" case: N = 3 and k = 2, where everything fits into place!
177(6)
4.2 A distinguished class of currents. Definition of F1
183(3)
4.3 The case N = 4 and k = 2; where complications appear
186(4)
4.4 The general case: N ≥ 2 and 1 ≤ k ≤ N - 1
190(6)
4.5 Complements and open problems
196(5)
4.6 Comments
201(4)
5 Lifting in fractional Sobolev spaces and in VMO
205(22)
5.0 Introduction
205(1)
5.1 Lifting of §1-valued maps in fractional Sobolev spaces
206(10)
5.2 Lifting of §1-valued maps in VMO and BMO
216(7)
5.3 Lifting in Ws,p, sp < 1, upgraded
223(1)
5.4 Complements and open problems
224(1)
5.5 Comments
225(2)
6 Uniqueness of lifting and beyond
227(26)
6.0 Introduction
227(1)
6.1 Constancy in VMO (Ω; Z)
228(1)
6.2 Constancy in W1,1 (Ω; Z)
229(1)
6.3 Constancy in W1/p-p(Ω; Z), 1 < p < ∞
229(5)
6.4 Connectedness of the essential range
234(1)
6.5 A new function space. Applications to sums
235(5)
6.6 Proof of Theorem 6.2 (the BBM formula)
240(2)
6.7 Complements and open problems
242(8)
6.8 Comments
250(3)
7 Factorization
253(26)
7.0 Introduction
253(2)
7.1 Proof of Theorem 7.2
255(3)
7.2 Outline of the proof of Theorem 7.1
258(1)
7.3 A glimpse of the theory of weighted Sobolev spaces
259(1)
7.4 Proof of Theorem 7.1
260(14)
7.5 Complements and open problems
274(3)
7.6 Comments
277(2)
8 Applications of the factorization
279(20)
8.0 Introduction
279(1)
8.1 Existence of Ju for u in W1/p,p
280(5)
8.2 Lifting revisited
285(3)
8.3 Least energy with prescribed Jacobian
288(1)
8.4 Relaxed energy
289(2)
8.5 Square root
291(1)
8.6 Minimizing the BV part of the phase
292(2)
8.7 Complements and open problems
294(3)
8.8 Comments
297(2)
9 Estimates of phases: positive and negative results
299(12)
9.0 Introduction
299(1)
9.1 N ≥ 1 and s ≥ 1
300(1)
9.2 N ≥ 1 and sp < 1
300(2)
9.3 N = 1, s = 1/p, and p < 1
302(1)
9.4 N = 1, 0 < 5 < 1, and sp < 1
303(1)
9.5 N ≥ 2, 0 < 5 < 1, and sp ≤ N
304(1)
9.6 N ≥ 2, 0 < 5 < 1, and 1 ≤ sp < N
304(2)
9.7 Complements and open problems
306(2)
9.8 Comments
308(3)
10 Density
311(20)
10.0 Introduction
311(1)
10.1 When are smooth maps dense?
312(1)
10.2 Density of 3%. Answer to Question 1
313(5)
10.3 Characterization of C00(Q; S1) Answer to Question 2
318(2)
10.4 Weak sequential density. Answer to Question 3
320(1)
10.5 Distance to smooth maps
320(4)
10.6 Complements and open problems
324(3)
10.7 Comments
327(4)
11 Traces
331(8)
11.0 Introduction
331(1)
11.1 Proof of Theorem 11.1
332(2)
11.2 A sharp form of the extension problem
334(1)
11.3 Complements and open problems
334(2)
11.4 Comments
336(3)
12 Degree
339(42)
12.0 Introduction
339(1)
12.1 Degree and VMO
340(4)
12.2 Degree, lifting, and traces in Ws,p(§; §1)
344(1)
12.3 Integral representations for the degree
345(5)
12.4 A "distributional degree"
350(2)
12.5 Estimates for the degree
352(6)
12.6 Homotopy classes
358(1)
12.7 Degree and Fourier coefficients
358(6)
12.8 Complements and open problems
364(15)
12.9 Comments
379(2)
13 Dirichlet problems. Gaps. Infinite energies
381(22)
13.0 Introduction
381(1)
13.1 Minimizing the W1,p energy when p ≥ 2
382(1)
13.2 Minimizing the W1,p energy when 1 < p < 2
383(3)
13.3 Gaps
386(2)
13.4 More about minimizers when Ω = D and 1 < p < 2
388(2)
13.5 Further regularity and no gap for p < 2, near p = 2, when N = 2
390(4)
13.6 Complements and open problems
394(7)
13.7 Comments
401(2)
14 Domains with topology
403(28)
14.0 Introduction
403(1)
14.1 Lifting in W1,p(Ω; §1) revisited
403(6)
14.2 Relaxed energy revisited
409(8)
14.3 Lifting in Ws,p(Ω; §1) revisited
417(4)
14.4 Density in Ws,p(Ω; §1) revisited
421(2)
14.5 Homotopy classes
423(2)
14.6 Complements and open problems
425(5)
14.7 Comments
430(1)
15 Appendices
431(92)
15.1 Sobolev spaces
431(4)
15.2 Sobolev embeddings and Gagliardo--Nirenberg inequalities
435(6)
15.3 Composition in Sobolev spaces
441(2)
15.4 Standard (and non-standard) examples of maps in Sobolev spaces
443(8)
15.5 Further results on BMO and VMO
451(10)
15.6 Enlarging Ω
461(6)
15.7 Fine theory of BV maps
467(8)
15.8 Description of (minimal) connections associated with (a, d)
475(14)
15.9 A pathological case
489(4)
15.10 Jacobians of W1,p(Ω; §) maps
493(3)
15.11 When Ju is a measure
496(1)
15.12 Products in fractional Sobolev spaces
497(2)
15.13 Maximal estimates
499(1)
15.14 Smoothing
500(23)
References
507(16)
Symbol Index 523(2)
Subject Index 525(2)
Author Index 527
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