Nonlinear Analysis on Manifolds

The volume is organized into nine chapters. Chapter 1 offers a brief introduction to differential and Riemannian geometry. Chapter 2 deals with the general theory of Sobolev spaces for compact manifolds. Chapter 3 presents the general theory of Sobolev spaces for complete, noncompact manifolds. Best constants problems for compact manifolds are discussed in Chapters 4 and 5. Chapter 6 presents special types of Sobolev inequalities under constraints. Best constants problems for complete noncompact manifolds are discussed in Chapter 7. Chapter 8 deals with Euclidean-type Sobolev inequalities. And Chapter 9 discusses the influence of symmetries on Sobolev embeddings. An appendix offers brief notes on the case of manifolds with boundaries.

This topic is a field undergoing great development at this time. However, several important questions remain open. So a substantial part of the book is devoted to the concept of best constants, which appeared to be crucial for solving limiting cases of some classes of PDEs.

The volume is highly self-contained. No familiarity is assumed with differentiable manifolds and Riemannian geometry, making the book accessible to a broad audience of readers, including graduate students and researchers.

Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.

Contents

• Elements of Riemannian geometry
• Sobolev spaces: The compact setting
• Sobolev spaces: The noncompact setting
• Best constants in the compact setting I
• Best constants in the compact setting II
• Optimal inequalities with constraints
• Best constants in the noncompact setting
• Euclidean-type Sobolev inequalities
• The influence of symmetries
• Manifolds with boundary
• Bibliography