Author | Simon, Leon. author |
---|---|
Title | Theorems on Regularity and Singularity of Energy Minimizing Maps [electronic resource] / by Leon Simon |
Imprint | Basel : Birkhรคuser Basel, 1996 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-9193-6 |
Descript | VIII, 152 p. 6 illus. online resource |
1 Analytic Preliminaries -- 1.1 Hรถlder 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 Luckhausโ Lemma -- 2.8 Proof of the Reverse Poincarรฉ 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 โ 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