Cartesian currents in the calculus of variations

Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Dedication......Page 4
Preface......Page 6
Contents of Volume II......Page 12
Contents of Volume I......Page 18
1. Regular Variational Integrals......Page 24
1 The Direct Methods......Page 25
1.1 The Abstract Setting......Page 26
1.2 Some Classical Lower Semicontinuity Theorems......Page 33
1.3 A General Semicontinuity Theorem......Page 42
2 Polyconvex Envelops and Regular Parametric Integrals......Page 46
2.1 Polyconvexity and Polyconvex Envelops......Page 49
2.2 Parametric Polyconvex Envelops of Integrands......Page 60
2.3 The Parametric Extension of Regular Integrals......Page 67
2.4 The Polyconvex l.s.c. Extension of Some Lagrangians......Page 68
3 Regular Integrals in the Class of Cartesian Currents......Page 97
3.1 Parametric Integrands and Lower Semicontinuity......Page 98
3.2 Existence of Minimizers in Classes of Cartesian Currents......Page 105
3.3 Relaxed Energies in the Setting of Cartesian Currents......Page 111
3.4 Relaxed Energies in the Parametric Case......Page 113
4 Regular Integrals and Quasiconvexity......Page 129
4.1 Quasiconvexity......Page 130
4.2 Quasiconvexity and Lower Semicontinuity......Page 140
4.3 Ellipticity and Quasiconvexity......Page 150
5 Notes......Page 154
2. Finite Elasticity and Weak Diffeomorphisms......Page 160
1.1 Fields and Transformations......Page 162
1.2 Kinematics......Page 163
1.3 Local Deformations......Page 166
1.4 Perfectly Elastic Bodies: Stored Energy, Convexity and Coercivity......Page 170
1.5 Variations and Stress......Page 175
2 Physical Implications on Kinematics and Stored Energies......Page 178
2.1 Kinematical Principles in Elasticity: Weak Deformations......Page 179
2.2 Frame Indifference and Isotropy......Page 192
2.3 Convexity-like Conditions......Page 193
2.4 Coercivity Conditions......Page 198
2.5 Examples of Stored Energies......Page 202
3 Weak Diffeomorphisms......Page 205
3.1 The Classes dif \'q(Q,12)......Page 206
3.2 The Classes dif p\' (f2, IRn)......Page 214
3.3 Convergence Theorems for the Inverse Maps......Page 222
3.4 General Weak Diffeomorphisms......Page 227
3.5 The Dif-classes......Page 236
3.6 Volume Preserving Diffeomorphisms......Page 237
4 Connectivity Properties of the Range of Weak Diffeomorphisms......Page 239
4.1 Connectivity of the Range of Sobolev Maps......Page 240
4.2 Connectivity of the Range of Weak Diffeomorphisms......Page 242
4.3 Regularity Properties of Locally Weak Invertible Maps......Page 252
4.4 Global Invertibility of Weak Maps......Page 261
4.5 An a.e. Open Map Theorem......Page 266
5 Composition......Page 270
5.1 Composition of weak deformations......Page 271
5.2 On the Summability of Compositions......Page 273
5.3  Composition of Weak Diffeomorphisms......Page 277
6 Existence of Equilibrium Configurations......Page 283
6.1 Existence Theorems......Page 284
6.2 Equilibrium and Conservation Equations......Page 287
6.3 The Cavitation Problem......Page 291
7 Notes......Page 301
1 Harmonic Maps Between Manifolds......Page 304
1.1 First Variation and Inner Variations......Page 306
1.2 Finding Harmonic Maps by Variational Methods......Page 316
2 Energy Minimizing Weak Harmonic Maps: Regularity Theory......Page 319
2.1 Some Preliminaries. Reverse Holder Inequalities......Page 320
2.2 Classical Regularity Results......Page 326
2.3 An Optimal Regularity Theorem......Page 330
2.4 The Partial Regularity Theorem......Page 342
3 Harmonic Maps in Homotopy Classes......Page 356
3.1 The Action of Wl\'2-maps on Loops......Page 357
3.2 Minimizing Energy with Homotopic Constraints......Page 359
3.3 Local Replacement by Harmonic Mappings: Bubbling......Page 360
4.1 The Partial Regularity Theory......Page 362
4.2 Stationary Harmonic Maps......Page 368
5 Notes......Page 373
4. The Dirichlet Energy for Maps into S2......Page 376
1.1 Harmonic Maps with Prescribed Degree......Page 377
1.2 The Structure Theorem in cart2\'1(fl x S2), fl C R^2......Page 385
1.3 Existence and Regularity of Minimizers......Page 389
2 Variational Problems from a Domain of R3 into S2......Page 406
2.1 The Class cart2\'1((2 x S2), Sl C R3......Page 408
2.2 Density Results in W1,2 (B3, S2)......Page 415
2.3 Dipoles and Gap Phenomenon......Page 423
2.4 The Structure Theorem in cart 2\'1(Sl x S2), SZ C3......Page 432
2.5 Approximation by Smooth Graphs: Dirichlet Data......Page 435
2.6 Approximation by Smooth Graphs: No Boundary Data......Page 442
2.7 The Dirichlet Integral in cart2\'1(fl x S2), ,(l C II......Page 446
2.8 Minimizers of Variational Problems......Page 452
2.9 A Partial Regularity Result......Page 456
2.10 The General Dipole Problem......Page 472
2.11 Singular Perturbations......Page 475
3 Notes......Page 481
1 The Liquid Crystal Energy......Page 490
1.2 The Relaxed Energy......Page 493
1.3 The Dipole Problem......Page 500
2.1 Maps with Values in S2......Page 508
2.2 The Dipole Problem......Page 512
2.3 The Structure Theorem......Page 517
3 The Dirichlet Integral in the Regular Case: Maps into a Manifold......Page 519
3.1 The Class cart2,1(Q x y)......Page 520
3.2 Spherical Vertical Parts and a Closure Theorem......Page 524
3.3 The Dirichlet Integral and Minimizers......Page 529
4 The Dirichlet Integral in the Non Regular Case: a Homological Theory......Page 531
4.1 (n,p)-Currents......Page 532
4.2 Graphs of Sobolev Maps......Page 539
4.3 p-Dirichlet Graphs and Cartesian Currents......Page 548
4.4 The Dirichlet Integral......Page 557
4.5 Prescribing Homological Singularities......Page 566
5 Notes......Page 569
6 The Non Parametric Area Functional......Page 586
1.1 Parametric Surfaces of Least Area......Page 587
1.2 Non Parametric Minimal Surfaces of Codimension One.......Page 602
2 Problems for Maps of Bounded Variation with Values in Sl......Page 613
2.1 Preliminaries......Page 617
2.2 The Class cart(,- x Sl)......Page 623
2.3 Relaxed Energies and Existence of Minimizers......Page 633
3.1 Plateau\'s Problem......Page 642
3.2 Existence of Two Dimensional Non Parametric Minimal Surfaces......Page 648
3.3 The Minimal Surface System......Page 650
4 Least Area Mappings and Least Mass Currents......Page 655
4.1 Topological Results......Page 656
4.2 Main Results......Page 658
5 The Non-parametric Area Integral......Page 662
5.1 The Mass of Cartesian Currents and the Relaxed Area......Page 664
5.2 Lebesgue\'s Area......Page 672
6 Notes......Page 674
Bibliography......Page 676
Index......Page 706
Symbols......Page 718