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 Calder�n-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, Poincar�, 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 Additional ISBNs3110482665, 3110484382, 9783110482669, 9783110484380Read More The Hodge-Laplacian: Boundary Value Problems on Riemannian Manifolds 1st Edition is written by Dorina Mitrea; Irina Mitrea; Marius Mitrea; Michael Taylor and published by De Gruyter. ISBNs for The Hodge-Laplacian are 9783110483390, 3110483394 and the print ISBNs are 9783110482669, 3110482665. Additional ISBNs include 3110482665, 3110484382, 9783110482669, 9783110484380.