Lecture Notes in Mathematics

By: GÃ¼nter Schwarz

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Paperback | 14 July 1995

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172 Pages

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23.5 x 15.24 x 1.27

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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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Introduction
Analysis of Differential Forms
Manifolds with boundary	p. 9
Differential forms	p. 19
Sobolev spaces	p. 30
Weighted Sobolev spaces	p. 42
Elements of the functional analysis	p. 47
Elliptic boundary value problems	p. 52
The Hodge Decomposition
Stokes' theorem, the Dirichlet integral and Gaffney's inequalities	p. 59
The Dirichlet and the Neumann potential	p. 67
Regularity of the potential	p. 73
Hodge decomposition on compact [delta]-manifolds	p. 80
Hodge decomposition on exterior domains	p. 94
Elements of de Rham cohomology theory	p. 103
Appendix: On the smooth deformation of Hilbert space decompositions	p. 107
Boundary Value Problems for Differential Forms
The Dirichlet problem for the exterior derivative	p. 113
First order boundary value problems on [actual symbol not reproducible]	p. 119
General inhomogeneous boundary conditions	p. 124
Harmonic fields, harmonic forms and the Poisson equation	p. 129
Vector analysis	p. 138
Bibliography	p. 147
Index	p. 153
Table of Contents provided by Blackwell. All Rights Reserved.

Lecture Notes in Mathematics

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Paperback

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