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; a good general background in mathematical analysis would be adequate preparation.
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; a good general background in mathematical analysis would be adequate preparation.
1 Analytic Preliminaries.- 1.1 Hölder Continuity.- 1.2 Smoothing.- 1.3 Functions with L2 Gradient.- 1.4 Harmonic Functions.- 1.5 Weakly Harmonic Functions.- 1.6 Harmonic Approximation Lemma.- 1.7 Elliptic regularity.- 1.8 A Technical Regularity Lemma.- 2 Regularity Theory for Harmonic Maps.- 2.1 Definition of Energy Minimizing Maps.- 2.2 The Variational Equations.- 2.3 The ?-Regularity Theorem.- 2.4 The Monotonicity Formula.- 2.5 The Density Function.- 2.6 A Lemma of Luckhaus.- 2.7 Corollaries of Luckhaus' Lemma.- 2.8 Proof of the Reverse Poincaré Inequality.- 2.9 The Compactness Theorem.- 2.10 Corollaries of the ?-Regularity Theorem.- 2.11 Remark on Upper Semicontinuity of the Density ?u(y).- 2.12 Appendix to Chapter 2.- 3 Approximation Properties of the Singular Set.- 3.1 Definition of Tangent Map.- 3.2 Properties of Tangent Maps.- 3.3 Properties of Homogeneous Degree Zero Minimizers.- 3.4 Further Properties of sing u.- 3.5 Definition of Top-dimensional Part of the Singular Set.- 3.6 Homogeneous Degree Zero ? with dim S(?) = n - 3.- 3.7 The Geometric Picture Near Points of sing*u.- 3.8 Consequences of Uniqueness of Tangent Maps.- 3.9 Approximation properties of subsets of ?n.- 3.10 Uniqueness of Tangent maps with isolated singularities.- 3.11 Functionals on vector bundles.- 3.12 The Liapunov-Schmidt Reduction.- 3.13 The ?ojasiewicz Inequality for ?.- 3.14 ?ojasiewicz for the Energy functional on Sn-1.- 3.15 Proof of Theorem 1 of Section 3.10.- 3.16 Appendix to Chapter 3.- 4 Rectifiability of the Singular Set.- 4.1 Statement of Main Theorems.- 4.2 A general rectifiability lemma.- 4.3 Gap Measures on Subsets of ?n.- 4.4 Energy Estimates.- 4.5 L2 estimates.- 4.6 The deviation function ?.- 4.7 Proof of Theorems 1, 2 of Section 4.1.- 4.8 The case when ?has arbitrary Riemannian metric.
1 Analytic Preliminaries.- 1.1 Hölder Continuity.- 1.2 Smoothing.- 1.3 Functions with L2 Gradient.- 1.4 Harmonic Functions.- 1.5 Weakly Harmonic Functions.- 1.6 Harmonic Approximation Lemma.- 1.7 Elliptic regularity.- 1.8 A Technical Regularity Lemma.- 2 Regularity Theory for Harmonic Maps.- 2.1 Definition of Energy Minimizing Maps.- 2.2 The Variational Equations.- 2.3 The ?-Regularity Theorem.- 2.4 The Monotonicity Formula.- 2.5 The Density Function.- 2.6 A Lemma of Luckhaus.- 2.7 Corollaries of Luckhaus' Lemma.- 2.8 Proof of the Reverse Poincaré Inequality.- 2.9 The Compactness Theorem.- 2.10 Corollaries of the ?-Regularity Theorem.- 2.11 Remark on Upper Semicontinuity of the Density ?u(y).- 2.12 Appendix to Chapter 2.- 3 Approximation Properties of the Singular Set.- 3.1 Definition of Tangent Map.- 3.2 Properties of Tangent Maps.- 3.3 Properties of Homogeneous Degree Zero Minimizers.- 3.4 Further Properties of sing u.- 3.5 Definition of Top-dimensional Part of the Singular Set.- 3.6 Homogeneous Degree Zero ? with dim S(?) = n - 3.- 3.7 The Geometric Picture Near Points of sing*u.- 3.8 Consequences of Uniqueness of Tangent Maps.- 3.9 Approximation properties of subsets of ?n.- 3.10 Uniqueness of Tangent maps with isolated singularities.- 3.11 Functionals on vector bundles.- 3.12 The Liapunov-Schmidt Reduction.- 3.13 The ?ojasiewicz Inequality for ?.- 3.14 ?ojasiewicz for the Energy functional on Sn-1.- 3.15 Proof of Theorem 1 of Section 3.10.- 3.16 Appendix to Chapter 3.- 4 Rectifiability of the Singular Set.- 4.1 Statement of Main Theorems.- 4.2 A general rectifiability lemma.- 4.3 Gap Measures on Subsets of ?n.- 4.4 Energy Estimates.- 4.5 L2 estimates.- 4.6 The deviation function ?.- 4.7 Proof of Theorems 1, 2 of Section 4.1.- 4.8 The case when ?has arbitrary Riemannian metric.
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