Theorems on Regularity and Singularity of Energy Minimizing Maps

by Leon Simon

Overview

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.

Product Details

ISBN-13:	9783764353971
Publisher:	Birkh�user Basel
Publication date:	03/28/1996
Series:	Lectures in Mathematics. ETH Z�rich
Edition description:	1996
Pages:	152
Product dimensions:	7.01(w) x 10.00(h) x 0.01(d)

Table of Contents

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.

Editorial Reviews

A self-contained introduction to the basic regularity theory for energy minimizing maps, including recent developments in the structure of the singular set and asymptotics on approach to the singular set. Based on the lecture notes of Norbert Hungerbuhler, the text is organized into four sections covering analytic preliminaries, regularity theory for harmonic maps, approximation properties of the singular set, and rectifiability of the singular set. The text does not assume any specialized knowledge in partial differential equations or the geometric calculus of variations. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews