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Hodge decomposition : a method for solving boundary value problems / Günter Schwarz.

By: Material type: TextTextSeries: Lecture notes in mathematics (Springer-Verlag) ; 1607.Publication details: Berlin ; New York : Springer-Verlag, 1995.Description: 1 online resource (155 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9783540494034
  • 3540494030
Subject(s): Additional physical formats: Print version:: Hodge decomposition.DDC classification:
  • 510 s 515/.353 20
LOC classification:
  • QA3 .L28 no. 1607 QA379
Other classification:
  • 31.45
Online resources:
Contents:
Ch. 1. Analysis of Differential Forms. 1.1. Manifolds with boundary. 1.2. Differential forms. 1.3. Sobolev spaces. 1.4. Weighted Sobolev spaces. 1.5. Elements of the functional analysis. 1.6. Elliptic boundary value problems -- Ch. 2. The Hodge Decomposition. 1.1. Stokes' theorem, the Dirichlet integral and Gaffney's inequalities. 2.2. The Dirichlet and the Neumann potential. 2.3. Regularity of the potential. 2.4. Hodge decomposition on compact [delta]-manifolds. 2.5. Hodge decomposition on exterior domains. 2.6. Elements of de Rham cohomology theory -- Appendix: On the smooth deformation of Hilbert space decompositions / J. Wenzelburger -- Ch. 3. Boundary Value Problems for Differential Forms. 3.1. The Dirichlet problem for the exterior derivative. 3.2. First order boundary value problems on [actual symbol not reproducible]. 3.3. General inhomogeneous boundary conditions. 3.4. Harmonic fields, harmonic forms and the Poisson equation. 3.5. Vector analysis.
Summary: Hodge theory is a standard tool in characterizing differ- ential complexes and the topology of manifolds. This book is a study of the Hodge-Kodaira and related decompositions on manifolds with boundary under mainly analytic aspects. It aims at developing a method for solving boundary value problems. Analysing a Dirichlet form on the exterior algebra bundle allows to give a refined version of the classical decomposition results of Morrey. A projection technique leads to existence and regularity theorems for a wide class of boundary value problems for differential forms and vector fields. The book links aspects of the geometry of manifolds with the theory of partial differential equations. It is intended to be comprehensible for graduate students and mathematicians working in either of these fields.
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Includes bibliographical references (pages 147-152) and index.

Hodge theory is a standard tool in characterizing differ- ential complexes and the topology of manifolds. This book is a study of the Hodge-Kodaira and related decompositions on manifolds with boundary under mainly analytic aspects. It aims at developing a method for solving boundary value problems. Analysing a Dirichlet form on the exterior algebra bundle allows to give a refined version of the classical decomposition results of Morrey. A projection technique leads to existence and regularity theorems for a wide class of boundary value problems for differential forms and vector fields. The book links aspects of the geometry of manifolds with the theory of partial differential equations. It is intended to be comprehensible for graduate students and mathematicians working in either of these fields.

Ch. 1. Analysis of Differential Forms. 1.1. Manifolds with boundary. 1.2. Differential forms. 1.3. Sobolev spaces. 1.4. Weighted Sobolev spaces. 1.5. Elements of the functional analysis. 1.6. Elliptic boundary value problems -- Ch. 2. The Hodge Decomposition. 1.1. Stokes' theorem, the Dirichlet integral and Gaffney's inequalities. 2.2. The Dirichlet and the Neumann potential. 2.3. Regularity of the potential. 2.4. Hodge decomposition on compact [delta]-manifolds. 2.5. Hodge decomposition on exterior domains. 2.6. Elements of de Rham cohomology theory -- Appendix: On the smooth deformation of Hilbert space decompositions / J. Wenzelburger -- Ch. 3. Boundary Value Problems for Differential Forms. 3.1. The Dirichlet problem for the exterior derivative. 3.2. First order boundary value problems on [actual symbol not reproducible]. 3.3. General inhomogeneous boundary conditions. 3.4. Harmonic fields, harmonic forms and the Poisson equation. 3.5. Vector analysis.

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