Global Carleman Estimates for Degenerate Parabolic Operators with Applications

Häftad, Engelska, 2016

Av P. Cannarsa, P. Martinez, J. Vancostenoble

1 479 kr

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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.

Produktinformation

Utgivningsdatum2016-01-01
Mått178 x 254 x undefined mm
Vikt317 g
FormatHäftad
SpråkEngelska
SerieMemoirs of the American Mathematical Society
FörlagAmerican Mathematical Society
ISBN9781470414962

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IntroductionPart 1. Weakly degenerate operators with Dirichlet boundary conditionsControllability and inverse source problemsNotation and main resultsGlobal Carleman estimates for weakly degenerate operatorsSome 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 3.21Outlines of the proof of Theorems 3.23 and 3.26Step 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 7.19Step 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 resultsApplication to some inverse source problems: proof of Theorems 2.9 and 2.11Part 2. Strongly degenerate operators with Neumann boundary conditionsControllability and inverse source problems: notation and main resultsGlobal Carleman estimates for strongly degenerate operatorsHardy-type inequalities: proof of Lemma 17.10 and applicationsGlobal Carleman estimates in the strongly degenerate case: proof of Theorem 17.7Proof of Theorem 17.6 (observability inequality)Lack of null controllability when $\alpha\geq 2$: proof of Proposition 16.5Explosion of the controllability cost as $\alpha\to 2^-$ in space dimension $1$: proof of Proposition 16.7Part 3. Some open problemsSome open problemsBibliographyIndex