The Hodge-Laplacian

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