Theorems on Regularity and Singularity of Energy Minimizing Maps [Paperback]

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 H?lder 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 Poincar? 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.- 2.12.1 Absolute Continuity Properties of Functions in W1,2.- 2.12.2 Proof of Luckhaus Lemma (Lemma 1 of Section 2.6).- 2.12.3 Nearest point projection.- 2.12.4 Proof of the ?-regularity theorem in case n = 2.- 3 Approximation Properties of the Singular Set.- 3.1 Definition of Tangent Map.- 3.2 Properties of Tangent Maps.- 3.3 Properties of Hl£)