Elliptic Carleman estimates and applications to stabilization and controllability. , Volume II, General boundary conditions on Riemannian manifolds / Jérôme Le Rousseau, Gilles Lebeau, Luc Robbiano.
Le Rousseau, Jérôme, author.; Lebeau, Gilles, author.; Robbiano, Luc, author.
2022
QA331
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Title Elliptic Carleman estimates and applications to stabilization and controllability. , Volume II, General boundary conditions on Riemannian manifolds / Jérôme Le Rousseau, Gilles Lebeau, Luc Robbiano.
Author Le Rousseau, Jérôme, author.
ISBN 9783030886707 (electronic bk.)
3030886700 (electronic bk.)
9783030886691 (print)
DOI https://doi.org/10.1007/978-3-030-88670-7
Published Cham, Switzerland : Birkhäuser, 2022.
Language English
Description 1 online resource (ix, 547 pages) : illustrations (some color).
Item Number 10.1007/978-3-030-88670-7 doi
Call Number QA331
Dewey Decimal Classification 512.7/3
Summary This monograph explores applications of Carleman estimates in the study of stabilization and controllability properties of partial differential equations, including quantified unique continuation, logarithmic stabilization of the wave equation, and null-controllability of the heat equation. Where the first volume derived these estimates in regular open sets in Euclidean space and Dirichlet boundary conditions, here they are extended to Riemannian manifolds and more general boundary conditions. The book begins with the study of Lopatinskii-Sapiro boundary conditions for the Laplace-Beltrami operator, followed by derivation of Carleman estimates for this operator on Riemannian manifolds. Applications of Carleman estimates are explored next: quantified unique continuation issues, a proof of the logarithmic stabilization of the boundary-damped wave equation, and a spectral inequality with general boundary conditions to derive the null-controllability result for the heat equation. Two additional chapters consider some more advanced results on Carleman estimates. The final part of the book is devoted to exposition of some necessary background material: elements of differential and Riemannian geometry, and Sobolev spaces and Laplace problems on Riemannian manifolds.
Bibliography, etc. Note Includes bibliographical references and indexes.
Access Note Access limited to authorized users.
Source of Description Online resource; title from PDF title page (SpringerLink, viewed April 26, 2022).
Added Author Lebeau, Gilles, author.
Robbiano, Luc, author.
Series Progress in nonlinear differential equations and their applications. PNLDE subseries in control ; v. 98.
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Introduction
Part 1: General Boundary Conditions
Lopatinskii-Sapiro Boundary Conditions
Fredholm Properties of Second-Order Elliptic Operators
Selfadjoint Operators under General Boundary Conditions
Part 2: Carleman Estimates on Riemannian Manifolds
Estimates on Riemannian Manifolds for Dirichlet Boundary Conditions
Pseudo-Differential Operators on a Half-Space
Sobolev Norms with a Large Parameter on a Manifold
Estimates for General Boundary Conditions
Part 3: Applications
Quantified Unique Continuation on a Riemannian Manifold
Stabilization of Waves under Neumann Boundary Damping
Spectral Inequality for General Boundary Conditions and Applications
Part 4: Further Aspects of Carleman Estimates
Carleman Estimates with Source Terms of Weaker Regularity
Optimal Estimates at the Boundary
Background Material: Geometry
Elements of Differential Geometry
Integration and Differential Operators on Manifolds
Elements of Riemannian Geometry
Sobolev Spaces and Laplace Problems on a Riemannian Manifold
Bibliography
Index
Index of Notation.

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