Global Carleman estimates for degenerate parabolic operators with applications
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Bibliographic Information
Global Carleman estimates for degenerate parabolic operators with applications
（Memoirs of the American Mathematical Society, no. 1133）
American Mathematical Society, [2016], c2015
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"January 2016, volume 239, number 1133 (fifth of 6 numbers)."
Includes bibliographical references and index
Description and Table of Contents
Description
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.
Table of Contents
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
by "Nielsen BookData"
