1 Introduction ................................................. 5
2 Preliminaries ............................................... 13
2.1 Sobolev spaces ......................................... 13
2.2 Invertibility of Sobolev mappings ...................... 14
2.3 The Hardy space  1( ) ................................. 15
2.4 Jacobians of Sobolev mappings .......................... 17
2.5 VMO() and BMO() ..................................... 18
2.6 The Cauchy transform and the Beurling transform ........ 20
2.7 Commutators ............................................ 21
2.8 An equivalent norm in BMO() ........................... 22
2.9 Banach spaces .......................................... 24
3 The Jacobian Equation with Radial Data ...................... 26
3.1 Radial stretchings as solutions ........................ 26
3.2 Generalized radially symmetric solutions ............... 31
4 Energy-Minimal Solutions and Lagrange Multipliers ........... 35
4.1 Energy-minimal solutions ............................... 35
4.2 Lagrange multipliers ................................... 37
4.3 Operator theoretical treatment of Lagrange
multipliers ............................................ 38
5 On the Existence of Lagrange Multipliers .................... 39
5.1 The norm ||•|| bmos and Lagrange multipliers ........... 39
5.2 Lagrange multipliers in VMOS() ....................... 40
5.3 Characterizations of the norms ||•||BMOS and ||•||H1S ... 41
5.4 Connection to commutators ............................. 42
6 On the Class  L2S ........................................... 44
6.1 Topological properties of {|Sƒ|2 - |ƒ|2 : ƒ ϵ L2S} ..... 44
6.2 Extreme points of  H1S ................................. 45
6.3 The duality mapping D: VMOS → H1S ..................... 45
7 Local Study of Energy-Minimal Solutions and Lagrange
Multipliers ................................................. 48
7.1 Lagrange multipliers in a bounded domain ............... 48
7.2 Smooth, compactly supported Lagrange multipliers ....... 51
8 Behavior of Solutions in Domains Where the Jacobian
Vanishes .................................................... 54
8.1 The Hopf-Laplace equation .............................. 54
8.2 Harmonicity of solutions ............................... 56
8.3 Solution of a problem of Lou, Yang and Song ............ 58
References .................................................. 62
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