The Hodge-Laplacian: Boundary Value Problems on Riemannian Manifolds

The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators.

A new generation of Caldern-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains.

Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of Ph D students, researchers, and professionals.

Contents: Preface Introduction and Statement of Main Results Geometric Concepts and Tools Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains Fatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT Domains Solvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham Formalism Additional Results and Applications Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis Bibliography Index

Book Details

• Authors: Dorina Mitrea, Irina Mitrea, Marius Mitrea, Michael Taylor
• Publisher: De Gruyter
• eBook ISBN: 9783110483390, 3110483394
• Print ISBN: 9783110482669, 3110482665
• Other ISBNs: 3110484382, 9783110484380