Partial Regularity for Harmonic Maps and Related Problems

The following sections are included:

• Preface

• Contents

The following sections are included:

• Harmonic Maps

• Related Elliptic Equations

• Evolution Problems

• Notation

The following sections are included:

• A Few Basic Inequalities Involving Sobolev Spaces

• The Hodge Decomposition

• Bounded Mean Oscillation

• The Hardy Space

• A Gagliardo-Nirenberg Type Inequality

• A Gehring Type Lemma

The following sections are included:

• The Nearest Point Projection and the Second Fundamental Form

• The Euler-Lagrange Equation

• Stationary Weakly Harmonic Maps and the Monotonicity Formula

• Higher Regularity

• The Target Manifold 𝕊ⁿ⁻¹

• A Moving Tangent Frame Field

• General Target Manifolds

• Generalized Harmonic Maps

• Further Remarks
  • Homogeneous Manifolds
  • Energy Minimizing Maps

The following sections are included:

• The Monotonicity Formula

• Hölder Continuity

• A Lemma of S. Luckhaus

• More About the Tangent Frame Field

• An Energy Inequality

• Small Dimensions

The following sections are included:

• The Stability Condition for the Harmonic Map Heat Flow

• Higher Regularity for Evolution Problems

• The Heat Flow into Spheres

• The Heat Flow into General Target Manifolds

• The Landau-Lifshitz Equation

• Further Remarks
  • The Harmonic Map Heat Flow into Homogeneous Manifolds
  • The Landau-Lifshitz Equation for Almost-Kähler Manifolds

The following sections are included:

• Bibliography

• Index