Hodge decomposition : a method for solving boundary value problems

Abstract

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.