Global Carleman Estimates for Degenerate Parabolic Operators with Applications

ISBN-13: 9781470414962 / Angielski

Piermarco Cannarsa;P. Martinez;J. Vancostenoble

Opis

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.

Spis treści

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