Gilles Lebeau's Books

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I

Dirichlet Boundary Conditions on Euclidean Space

By: Jérôme Le Rousseau,Gilles Lebeau,Luc Robbiano

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Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I

Dirichlet Boundary Conditions on Euclidean Space

By: Jérôme Le Rousseau,Gilles Lebeau,Luc Robbiano

This monograph explores applications of Carleman estimates in the study of stabilization and controllability properties of partial differential equations, including the stabilization property of the damped wave equation and the null-controllability of the heat equation. All analysis is performed in the case of open sets in the Euclidean space; a second volume will extend this treatment to Riemannian manifolds. The first three chapters illustrate the derivation of Carleman estimates using pseudo-differential calculus with a large parameter. Continuation issues are then addressed, followed by a proof of the logarithmic stabilization of the damped wave equation by means of two alternative proofs of the resolvent estimate for the generator of a damped wave semigroup. The authors then discuss null-controllability of the heat equation, its equivalence with observability, and how the spectral inequality allows one to either construct a control function or prove the observability inequality. The final part of the book is devoted to the exposition of some necessary background material: the theory of distributions, invariance under change of variables, elliptic operators with Dirichlet data and associated semigroup, and some elements from functional analysis and semigroup theory.

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410

Publisher

Springer Nature

Published Date

ISBN 10

3030886743

ISBN 13

9783030886745

The Hypoelliptic Laplacian and Ray-Singer Metrics. (AM-167)

By: Jean-Michel Bismut,Gilles Lebeau

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The Hypoelliptic Laplacian and Ray-Singer Metrics. (AM-167)

By: Jean-Michel Bismut,Gilles Lebeau

This book presents the analytic foundations to the theory of the hypoelliptic Laplacian. The hypoelliptic Laplacian, a second-order operator acting on the cotangent bundle of a compact manifold, is supposed to interpolate between the classical Laplacian and the geodesic flow. Jean-Michel Bismut and Gilles Lebeau establish the basic functional analytic properties of this operator, which is also studied from the perspective of local index theory and analytic torsion. The book shows that the hypoelliptic Laplacian provides a geometric version of the Fokker-Planck equations. The authors give the proper functional analytic setting in order to study this operator and develop a pseudodifferential calculus, which provides estimates on the hypoelliptic Laplacian's resolvent. When the deformation parameter tends to zero, the hypoelliptic Laplacian converges to the standard Hodge Laplacian of the base by a collapsing argument in which the fibers of the cotangent bundle collapse to a point. For the local index theory, small time asymptotics for the supertrace of the associated heat kernel are obtained. The Ray-Singer analytic torsion of the hypoelliptic Laplacian as well as the associated Ray-Singer metrics on the determinant of the cohomology are studied in an equivariant setting, resulting in a key comparison formula between the elliptic and hypoelliptic analytic torsions.

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377

Publisher

Princeton University Press

Published Date

ISBN 10

0691137323

ISBN 13

9780691137322

Diffraction by an Immersed Elastic Wedge

By: Jean-Pierre Croisille,Gilles Lebeau

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Diffraction by an Immersed Elastic Wedge

By: Jean-Pierre Croisille,Gilles Lebeau

This series reports on new developments in mathematical research and teaching - quickly, informally and at a high level. The type of material considered for publication includes 1. Research monographs 2. Lectures on a new field or presentations of a new angle in a classical field 3. Summer schools and intensive courses on topics of current research. Texts which are out of print but still in demand may also be considered. The timeliness of a manuscript is sometimes more important than its form, which might be preliminary or tentative. Details of the editorial policy can be found on the inside front-cover of a current volume. Manuscripts should be submitted in camera-ready form according to Springer-Verlag's specification: technical instructions will be sent on request. TEX macros may be found at: http://www.springer.de/math/authors/b- tex.html Select the version of TEX you use and then click on "Monographs". A subject index should be included. We recommend contacting the publisher or the series editors at an early stage of your project. Addresses are given on the inside back-cover.

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146

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Springer Science & Business Media

Published Date

ISBN 10

3540668101

ISBN 13

9783540668107

Integral Fourier Operators

Proceedings of a Summer School, Ouagadougou 14–25 September 2015

By: Michèle Audin,Catherine Ducourtioux,Françoise Ouédraogo,René Schulz,Julio Delgado,Michael Ruzhansky,Gilles Lebeau

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Integral Fourier Operators

Proceedings of a Summer School, Ouagadougou 14–25 September 2015

By: Michèle Audin,Catherine Ducourtioux,Françoise Ouédraogo,René Schulz,Julio Delgado,Michael Ruzhansky,Gilles Lebeau

This volume of contributions based on lectures delivered at a school on Fourier Integral Operators held in Ouagadougou, Burkina Faso, 14–26 September 2015, provides an introduction to Fourier Integral Operators (FIO) for a readership of Master and PhD students as well as any interested layperson. Considering the wide spectrum of their applications and the richness of the mathematical tools they involve, FIOs lie the cross-road of many a field. This volume offers the necessary background, whether analytic or geometric, to get acquainted with FIOs, complemented by more advanced material presenting various aspects of active research in that area.

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252

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Universitätsverlag Potsdam

Published Date

ISBN 10

386956413X

ISBN 13

9783869564135

Between France and New France

Life Aboard the Tall Sailing Ships

By: Gilles Proulx

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Between France and New France

Life Aboard the Tall Sailing Ships

By: Gilles Proulx

Between France and New France is an absorbing look at life abroad the sailing vessels which plied the North Atlantic during the French colonial era in North America. Focusing on the first half of the eighteenth century and the Seven Years' War period, this book analyses four major aspects of the crossing: martime traffic and the outfit of vessels; the Atlantic course and navigation; the people and their occupations; and life aboard the ship. Together they present a fascinating view of sea life. Gilles Proulx has used official correspondence between the Minister of marine and the Canadian colonial authorities, and the papers seized on boarded vessels, as well as over one hundred log-books and personal diaries, to obtain a wealth of detail about the rigours of the colonial shipboard experience. In addition, many photographs, both colour and black and white, have been included to illustrate this exciting period in Canadian history.

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173

Publisher

Dundurn

Published Date

ISBN 10

0919670814

ISBN 13

9780919670815

Similarity Problems and Completely Bounded Maps

By: Gilles Pisier

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Similarity Problems and Completely Bounded Maps

By: Gilles Pisier

These notes revolve around three similarity problems, appearing in three different contexts, but all dealing with the space B(H) of all bounded operators on a complex Hilbert space H. The first one deals with group representations, the second one with C* -algebras and the third one with the disc algebra. We describe them in detail in the introduction which follows. This volume is devoted to the background necessary to understand these three problems, to the solutions that are known in some special cases and to numerous related concepts, results, counterexamples or extensions which their investigation has generated. While the three problems seem different, it is possible to place them in a common framework using the key concept of "complete boundedness", which we present in detail. Using this notion, the three problems can all be formulated as asking whether "boundedness" implies "complete boundedness" for linear maps satisfying certain additional algebraic identities.

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210

Publisher

Springer

Published Date

ISBN 10

3540445633

ISBN 13

9783540445630

The Hypoelliptic Laplacian and Ray-Singer Metrics

By: Jean-Michel Bismut,Gilles Lebeau

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The Hypoelliptic Laplacian and Ray-Singer Metrics

By: Jean-Michel Bismut,Gilles Lebeau

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378

Publisher

Princeton University Press

Published Date

ISBN 10

1400829062

ISBN 13

9781400829064

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I

Dirichlet Boundary Conditions on Euclidean Space

By: Jérôme Le Rousseau,Gilles Lebeau,Luc Robbiano

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Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I

Dirichlet Boundary Conditions on Euclidean Space

By: Jérôme Le Rousseau,Gilles Lebeau,Luc Robbiano

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410

Publisher

Springer Nature

Published Date

ISBN 10

3030886743

ISBN 13

9783030886745

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II

General Boundary Conditions on Riemannian Manifolds

By: Jérôme Le Rousseau,Gilles Lebeau,Luc Robbiano

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Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II

General Boundary Conditions on Riemannian Manifolds

By: Jérôme Le Rousseau,Gilles Lebeau,Luc Robbiano

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.

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542

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Springer Nature

Published Date

ISBN 10

3030886700

ISBN 13

9783030886707

Diffraction by an Immersed Elastic Wedge

By: Jean-Pierre Croisille,Gilles Lebeau

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Diffraction by an Immersed Elastic Wedge

By: Jean-Pierre Croisille,Gilles Lebeau

This monograph presents the mathematical description and numerical computation of the high-frequency diffracted wave by an immersed elastic wave with normal incidence. The mathematical analysis is based on the explicit description of the principal symbol of the pseudo-differential operator connected with the coupled linear problem elasticity/fluid by the wedge interface. This description is subsequently used to derive an accurate numerical computation of diffraction diagrams for different incoming waves in the fluid, and for different wedge angles. The method can be applied to any problem of coupled waves by a wedge interface. This work is of interest for any researcher concerned with high frequency wave scattering, especially mathematicians, acousticians, engineers.

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138

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Springer

Published Date

ISBN 10

3540466983

ISBN 13

9783540466987

Integral Fourier Operators

Proceedings of a Summer School, Ouagadougou 14–25 September 2015

By: Michèle Audin,Catherine Ducourtioux,Françoise Ouédraogo,René Schulz,Julio Delgado,Michael Ruzhansky,Gilles Lebeau

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Integral Fourier Operators

Proceedings of a Summer School, Ouagadougou 14–25 September 2015

By: Michèle Audin,Catherine Ducourtioux,Françoise Ouédraogo,René Schulz,Julio Delgado,Michael Ruzhansky,Gilles Lebeau

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Solve jigsaw puzzle

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Page Count

252

Publisher

Universitätsverlag Potsdam

Published Date

ISBN 10

386956413X

ISBN 13

9783869564135

Book's by Gilles Lebeau

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I

The Hypoelliptic Laplacian and Ray-Singer Metrics. (AM-167)

The Hypoelliptic Laplacian and Ray-Singer Metrics. (AM-167)

Diffraction by an Immersed Elastic Wedge

Diffraction by an Immersed Elastic Wedge

Integral Fourier Operators

Integral Fourier Operators

Between France and New France

Between France and New France

Similarity Problems and Completely Bounded Maps

Similarity Problems and Completely Bounded Maps

The Hypoelliptic Laplacian and Ray-Singer Metrics

The Hypoelliptic Laplacian and Ray-Singer Metrics

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II

Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume II

Diffraction by an Immersed Elastic Wedge

Diffraction by an Immersed Elastic Wedge

Integral Fourier Operators

Integral Fourier Operators

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