Preface
Overview
Notation
Contents
1 Lifting in W1,p
1.0 Introduction
1.1 Warm up. Lifting of continuous maps: local and global aspects
1.2 Lifting of mathbbS1-valued maps in W1,p. Definition of Ju
1.3 Square root
1.4 Lifting of mathbbS1-valued maps in W1,1 and BV . Definition of Σ(u)
1.4.1 BV maps
1.4.2 W1,1 maps. Minimal liftings. Definition of Σ(u)
1.5 Σ(u) computed via duality: the basic relation between Σ(u) and Ju
1.6 An excursion into Monge–Kantorovich (=MK) territory
1.7 Relaxed energy
1.8 Least energy with prescribed Jacobian
1.9 Complements and open problems
1.9.1 Minimal liftings of BV maps (following Ignat) aut]Ignat
1.9.2 Duality and density revisited
1.9.3 Maps on Ω
1.9.4 Open problems
1.10 Comments
2 The geometry of Ju and Σ(u) in 2D; point singularities and minimal connections
2.0 Introduction
2.1 Ju as a sum of Dirac masses
2.2 Where optimal transport (=OT) enters: the quantity L(a,d) and minimal configurations
2.3 Returning to u: Σ(u)=2πL(a,d)
2.4 Σ(u)=S1(u)ge2π L(a,d) via the coarea formula
2.5 Σ(u)=S1(u)le2πL(a,d) via the dipole construction
2.6 E(u)leintΩ|u|+2πL(a,d) via minimal configurations
2.7 Connections associated with (a,d). Minimal connections
2.8 Returning to u: a bijective correspondence between BV liftings and connections
2.9 Describing Ju and Σ(u) for a general uinW1,1
2.9.1 The class calE
2.9.2 J u is an infinite sum of Dirac masses. And conversely
2.9.3 Connections and minimal connections revisited
2.9.4 The same story told in the language of geometric measure theory (= GMT)
2.9.5 The case where Ju is a measure
2.10 An integral representation of the distribution Ju
2.11 Complements and open problems
2.11.1 Minimizing the W1,1 energy with prescribed Jacobian
2.11.2 Describing J(W1,p(Ω; mathbbS1)) (following Bousquet) aut]Bousquet
2.11.3 Micromagnetics and mathbbS1-valued maps
2.11.4 Open problems
2.12 Comments
3 The geometry of Ju and Σ(u) in 3D (and higher); line singularities and minimal surfaces
3.0 Introduction
3.1 Examples. Ju as path integration
3.2 Σ(u) as a least area: Σ(u)=2πA0(Γ)
3.3 Σ(u)ge2πA0(Γ) via the coarea formula
3.4 Σ(u)le2πA0(Γ) via the dipole construction
3.5 Least area spanned by a contour from the perspective of MK
3.6 The structure of Ju for a general uinW1,1. Where Federer encounters Kantorovich
3.7 Further properties when Ju is a measure
3.8 Complements and open problems
3.8.1 More on the case N=3
3.8.2 Nge4
3.8.3 Describing J(W1,p(Ω; mathbbS1))
3.8.4 Open problems
3.9 Comments
4 A digression: sphere-valued maps
4.0 Introduction
4.1 The ``historical'' case: N=3 and k=2, where everything fits into place!
4.1.1 The structure of Ju for a general uinH1(Ω; mathbbS2) and one more formula for Σ(u)
4.1.2 Least energy with prescribed Jacobian: S2(u)=2Σ(u)
4.1.3 The relaxed energy: R2(u)=intΩ|u|2+2 Σ(u)
4.2 A distinguished class of currents. Definition of calFell
4.3 The case N=4 and k=2; where complications appear
4.3.1 A typical example
4.3.2 The structure of Ju for a general uinH1(Ω; mathbbS2)
4.3.3 The least energy S2(u) as a least area: S2(u)=2 Σast(u)
4.3.4 Relaxed energy: R2(u)=intΩ|u|2+S2(u)
4.4 The general case: Nge2 and 1lekleN-1
4.4.1 A typical example
4.4.2 The structure of Ju for a general uinW1,k(Ω; mathbbSk)
4.4.3 The least energy Sk(u) as a least area/mass
4.4.4 The relaxed energy
4.5 Complements and open problems
4.5.1 Density/non-density of Cinfty(overlineΩ; mathbbSk) in W1,p(Ω; mathbbSk)
4.5.2 More on the least energy with prescribed Jacobian
4.5.3 Open problems
4.6 Comments
5 Lifting in fractional Sobolev spaces and in VMO
5.0 Introduction
5.1 Lifting of mathbbS1-valued maps in fractional Sobolev spaces
5.1.1 Case 1: N=1
5.1.2 Case 2: N 2, 0 < s < 1
5.1.3 Case 3: N2, s1
5.2 Lifting of mathbbS1-valued maps in VMO and BMO
5.2.1 BMO and VMO
5.2.2 Lifting in BMO and VMO
5.3 Lifting in Ws,p, sp<1, upgraded
5.4 Complements and open problems
5.4.1 Lifting in covering spaces (following Bethuel and Chiron, Mironescu and Van Schaftingen)
5.4.2 Open problems
5.5 Comments
6 Uniqueness of lifting and beyond
6.0 Introduction
6.1 Constancy in VMO (Ω; mathbbZ)
6.2 Constancy in W1,1(Ω; mathbbZ)
6.3 Constancy in W1/p,p(Ω; mathbbZ), 1
6.4 Connectedness of the essential range
6.5 A new function space. Applications to sums
6.6 Proof of Theorem 6.2 (the BBM formula)
6.7 Complements and open problems
6.7.1 Non-local energies and perimeter
6.7.2 Γ-convergence of Iε
6.7.3 Another convex non-local approximation of the BV norm
6.7.4 A non-convex non-local approximation of the BV norm
6.7.5 Lp versions
6.7.6 Open problems
6.8 Comments
7 Factorization
7.0 Introduction
7.1 Proof of Theorem 7.2
7.2 Outline of the proof of Theorem 7.1
7.3 A glimpse of the theory of weighted Sobolev spaces
7.4 Proof of Theorem 7.1
7.5 Complements and open problems
7.5.1 An improvement of Theorem 7.1 when sp is an integer
7.5.2 Approach by duality
7.5.3 Open problems
7.6 Comments
8 Applications of the factorization
8.0 Introduction
8.1 Existence of Ju for u in W1/p,p
8.2 Lifting revisited
8.2.1 What we already know
8.2.2 Old result, new proof: lifting in Ws,p when sp<1
8.2.3 The new results
8.2.4 Where Ju=0 enters
8.3 Least energy with prescribed Jacobian
8.4 Relaxed energy
8.5 Square root
8.6 Minimizing the BV part of the phase
8.7 Complements and open problems
8.7.1 Describing J(Ws,p(Ω; mathbbS1))
8.7.2 Sphere-valued maps
8.8 Comments
9 Estimates of phases: positive and negative results
9.0 Introduction
9.1 Nge1 and sge1
9.2 Nge1 and sp<1
9.3 N=1, s=1/p, and p>1
9.4 N=1, 0
1
9.5 Nge2, 0
9.6 Nge2, 0
9.7 Complements and open problems
9.7.1 Further BMO -type estimates for the phase
9.7.2 Profile decomposition in W1/p, p(mathbbS1; mathbbS1)
9.7.3 Open problems
9.8 Comments
10 Density
10.0 Introduction
10.1 When are smooth maps dense?
10.2 Density of mathcalR. Answer to Question 1
10.3 Characterization of overlineCinfty(overlineΩ; mathbbS1)Ws,p. Answer to Question 2
10.4 Weak sequential density. Answer to Question 3
10.5 Distance to smooth maps
10.6 Complements and open problems
10.6.1 Density in W1,pg(Ω; mathbbS1)
10.6.2 Open problems
10.7 Comments
11 Traces
11.0 Introduction
11.1 Proof of Theorem 11.1
11.2 A sharp form of the extension problem
11.3 Complements and open problems
11.3.1 An intermission
11.3.2 Sphere-valued maps
11.4 Comments
12 Degree
12.0 Introduction
12.1 Degree and VMO
12.2 Degree, lifting, and traces in Ws,p(mathbbS1; mathbbS1)
12.3 Integral representations for the degree
12.4 A ``distributional degree''
12.5 Estimates for the degree
12.6 Homotopy classes
12.7 Degree and Fourier coefficients
12.7.1 Some preliminaries
12.7.2 Computing the degree using Fourier coefficients
12.8 Complements and open problems
12.8.1 Least energy with prescribed degree
12.8.2 Existence of least energy maps
12.8.3 Distances between homotopy classes
12.8.4 Degree of maps u:ΩtimesmathbbS1tomathbbS1
12.8.5 Trace theory revisited
12.8.6 Maps from mathbbSN to mathbbSN
12.8.7 Open problems
12.9 Comments
13 Dirichlet problems. Gaps. Infinite energies
13.0 Introduction
13.1 Minimizing the W1,p energy when pge2
13.2 Minimizing the W1,p energy when 1
13.3 Gaps
13.4 More about minimizers when Ω=mathbbD and 1
13.5 Further regularity and no gap for p<2, near p=2, when N=2
13.6 Complements and open problems
13.6.1 A generalization of Theorem 13.6 to dimension Nge2
13.6.2 Minimizing the W1,p energy with prescribed Jacobian
13.6.3 Minimizing infinite energies when N=2 and pge2. The BBH theory of Ginzburg–Landau vortices
13.6.4 Minimizing infinite energies when Nge3 and 2lep
13.6.5 Open problems
13.7 Comments
14 Domains with topology
14.0 Introduction
14.1 Lifting in W1,p(Ω; mathbbS1) revisited
14.2 Relaxed energy revisited
14.3 Lifting in Ws,p(Ω; mathbbS1) revisited
14.4 Density in Ws,p(Ω; mathbbS1) revisited
14.5 Homotopy classes
14.6 Complements and open problems
14.6.1 Domains that admit global liftings
14.6.2 What happens when Ω is a manifold?
14.6.3 Replacing mathbbS1 by a general target manifold mathscrN
14.6.4 Open problems
14.7 Comments
15 Appendices
15.1 Sobolev spaces
15.2 Sobolev embeddings and Gagliardo–Nirenberg inequalities
15.3 Composition in Sobolev spaces
15.4 Standard (and non-standard) examples of maps in Sobolev spaces
15.5 Further results on BMO and VMO
15.6 Enlarging Ω
15.7 Fine theory of BV maps
15.8 Description of (minimal) connections associated with (a,d)
15.8.1 Structure of general connections associated with (a,d)
15.8.2 Structure of minimal connections associated with (a,d)
15.9 A pathological case
15.10 Jacobians of W1,p(Ω; mathbbS1) maps
15.11 When Ju is a measure
15.12 Products in fractional Sobolev spaces
15.13 Maximal estimates
15.14 Smoothing
References
Symbol Index
Subject Index
Author Index 
