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Stability Estimates for Some Parabolic Inverse Problems With the Final Overdetermination via a New Carleman Estimate

Michael V. Klibanov

Abstract

This paper is about Holder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time {t=T}. In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem.

Stability Estimates for Some Parabolic Inverse Problems With the Final Overdetermination via a New Carleman Estimate

Abstract

This paper is about Holder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time {t=T}. In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem.
Paper Structure (26 sections, 255 equations)

This paper contains 26 sections, 255 equations.

Table of Contents

  1. Introduction
  2. New Elements of This Paper and Known Results
  3. New elements
  4. Known results for CIPs for parabolic equations with the final overdetermination
  5. The main differences between our paper and ImYam2IsakovPrilepko
  6. Three types of other known stability and uniqueness results for CIPs and ISPs for parabolic PDEs
  7. Statements of the Coefficient Inverse Problems and the Inverse Source Problems
  8. Theorems
  9. The new Carleman estimate
  10. Comments about Theorem 1
  11. Formulations of Theorems 2-4
  12. Proof of Theorem 2
  13. Proof of Theorem 3
  14. Proof of Theorem 4
  15. Appendix: Proof of Theorem 1
  16. ...and 11 more sections