The Hodge-Laplacian

Boundary Value Problems on Riemannian Manifolds

Dorina Mitrea, Irina Mitrea, Marius Mitrea , Michael Taylor

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English

De Gruyter
10 October 2016

Differential & Riemannian geometry

Series: De Gruyter Studies in Mathematics

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 Calderón-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 PhD 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

By: Dorina Mitrea, Irina Mitrea, Marius Mitrea, Michael Taylor
Imprint: De Gruyter
Country of Publication: Germany
Volume: 64
Dimensions: Height: 240mm, Width: 170mm, Spine: 33mm
Weight: 1.019kg
ISBN: 9783110482669
ISBN 10: 3110482665
Series: De Gruyter Studies in Mathematics
Pages: 528
Publication Date: 10 October 2016
Recommended Age: College Graduate Student
Audience: Professional and scholarly , Undergraduate , Undergraduate
Format: Hardback
Publisher's Status: Active

D. Mitrea and M. Mitrea, Univ. of Missouri, USA; I. Mitrea, Temple Univ., Philadelphia, USA; M. Taylor, Univ. of North Carolina, USA.

Reviews for The Hodge-Laplacian: Boundary Value Problems on Riemannian Manifolds

The book represents the cumulation of a large body of work of the authors. Nonetheless, it is essentially self-contained, including the main geometric and analytic preliminaries. There are a large number of variations of settings. But the book is very well structured, avoiding potential confusions here. Mathematical Reviews