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About this Title

P. Cannarsa, P. Martinez and J. Vancostenoble

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 239, Number 1133
ISBNs: 978-1-4704-1496-2 (print); 978-1-4704-2749-8 (online)
Published electronically: June 30, 2015
Keywords:Degenerate parabolic equations, controllability, inverse problems, Carleman estimates

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Controllability and inverse source problems: Notation and main results
  • Chapter 3. Global Carleman estimates for weakly degenerate operators
  • Chapter 4. Some Hardy-type inequalities (proof of Lemma 3.18)
  • Chapter 5. Asymptotic properties of elements of $H^2 (\Omega ) \cap H^1 _{A,0}(\Omega )$
  • Chapter 6. Proof of the topological lemma 3.21
  • Chapter 7. Outlines of the proof of Theorems 3.23 and 3.26
  • Chapter 8. Step 1: computation of the scalar product on subdomains (proof of Lemmas 7.1 and 7.16)
  • Chapter 9. Step 2: a first estimate of the scalar product: proof of Lemmas 7.2, 7.4, 7.18 and 7.19
  • Chapter 10. Step 3: the limits as $\Omega ^\delta \to \Omega $ (proof of Lemmas 7.5 and 7.20)
  • Chapter 11. Step 4: partial Carleman estimate (proof of Lemmas 7.6 and 7.21)
  • Chapter 12. Step 5: from the partial to the global Carleman estimate (proof of Lemmas 7.9–7.11)
  • Chapter 13. Step 6: global Carleman estimate (proof of Lemmas 7.12, 7.14 and 7.15)
  • Chapter 14. Proof of observability and controllability results
  • Chapter 15. Application to some inverse source problems: proof of Theorems 2.9 and 2.11
  • Chapter 16. Controllability and inverse source problems: notation and main results
  • Chapter 17. Global Carleman estimates for strongly degenerate operators
  • Chapter 18. Hardy-type inequalities: proof of Lemma 17.10 and applications
  • Chapter 19. Global Carleman estimates in the strongly degenerate case: proof of Theorem 17.7
  • Chapter 20. Proof of Theorem 17.6 (observability inequality)
  • Chapter 21. Lack of null controllability when $\alpha \geq 2$: proof of Proposition 16.5
  • Chapter 22. Explosion of the controllability cost as $\alpha \to 2^-$ in space dimension $1$: proof of Proposition 16.7
  • Chapter 23. Some open problems


Degenerate parabolic operators have received increasing attention in recent years because they are associated with both important theoretical analysis, such as stochastic diffusion processes, and interesting applications to engineering, physics, biology, and economics. This manuscript has been conceived to introduce the reader to global Carleman estimates for a class of parabolic operators which may degenerate at the boundary of the space domain, in the normal direction to the boundary. Such a kind of degeneracy is relevant to study the invariance of a domain with respect to a given stochastic diffusion flow, and appears naturally in climatology models. Global Carleman estimates are a priori estimates in weighted Sobolev norms for solutions of linear partial differential equations subject to boundary conditions. Such estimates proved to be extremely useful for several kinds of uniformly parabolic equations and systems. This is the first work where such estimates are derived for degenerate parabolic operators in dimension higher than one. Applications to null controllability with locally distributed controls and inverse source problems are also developed in full detail. Compared to nondegenerate parabolic problems, the current context requires major technical adaptations and a frequent use of Hardy type inequalities. On the other hand, the treatment is essentially self-contained, and only calls upon standard results in Lebesgue measure theory, functional analysis and ordinary differential equations.

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