Fully Nonlinear PDEs in Real and Complex Geometry and Optics: Cetraro, Italy 2012, Editors: Cristian E. Gutiérrez, Ermanno Lanconelli

Preface......Page 6
Contents......Page 10
Contributors......Page 12
1 Introduction......Page 13
2.1  Notation:  Topology......Page 15
2.2 Notation: Differentials and Dilation of Mappings......Page 16
2.3 Notation: Complex Analysis......Page 17
3 Conformal Deformations......Page 19
4 Grötzsch Problem and Quasiconformal Deformations......Page 21
4.1 Grötzsch Problem......Page 23
5 Teichmüller Theorem and Extremal Quasiconformal Mappings......Page 25
5.1 Riemann Surfaces......Page 27
5.2 Teichmüller Theorem......Page 28
5.3 Coverings and Group Action......Page 29
5.4 Quadratic Differentials......Page 30
5.5 Ahlfors\' Proof of Existence and Uniqueness......Page 31
5.6 Normal Family of Mappings with Integrable Distortion......Page 33
5.7 Teichmüller Mappings in Local Parameters......Page 35
6 A Variation on the Theme: Extremal Mappings of Finite Distortion......Page 37
6.1 The Finite Distortion Version of Grötzsch Problem......Page 38
6.2 Trace Norm vs. Operator Norm......Page 39
6.3 Affine Boundary Data......Page 40
6.4 More General Boundary Data......Page 41
7 Minimal Lipschitz Extensions......Page 43
7.1 Aronsson\'s Approach in the Scalar Case N=1......Page 44
7.2 Aronsson\'s Approach in the Vector-Valued Case N> 1......Page 47
7.3 A Refinement of the Aronsson Equation......Page 49
8 Aronsson\'s Approach for the Extremal Dilation Problem......Page 50
8.1 A Gradient Flow Approach......Page 53
References......Page 56
Curvature Measures, Isoperimetric Type Inequalities and Fully Nonlinear PDEs......Page 59
1 The Steiner Formula and Curvature Measures......Page 60
2 Some Properties of Elementary Symmetric Functions......Page 64
3 Prescribing Curvature Measures......Page 72
3.1 Uniqueness and C1-Estimates......Page 76
3.2 C2-Estimates and the Existence......Page 79
4 Isoperimetric Inequality for Quermassintegrals on Starshaped Domains......Page 83
4.1 Monotonicity Properties......Page 86
4.2 The Harnack Estimate......Page 88
4.3 C2 Estimates......Page 92
5 Appendix......Page 97
6 Notes......Page 102
References......Page 104
1 Introduction......Page 107
2.1 In Vector Form......Page 108
2.1.1 κ<1......Page 109
2.1.2 κ>1......Page 110
2.2 Derivation of the Snell Law......Page 111
2.3.1 Case κ<1......Page 113
2.3.2 Case κ>1......Page 114
2.4 Uniform Refraction: Near Field Case......Page 116
2.5 Case 0<κ<1......Page 117
2.6 Case κ>1......Page 119
3 Optimal Mass Transportation......Page 120
3.1 Application to the Refractor Problem κ<1......Page 126
3.2 Refractor Measure and Weak Solutions......Page 127
3.3 Solution of the Refractor Problem......Page 129
4 Solution to the Refractor Problem for κ<1 with the Minkowski Method......Page 130
4.1 Existence of Solutions in the Discrete Case......Page 132
4.2 Solution in the General Case......Page 137
4.3 Uniqueness Discrete Case......Page 140
5 Maxwell\'s Equations......Page 142
5.2 Maxwell Equations in Integral Form......Page 143
5.3 Boundary Conditions at a Surface of Discontinuity......Page 145
5.5 The Wave Equation......Page 148
5.6 Dispersion Equation......Page 149
5.7 Plane Waves......Page 150
5.8 Fresnel Formulas......Page 151
5.10 The Poynting Vector......Page 154
5.11 Polarization......Page 155
5.12 Estimation of the Fresnel Coefficients......Page 157
5.13 Application to the Far Field Refractor Problem with Loss of Energy......Page 159
References......Page 162
On the Levi Monge-Ampère Equation......Page 163
1 Introduction......Page 164
2 The Levi Form......Page 167
2.2.1 Standard Complex Structure......Page 168
2.3.2 Totally Real Part of the Tangent Space......Page 169
2.3.3 Immersed CR Manifold......Page 170
2.4.2 Levi Flat......Page 171
2.6 Levi Form of a Real Hypersurface......Page 172
2.7 Comparison with the Second Fundamental Form......Page 173
2.8 Strictly Pseudoconvexity......Page 175
2.9 Pseudoconvex Domains......Page 176
3 The Levi Problem and Levi Curvature for a Real Hypersurface......Page 177
3.1 Levi Curvature for a Real Hypersurface in C2......Page 178
3.1.2 Levi Curvature of a Graph in C2......Page 179
3.2 The Dirichlet Problem for the Prescribed Levi Curvature Equation......Page 180
3.4 Regularity of the Viscosity Solutions: Classical Solvability of the Dirichlet Problem......Page 181
3.5  L0\'s Sub-Riemannian Structure and the Smoothness Result......Page 182
3.7 Sketch of the Proof of Theorem 7......Page 183
3.7.2 Caccioppoli-Type Estimates......Page 184
3.7.4 Caccioppoli-Type Estimates......Page 185
3.9 Classical Solution......Page 186
3.10.3 Interpolation Inequalities......Page 187
3.10.7 Non Linear Equation......Page 188
4 Levi Curvatures in Higher Dimension......Page 189
4.2 Gauss-Levi Curvature for a Real Hypersurface in Cn+1......Page 190
4.4 General Expressions for K(m)......Page 191
4.6  Domains of Holomorphy and Levi Curvatures......Page 192
4.7.1 Structure of the Levi Curvature PDO\'s......Page 193
4.7.3 Some Results......Page 194
4.9 Sketch of the Proof of Theorem 13......Page 195
4.10 A More Explicit Form of the Gauss-Levi Curvature Equation......Page 196
4.13 Viscosity Solutions to the Prescribed Levi Curvature Equation......Page 197
4.15 Comparison Principle......Page 198
5 A Negative Regularity Result for LMA......Page 199
5.1 Notation and Main Theorem......Page 200
5.3 Existence Result......Page 202
5.4 Stability of Viscosity Solutions......Page 203
5.5 Proof of Theorem 16......Page 204
6 Symmetry Results......Page 207
6.1 Examples and Open Problems......Page 209
6.3 An Obstacle to the Boundary Comparison Principle......Page 210
6.4 Strong Comparison Principle......Page 211
6.5 Null Lagrangian Property for Elementary Symmetric Functions in the Eigenvalues of the Complex Hessian Matrix......Page 212
6.6 Integral Formulas......Page 213
6.7 Isoperimetric Inequality......Page 214
6.9 Curvature Lines and Geodesics. Second Alexandrov Type Result......Page 216
References......Page 218
List of Participants......Page 221