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Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds


About this Title

Dorina Mitrea, Marius Mitrea and Michael Taylor

Publication: Memoirs of the American Mathematical Society
Publication Year 2001: Volume 150, Number 713
ISBNs: 978-0-8218-2659-1 (print); 978-1-4704-0306-5 (online)
DOI: http://dx.doi.org/10.1090/memo/0713
MathSciNet review: 1809655
MSC (2000): Primary 58J05; Secondary 31C12, 35J45, 42B20, 45E05, 78A30

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Table of Contents


Chapters

  • Introduction
  • 1. Singular integrals on Lipschitz submanifolds of codimension one
  • 2. Estimates on fundamental solutions
  • 3. General second-order strongly elliptic systems
  • 4. The Dirichlet problem for the Hodge Laplacian and related operators
  • 5. Natural boundary problems for the Hodge Laplacian in Lipschitz domains
  • 6. Layer potential operators on Lipschitz domains
  • 7. Rellich type estimates for differential forms
  • 8. Fredholm properties of boundary integral operators on regular spaces
  • 9. Weak extensions of boundary derivative operators
  • 10. Localization arguments and the end of the proof of Theorem 6.2
  • 11. Harmonic fields on Lipschitz domains
  • 12. The proofs of the Theorems 5.1–5.5
  • 13. The proofs of the auxiliary lemmas
  • 14. Applications to Maxwell's equations on Lipschitz domains