The aim of these lecture notes is to give an essentially self-contained introduction to the basic regularity theory for energy minimizing maps, including recent developments concerning the structure of the singular set and asymptotics on approach to the singular set. Specialized knowledge in partial differential equations or the geometric calculus of variations is not required. Instead, a good general background in mathematical analysis would be adequate preparation.
Author(s): Leon Simon
Series: Lectures in Mathematics. ETH Zürich
Publisher: Birkhauser
Year: 1996
Language: English
Pages: 162
Contents......Page 6
Preface......Page 8
1.1 Holder Continuity ......Page 10
1.2 Smoothing ......Page 13
1.3 Functions with L2 Gradient ......Page 14
1.4 Harmonic Functions ......Page 17
1.6 Harmonic Approximation Lemma ......Page 19
1.7 Elliptic regularity ......Page 20
1.8 A Technical Regularity Lemma ......Page 22
2.1 Definition of Energy Minimizing Maps ......Page 28
2.2 The Variational Equations ......Page 29
2.3 The e-Regularity Theorem ......Page 31
2.4 The Monotonicity Formula ......Page 32
2.5 The Density Function ......Page 33
2.6 A Lemma of Luckhaus ......Page 34
2.7 Corollaries of Luckhaus' Lenuna ......Page 35
2.8 Proof of the Reverse Poincare Inequality ......Page 38
2.9 The Compactness Theorem ......Page 41
2.10 Corollaries of the e-Regularity Theorem ......Page 44
2.12.1 Absolute Continuity Properties of Functions in 1'6'1"2 ......Page 46
2.12.2 Proof of Luckhaus' Lemma (Lemma 1 of Section 2.6) ......Page 47
2.12.3 Nearest point projection . ......Page 51
2.12.4 Proof of the e-regularity theorem in case n = 2 ......Page 55
3.1 Definition of Tangent Map ......Page 60
3.3 Properties of Homogeneous Degree Zero Minimizers ......Page 61
3.4 Further Properties of sing u ......Page 63
3.6 Homogeneous Degree Zero V with dim S(jp) = n - 3 ......Page 67
3.7 The Geometric Picture Near Points of sing.u ......Page 68
3.8 Consequences of Uniqueness of Tangent Maps ......Page 70
3.9 Approximation properties of subsets of W ......Page 71
3.10 Uniqueness of Tangent maps with isolated singularities ......Page 76
3.11 Functionals on vector bundles ......Page 81
3.12 The Liapunov-Schmidt Reduction ......Page 83
3.13 The Lojasiewicz Inequality for F ......Page 87
3.14 Lojasiewicz for the Energy functional on S"-1 ......Page 89
3.15 Proof of Theorem 1 of Section 3.10 . ......Page 91
3.16.1 The Liapunov-Schmidt Reduction in a Finite Dimensional Setting ......Page 96
4.1 Statement of Main Theorems ......Page 100
4.2 A general rectifiability lemma ......Page 101
4.3 Gap Measures on Subsets of W ......Page 113
4.4 Energy Estimates ......Page 119
4.5 L2 estimates ......Page 129
4.6 The deviation function 0 ......Page 138
4.7 Proof of Theorems 1, 2 of Section 4.1 . ......Page 144
4.8 The case when St has arbitrary Riemannian metric ......Page 152
Bibliography ......Page 156
Index ......Page 159
