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GLOBAL CARLEMAN ESTIMATES FOR DEGENERATE PARABOLIC OPERATORS WITH APPLICATIONS
About this Title
P. Cannarsa, Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientifica, 00133 Roma, Italy, P. Martinez, Institut de Mathématiques de Toulouse, U.M.R. C.N.R.S. 5219, Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31 062 Toulouse Cedex 4, France and J. Vancostenoble, Institut de Mathématiques de Toulouse, U.M.R. C.N.R.S. 5219, Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31 062 Toulouse Cedex 4, France
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)
DOI: https://doi.org/10.1090/memo/1133
Published electronically: June 30, 2015
Keywords: Degenerate parabolic equations,
controllability,
inverse problems,
Carleman estimates,
Hardy type inequalities
MSC: Primary 35K65, 35R30, 93B05, 93B07, 93C20, 26D10
Table of Contents
Chapters
- 1. Introduction
1. Weakly degenerate operators with Dirichlet boundary conditions
- 2. Controllability and inverse source problems: Notation and main results
- 3. Global Carleman estimates for weakly degenerate operators
- 4. Some Hardy-type inequalities (proof of Lemma )
- 5. Asymptotic properties of elements of $H^2 (\Omega ) \cap H^1 _{A,0}(\Omega )$
- 6. Proof of the topological lemma
- 7. Outlines of the proof of Theorems and
- 8. Step 1: computation of the scalar product on subdomains (proof of Lemmas and )
- 9. Step 2: a first estimate of the scalar product: proof of Lemmas , , and
- 10. Step 3: the limits as $\Omega ^\delta \to \Omega$ (proof of Lemmas and )
- 11. Step 4: partial Carleman estimate (proof of Lemmas and )
- 12. Step 5: from the partial to the global Carleman estimate (proof of Lemmas –)
- 13. Step 6: global Carleman estimate (proof of Lemmas , and )
- 14. Proof of observability and controllability results
- 15. Application to some inverse source problems: proof of Theorems and
2. Strongly degenerate operators with Neumann boundary conditions
- 16. Controllability and inverse source problems: notation and main results
- 17. Global Carleman estimates for strongly degenerate operators
- 18. Hardy-type inequalities: proof of Lemma and applications
- 19. Global Carleman estimates in the strongly degenerate case: proof of Theorem
- 20. Proof of Theorem (observability inequality)
- 21. Lack of null controllability when $\alpha \geq 2$: proof of Proposition
- 22. Explosion of the controllability cost as $\alpha \to 2^-$ in space dimension $1$: proof of Proposition
3. Some open problems
- 23. Some open problems
Abstract
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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