GLOBAL CARLEMAN ESTIMATES FOR DEGENERATE PARABOLIC OPERATORS WITH APPLICATIONS

Abstract

* 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