Inverse Problems

by Joel Feldman and Gunther Uhlmann

This book contains a mathematical analysis of the following problem.

The mathematical background of the reader is assumed to be at least at the level of

• Sections 1.1-1.3 and 2.1-2.3 of Reed and Simon, Methods of Modern Mathematical Physics, I: Functional Analysis.
• Chapter 6 and Sections 10.1, 10.2, 10.3, 10.8 of Royden Real Analysis.
• Beginning with section 4.4, familiarity with complex analysis at the level of the books of Ahlfors or Conway is also assumed.

• §1 Introduction [version of Oct 19, 2004]
  • §1.1 Dirichlet to Neumann Problems
  • §1.2 X-ray Tomography
  • §1.3 Inverse Scattering
• §2 Sobolev Spaces [version of Dec 4, 2004]
  • §2.1 Definitions of Sobolev Spaces
  • §2.2 Traces - Restriction to the Boundary
  • §2.3 Inequalities
  • §2.A Appendix - Pseudodifferential Operators [version of Dec 2, 2003]
• §3 Elliptic Boundary Value Problems and the Dirichlet to Neumann Map [version of Dec 1, 2004]
  • §3.1 Second Order Elliptic Equations in Rⁿ
  • §3.2 Second Order Elliptic Equations in R+ⁿ
  • §3.3 The Dirichlet Problem for Conductivity Operators
  • §3.4 The Dirichlet to Neumann Map
  • §3.5 Schrödinger Operators
• §4 Identifiability of Isotropic Conductivities [version of Nov 30, 2004]
  • §4.1 Identification of Boundary Values of Isotropic Conductivities
  • §4.2 Exponentially Growing Solutions
  • §4.3 Global Identification of Isotropic Conductivities, n>2
  • §4.4 Global Identification of Isotropic Conductivities, n=2
• §5 Anisotropic Conductors [not released yet]
• §6 Scattering Theory [not released yet]
• Appendix A: Compact Operators [version of Sept 26, 2003]
• Appendix B: Integation on Manifolds [version of Dec 16, 2004]
• Appendix S1: Problem Solutions for §1 [version of Dec 4, 2004]
• Appendix S2: Problem Solutions for §2 [version of Dec 4, 2004]
• Appendix S2A: Problem Solutions for §2.A [version of Dec 2, 2003]
• Appendix S3: Problem Solutions for §3 [version of Dec 1, 2004]
• Appendix S4: Problem Solutions for §4 [version of Nov 30, 2004]
• Appendix S5: Problem Solutions for §5 [not released yet]
• Appendix S6: Problem Solutions for §6 [not released yet]
• Appendix SA: Problem Solutions for Appendix A [version of Oct 28, 2004]
• Appendix SA: Problem Solutions for Appendix B [version of Dec 16, 2004]
• References [version of Dec 29, 2005]
• Notation

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