This monograph (in two volumes) deals with non scalar variational problems arising in geometry, as harmonic mappings between Riemannian manifolds and minimal graphs, and in physics, as stable equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and accessible to non specialists. Topics are treated as far as possible in an elementary way, illustrating results with simple examples; in principle, chapters and even sections are readable independently of the general context, so that parts can be easily used for graduate courses. Open questions are often mentioned and the final section of each chapter discusses references to the literature and sometimes supplementary results. Finally, a detailed Table of Contents and an extensive Index are of help to consult this monograph
Author(s): Mariano Giaquinta, Giuseppe Modica, Jiri Soucek
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics
Publisher: Springer
Year: 1998
Language: English
Pages: 734
Cover......Page 1
Title......Page 2
Copyright......Page 3
Dedication......Page 4
Preface......Page 6
Contents of Volume I......Page 12
Contents of Volume II......Page 18
1.1 Measures and Integrals......Page 24
1.2 Borel Regular and Radon Measures.......Page 33
1.3 Hausdorff Measures.......Page 35
1.4 Lebesgue's, Radon-Nikodym's and Riesz's Theorems......Page 46
1.5 Covering Theorems, Differentiation and Densities.......Page 52
2.1 Weak Convergence of Vector Valued Measures.......Page 59
2.2 Typical Behaviours of Weakly Converging Sequences......Page 63
2.3 Weak Convergence in L4, q > 1......Page 67
2.4 Weak Convergence in L'......Page 73
2.5 Concentration: Weak Convergence of Measures.......Page 78
2.6 Oscillations: Young Measures......Page 81
2.7 More on Weak Convergence in Ll......Page 87
3 Notes.......Page 90
1.1 Area and Coarea Formulas for Linear Maps......Page 92
1.2 Area Formula for Lipschitz Maps.......Page 97
1.3 Coarea Formula for Lipschitz Maps.......Page 105
1.4 Rectifiable Sets and the Structure Theorem......Page 113
1.5 The General Area and Coarea Formulas.......Page 122
2 Currents......Page 126
2.1 Multivectors and Covectors......Page 127
2.2 Differential Forms.......Page 141
2.3 Currents: Basic Facts.......Page 145
2.4 Integer Multiplicity Rectifiable Currents.......Page 159
2.5 Slicing......Page 174
2.6 The Deformation Theorem and Approximations......Page 180
2.7 The Closure Theorem.......Page 184
3 Notes.......Page 196
3. Cartesian Maps.......Page 198
1 Differentiability of Non Smooth Functions.......Page 202
1.1 The Maximal Function and Lebesgue's Differentiation Theorem......Page 203
1.2 Differentiability Properties of W 1,P Functions......Page 215
1.3 Lusin Type Properties of W''P Functions......Page 225
1.4 Approximate Differential and Lusin Type Properties......Page 233
1.5 Area Formulas, Degree, and Graphs of Non Smooth Maps......Page 241
2 Maps with Jacobian Minors in L'......Page 251
2.1 The Class AI (.(l, RN) , Graphs and Boundaries.......Page 252
2.2 Examples.......Page 256
2.3 Boundaries and Integration by Parts......Page 261
2.4 More on the Jacobian Determinant.......Page 270
2.5 Boundaries and Traces......Page 288
3 Cartesian Maps......Page 299
3.1 Weak Continuity of Minors......Page 300
3.2 The Class cart' (.fl, RN): Closure and Compactness.......Page 308
3.3 The Classes cartP(f?, R'`'), p > 1......Page 316
4 Approximability of Cartesian Maps.......Page 319
4.1 The Transfinite Inductive Process......Page 322
4.2 Weak and Strong Approximation of Minors......Page 326
4.3 The Join of Cartesian Maps.......Page 336
5 Notes.......Page 341
4. Cartesian Currents in Euclidean Spaces.......Page 346
1 Functions of Bounded Variation......Page 350
1.1 The Space BV (Q, R).......Page 352
1.2 Caccioppoli Sets.......Page 363
1.3 De Giorgi's Rectifiability Theorem......Page 369
1.4 The Structure Theorem for BV Functions.......Page 377
1.5 Subgraphs of BV Functions.......Page 394
2 Cartesian Currents in Euclidean Spaces.......Page 402
2.1 Limit Currents of Smooth Graphs......Page 403
2.2 The Classes cart (,fl x IRN) and graph(.f2 x ISBN).......Page 407
2.3 The Structure Theorem.......Page 414
2.4 Cartesian Currents in Codimension One.......Page 426
2.5 Examples of Cartesian Currents......Page 434
2.6 Radial Currents......Page 462
3 Degree Theory.......Page 473
3.1 n-Dimensional Currents and BV Functions......Page 474
3.2 Degree Mapping and Degree of Cartesian Currents.......Page 483
3.3 The Degree of Continuous Maps......Page 494
3.4 h-Connected Components and the Degree.......Page 497
4 Notes.......Page 502
5. Cartesian Currents in Riemannian Manifolds.......Page 516
1.1 The Deformation Theorem......Page 517
1.2 Mollifying Currents.......Page 528
1.3 Flat Chains......Page 535
2 Differential Forms and Cohomology.......Page 550
2.1 Forms on Manifolds.......Page 551
2.2 Hodge Operator......Page 554
2.3 Sobolev Spaces of Forms......Page 559
2.4 Harmonic Forms.......Page 561
2.5 Hodge and Hodge-Kodaira-Morrey Theorems......Page 566
2.6 Relative Cohomology: Hodge-Morrey Decomposition......Page 572
2.7 Weitzenbock Formula.......Page 582
2.8 Poincare and Poincare-Lefschetz Dualities in Cohomology......Page 588
3 Currents and Real Homology of Compact Manifolds......Page 593
3.1 Currents on Manifolds......Page 595
3.2 Poincare and de Rham Dualities.......Page 597
3.3 Poincare-Lefschetz and de Rham Dualities.......Page 612
3.4 Intersection of Currents and Kronecker Index......Page 622
3.5 Relative Homology and Cohomology Groups.......Page 631
4.1 Integral Homology Groups.......Page 638
4.2 Intersection in Integral Homology......Page 647
5 Maps Between Manifolds......Page 654
5.1 Sobolev Classes of Maps Between Riemannian Manifolds......Page 655
5.2 Cartesian Currents Between Manifolds......Page 663
5.3 Homology Induced Maps: Manifolds Without Boundary.......Page 671
5.4 Homology Induced Maps: Manifolds with Boundary......Page 681
6 Notes.......Page 686
Bibliography......Page 690
Index......Page 720
Symbols......Page 732
