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Diffraction by an immersed elastic wedge / / Jean-Pierre Croisille, Gilles Lebeau
Diffraction by an immersed elastic wedge / / Jean-Pierre Croisille, Gilles Lebeau
Autore Croisille Jean-Pierre <1961->
Edizione [1st ed. 1999.]
Pubbl/distr/stampa Berlin, Germany ; ; New York, New York : , : Springer, , [1999]
Descrizione fisica 1 online resource (VIII, 140 p.)
Disciplina 518
Collana Lecture Notes in Mathematics
Soggetto topico Waves - Diffraction
Wedges
Wave-motion, Theory of
ISBN 3-540-46698-3
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Notation and results -- The spectral function -- Proofs of the results -- Numerical algorithm -- Numerical results.
Record Nr. UNISA-996466650103316
Croisille Jean-Pierre <1961->  
Berlin, Germany ; ; New York, New York : , : Springer, , [1999]
Materiale a stampa
Lo trovi qui: Univ. di Salerno
Opac: Controlla la disponibilità qui
Diffraction by an immersed elastic wedge / / Jean-Pierre Croisille, Gilles Lebeau
Diffraction by an immersed elastic wedge / / Jean-Pierre Croisille, Gilles Lebeau
Autore Croisille Jean-Pierre <1961->
Edizione [1st ed. 1999.]
Pubbl/distr/stampa Berlin, Germany ; ; New York, New York : , : Springer, , [1999]
Descrizione fisica 1 online resource (VIII, 140 p.)
Disciplina 518
Collana Lecture Notes in Mathematics
Soggetto topico Waves - Diffraction
Wedges
Wave-motion, Theory of
ISBN 3-540-46698-3
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Notation and results -- The spectral function -- Proofs of the results -- Numerical algorithm -- Numerical results.
Record Nr. UNINA-9910146313403321
Croisille Jean-Pierre <1961->  
Berlin, Germany ; ; New York, New York : , : Springer, , [1999]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Elliptic Carleman estimates and applications to stabilization and controllability . Volume 1 : Dirichlet boundary conditions on Euclidean space / / Jérôme Le Rousseau, Gilles Lebeau, and Luc Robbiano
Elliptic Carleman estimates and applications to stabilization and controllability . Volume 1 : Dirichlet boundary conditions on Euclidean space / / Jérôme Le Rousseau, Gilles Lebeau, and Luc Robbiano
Autore Le Rousseau Jérôme
Pubbl/distr/stampa Cham, Switzerland : , : Birkhauser Verlag, , [2022]
Descrizione fisica 1 online resource (410 pages)
Disciplina 515.353
Collana Progress in Nonlinear Differential Equations and Their Applications
Soggetto topico Differential equations, Partial
Differential equations, Partial - Asymptotic theory
Equacions en derivades parcials
Soggetto genere / forma Llibres electrònics
ISBN 3-030-88674-3
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Contents -- Part 1. Calculus with a Large Parameter, Carleman Estimates Derivation -- Chapter 1. Introduction -- 1.1. Some Aspects of Unique Continuation -- 1.2. Form of Carleman Estimates and Quantification of Unique Continuation -- 1.3. Application to Stabilization and Controllability -- 1.4. Outline -- 1.5. Missing Subjects -- 1.6. Acknowledgement -- 1.7. Some Notation -- 1.7.1. Open Sets -- 1.7.2. Euclidean Inner Products and Norms -- 1.7.3. Differential Operators -- 1.7.4. Fourier Transformation -- 1.7.5. Function Norms -- 1.7.6. Homogeneity and Conic Sets -- 1.7.7. Miscellaneous -- Chapter 2. (Pseudo-)Differential Operators with a Large Parameter -- 2.1. Introduction -- 2.2. Classes of Symbols -- 2.2.1. Homogeneous and Polyhomogeneous Symbols -- 2.3. Classes of Pseudo-Differential Operators -- 2.4. Oscillatory Integrals -- 2.5. Symbol Calculus -- 2.6. Sobolev Spaces and Operator Bound -- 2.7. Positivity Inequalities of Gårding Type -- 2.8. Parametrices -- 2.9. Action of Change of Variables -- 2.10. Tangential Operators -- 2.11. Semi-Classical Operators -- 2.12. Standard Pseudo-Differential Operators -- 2.13. Notes -- Appendix -- 2.A. Technical Proofs for Pseudo-Differential Calculus -- 2.A.1. Symbol Asymptotic Series: Proof of Lemma 2.4 -- 2.A.2. Action on the Schwartz Space: Proof of Proposition 2.10 -- 2.A.3. Proofs of Results on Oscillatory Integrals -- 2.A.3.1. Definitions of Oscillatory Integrals: Proof of Theorem 2.11 -- 2.A.3.2. Definitions of Oscillatory Integrals: Proof of Theorem 2.16 -- 2.A.4. Proofs of the Results on Symbol Calculus -- 2.A.5. Proof of Theorem 2.26: Sobolev Bound -- 2.A.6. Proofs of the Gårding Inequalities -- 2.A.6.1. Proof of the Local Gårding Inequality of Theorem 2.28 -- 2.A.6.2. Proof of the Microlocal Gårding Inequality of Theorem 2.29 -- 2.A.6.3. Proof of the Gårding Inequalities for Systems.
2.A.7. Parametrix Construction and Properties -- 2.A.8. A Characterization of Ellipticity -- Chapter 3. Carleman Estimate for a Second-Order Elliptic Operator -- 3.1. Setting -- 3.2. Weight Function and Conjugated Operator -- 3.2.1. Conjugated Operator -- 3.2.2. Characteristic Set and Sub-ellipticity Property -- 3.2.3. Invariance Under Change of Variables -- 3.3. Local Estimate Away from Boundaries -- 3.4. Local Estimates at the Boundary -- 3.4.1. Some Remarks -- 3.4.2. Proofs in Adapted Local Coordinates -- 3.5. Patching Estimates -- 3.6. Global Estimates with Observation Terms -- 3.6.1. A Global Estimate with an Inner Observation Term -- 3.6.2. A Global Estimate with a Boundary Observation Term -- 3.7. Alternative Approach -- 3.7.1. A Modified Carleman Estimate Derivation Away from Boundaries -- 3.7.2. A Modified Carleman Estimate Derivation at a Boundary -- 3.7.3. Alternative Derivation in the Case of Limited Smoothness -- 3.7.4. Valuable Aspects of the Different Approaches -- 3.8. Notes -- Appendices -- 3.A. Poisson Bracket and Weight Function -- 3.A.1. Smoothness of the Characteristic Set -- 3.A.2. Expression of the Poisson Bracket -- 3.A.3. Construction of a Weight Function -- 3.A.4. Local Extension of the Domain Where Sub-ellipticity Holds -- 3.B. Symbol Positivity -- 3.B.1. Symbol Positivity Away from a Boundary -- 3.B.2. Tangential Symbol Positivity Near a Boundary -- 3.B.3. Proof of Lemma 3.27 -- 3.B.4. Symbol Positivity in the Modified Approach -- 3.C. An Explicit Computation -- Chapter 4. Optimality Aspects of Carleman Estimates -- 4.1. On the Necessity of the Sub-ellipticity Property -- 4.1.1. Bracket Nonnegativity -- 4.1.2. Optimal Strength in the Large Parameter and Bracket Positivity -- 4.2. Limiting Weights and Limiting Carleman Estimates -- 4.2.1. Limiting Weights -- 4.2.2. Convexification.
4.2.3. Limiting Carleman Estimates Away from a Boundary -- 4.2.4. Global Limiting Carleman Estimates -- 4.3. Carleman Weight Behavior at a Boundary -- 4.4. Notes -- Appendix -- 4.A. Some Technical Results -- 4.A.1. A Linear Algebra Lemma -- 4.A.2. Sub-ellipticity for First-Order Operators with Linear Symbols -- 4.A.3. A Particular Class of Limiting Weights -- Part 2. Applications of Carleman Estimates -- Chapter 5. Unique Continuation -- 5.1. Introduction -- 5.2. Local and Global Unique Continuation -- 5.3. Quantification of Unique Continuation -- 5.3.1. Quantified Unique Continuation Away from a Boundary -- 5.3.2. Quantified Unique Continuation Up to a Boundary -- 5.4. Unique Continuation Initiated at the Boundary -- 5.5. Unique Continuation Without Any Prescribed Boundary Condition -- 5.6. Notes -- Appendix -- 5.A. A Hardy Inequality -- Chapter 6. Stabilization of the Wave Equation with an Inner Damping -- 6.1. Introduction and Setting -- 6.2. Preliminaries on the Damped Wave Equation -- 6.3. Stabilization and Resolvent Estimate -- 6.4. Remarks and Non-Quantified Stabilization Results -- 6.4.1. Comparison with Exponential Stability -- 6.4.2. Zero Eigenvalue -- 6.4.3. Non-Quantified Stabilization Results -- 6.5. Resolvent Estimate for the Damped Wave Generator -- 6.5.1. Estimations Through an Interpolation Inequality -- 6.5.2. Estimations Through the Derivation of a Global Carleman Estimate -- 6.6. Alternative Proof Scheme of the Resolvent Estimate -- 6.7. Notes -- Appendices -- 6.A. The Generator of the Damped-Wave Semigroup -- 6.B. Well-Posedness of the Damped Wave Equation -- 6.B.1. Proof of Well-Posedness -- 6.B.2. Other Formulations of Weak Solutions -- 6.C. From a Resolvent to a Semigroup Stabilization Estimate -- 6.D. Proofs of Non-Quantified Stabilization Results -- 6.D.1. Proof of Proposition 6.12 -- 6.D.2. Proof of Proposition 6.14.
6.D.3. Proof of Proposition 6.15 -- Chapter 7. Controllability of Parabolic Equations -- 7.1. Introduction and Setting -- 7.2. Exact Controllability for a Parabolic Equation -- 7.3. Null-Controllability for Semigroup Operators -- 7.4. Observability for the Semigroup Parabolic Equation -- 7.5. A Spectral Inequality -- 7.5.1. Spectral Inequality Through an Interpolation Inequality -- 7.5.2. Spectral Inequality Through the Derivation of a Global Carleman Estimate -- 7.5.3. Sharpness of the Spectral Inequality -- 7.6. Partial Observability and Partial Control -- 7.7. Construction of a Control Function for a Parabolic Equation -- 7.8. Dual Approach for Observability and Control Cost -- 7.9. Properties of the Reachable Set and Generalizations -- 7.10. Boundary Null-Controllability for Parabolic Equations -- 7.11. Notes -- Part 3. Background Material: Analysis and Evolution Equations -- Chapter 8. A Short Review of Distribution Theory -- 8.1. Distributions on an Open Set of Rd and on a Manifold -- 8.1.1. Test Functions -- 8.1.2. Definition of Distributions and Basic Properties -- 8.1.2.1. Localization and Support -- 8.1.2.2. Distributions with Compact Support -- 8.1.3. Composition by Diffeomorphisms, Distributions on aManifold -- 8.2. Temperate Distributions on Rd and Fourier Transformation -- 8.2.1. The Schwartz Space and Temperate Distributions -- 8.2.2. The Fourier Transformation on S(Rd), S'(Rd), and L2(Rd) -- 8.3. Distributions on a Product Space -- 8.3.1. Tensor Products of Functions -- 8.3.2. Tensor Products of Distributions -- 8.3.3. Convolution -- 8.3.4. The Kernel Theorem (Various Forms) -- 8.4. Notes -- Chapter 9. Invariance Under Change of Variables -- 9.1. A Review of the Actions of Change of Variables -- 9.1.1. Pullbacks and Push-Forwards -- 9.1.2. Action of a Change of Variables on a Differential Operator.
9.2. Action on Symplectic Structures -- 9.2.1. The Symplectic Two-Form -- 9.2.2. Hamiltonian Vector Fields -- 9.2.3. Poisson Bracket -- 9.3. Invariance of the Sub-ellipticity Condition -- 9.3.1. Action of a Change of Variables on the Conjugated Operator -- 9.3.2. The Sub-ellipticity Condition -- 9.4. Normal Geodesic Coordinates -- Chapter 10. Elliptic Operator with Dirichlet Data and Associated Semigroup -- 10.1. Resolvent and Spectral Properties of Elliptic Operators -- 10.1.1. Basic Properties of Second-Order Elliptic Operators -- 10.1.2. Spectral Properties -- 10.1.3. A Sobolev Scale and Operator Extensions -- 10.2. The Parabolic Semigroup -- 10.2.1. Spectral Representation of the Semigroup -- 10.2.2. Well-Posedness: An Elementary Proof -- 10.2.3. Additional Properties of the Parabolic Semigroup -- 10.2.4. Properties of the Parabolic Kernel -- 10.3. The Nonhomogeneous Parabolic Cauchy Problem -- 10.3.1. Properties of the Duhamel Term -- 10.3.2. Abstract Solutions of the Nonhomogeneous Semigroup Equations -- 10.3.3. Strong Solutions -- 10.3.4. Weak Solutions -- 10.4. Elementary Form of the Maximum Principle -- 10.5. The Dirichlet Lifting Map -- 10.6. Parabolic Equation with Dirichlet Boundary Data -- Chapter 11. Some Elements of Functional Analysis -- 11.1. Linear Operators in Banach Spaces -- 11.2. Continuous and Bounded Operators -- 11.3. Spectrum of a Linear Operator in a Banach Space -- 11.4. Adjoint Operator -- 11.5. Fredholm Operators -- 11.5.1. Characterization of Bounded Fredholm Operators -- 11.6. Linear Operators in Hilbert Spaces -- Chapter 12. Some Elements of Semigroup Theory -- 12.1. Strongly Continuous Semigroups -- 12.1.1. Definition and Basic Properties -- 12.1.2. The Hille-Yosida Theorem -- 12.1.3. The Lumer-Phillips Theorem -- 12.2. Differentiable and Analytic Semigroups -- 12.3. Mild Solution of the Inhomogeneous Cauchy Problem.
12.4. The Case of a Hilbert Space.
Record Nr. UNISA-996466420403316
Le Rousseau Jérôme  
Cham, Switzerland : , : Birkhauser Verlag, , [2022]
Materiale a stampa
Lo trovi qui: Univ. di Salerno
Opac: Controlla la disponibilità qui
Elliptic Carleman estimates and applications to stabilization and controllability . Volume II : general boundary conditions on Riemannian manifolds / / Jérôme Le Rousseau, Gilles Lebeau, and Luc Robbiano
Elliptic Carleman estimates and applications to stabilization and controllability . Volume II : general boundary conditions on Riemannian manifolds / / Jérôme Le Rousseau, Gilles Lebeau, and Luc Robbiano
Autore Le Rousseau Jérôme
Pubbl/distr/stampa Cham, Switzerland : , : Springer, , [2022]
Descrizione fisica 1 online resource (542 pages)
Disciplina 512.73
Collana Progress in Nonlinear Differential Equations and Their Applications
Soggetto topico Riemannian manifolds
Boundary value problems
Varietats de Riemann
Problemes de contorn
Soggetto genere / forma Llibres electrònics
ISBN 3-030-88670-0
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Record Nr. UNISA-996472038003316
Le Rousseau Jérôme  
Cham, Switzerland : , : Springer, , [2022]
Materiale a stampa
Lo trovi qui: Univ. di Salerno
Opac: Controlla la disponibilità qui
Elliptic Carleman estimates and applications to stabilization and controllability . Volume II : general boundary conditions on Riemannian manifolds / / Jérôme Le Rousseau, Gilles Lebeau, and Luc Robbiano
Elliptic Carleman estimates and applications to stabilization and controllability . Volume II : general boundary conditions on Riemannian manifolds / / Jérôme Le Rousseau, Gilles Lebeau, and Luc Robbiano
Autore Le Rousseau Jérôme
Pubbl/distr/stampa Cham, Switzerland : , : Springer, , [2022]
Descrizione fisica 1 online resource (542 pages)
Disciplina 512.73
Collana Progress in Nonlinear Differential Equations and Their Applications
Soggetto topico Riemannian manifolds
Boundary value problems
Varietats de Riemann
Problemes de contorn
Soggetto genere / forma Llibres electrònics
ISBN 3-030-88670-0
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Record Nr. UNINA-9910564679003321
Le Rousseau Jérôme  
Cham, Switzerland : , : Springer, , [2022]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Elliptic Carleman estimates and applications to stabilization and controllability . Volume 1 : Dirichlet boundary conditions on Euclidean space / / Jérôme Le Rousseau, Gilles Lebeau, and Luc Robbiano
Elliptic Carleman estimates and applications to stabilization and controllability . Volume 1 : Dirichlet boundary conditions on Euclidean space / / Jérôme Le Rousseau, Gilles Lebeau, and Luc Robbiano
Autore Le Rousseau Jérôme
Pubbl/distr/stampa Cham, Switzerland : , : Birkhauser Verlag, , [2022]
Descrizione fisica 1 online resource (410 pages)
Disciplina 515.353
Collana Progress in Nonlinear Differential Equations and Their Applications
Soggetto topico Differential equations, Partial
Differential equations, Partial - Asymptotic theory
Equacions en derivades parcials
Soggetto genere / forma Llibres electrònics
ISBN 3-030-88674-3
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Contents -- Part 1. Calculus with a Large Parameter, Carleman Estimates Derivation -- Chapter 1. Introduction -- 1.1. Some Aspects of Unique Continuation -- 1.2. Form of Carleman Estimates and Quantification of Unique Continuation -- 1.3. Application to Stabilization and Controllability -- 1.4. Outline -- 1.5. Missing Subjects -- 1.6. Acknowledgement -- 1.7. Some Notation -- 1.7.1. Open Sets -- 1.7.2. Euclidean Inner Products and Norms -- 1.7.3. Differential Operators -- 1.7.4. Fourier Transformation -- 1.7.5. Function Norms -- 1.7.6. Homogeneity and Conic Sets -- 1.7.7. Miscellaneous -- Chapter 2. (Pseudo-)Differential Operators with a Large Parameter -- 2.1. Introduction -- 2.2. Classes of Symbols -- 2.2.1. Homogeneous and Polyhomogeneous Symbols -- 2.3. Classes of Pseudo-Differential Operators -- 2.4. Oscillatory Integrals -- 2.5. Symbol Calculus -- 2.6. Sobolev Spaces and Operator Bound -- 2.7. Positivity Inequalities of Gårding Type -- 2.8. Parametrices -- 2.9. Action of Change of Variables -- 2.10. Tangential Operators -- 2.11. Semi-Classical Operators -- 2.12. Standard Pseudo-Differential Operators -- 2.13. Notes -- Appendix -- 2.A. Technical Proofs for Pseudo-Differential Calculus -- 2.A.1. Symbol Asymptotic Series: Proof of Lemma 2.4 -- 2.A.2. Action on the Schwartz Space: Proof of Proposition 2.10 -- 2.A.3. Proofs of Results on Oscillatory Integrals -- 2.A.3.1. Definitions of Oscillatory Integrals: Proof of Theorem 2.11 -- 2.A.3.2. Definitions of Oscillatory Integrals: Proof of Theorem 2.16 -- 2.A.4. Proofs of the Results on Symbol Calculus -- 2.A.5. Proof of Theorem 2.26: Sobolev Bound -- 2.A.6. Proofs of the Gårding Inequalities -- 2.A.6.1. Proof of the Local Gårding Inequality of Theorem 2.28 -- 2.A.6.2. Proof of the Microlocal Gårding Inequality of Theorem 2.29 -- 2.A.6.3. Proof of the Gårding Inequalities for Systems.
2.A.7. Parametrix Construction and Properties -- 2.A.8. A Characterization of Ellipticity -- Chapter 3. Carleman Estimate for a Second-Order Elliptic Operator -- 3.1. Setting -- 3.2. Weight Function and Conjugated Operator -- 3.2.1. Conjugated Operator -- 3.2.2. Characteristic Set and Sub-ellipticity Property -- 3.2.3. Invariance Under Change of Variables -- 3.3. Local Estimate Away from Boundaries -- 3.4. Local Estimates at the Boundary -- 3.4.1. Some Remarks -- 3.4.2. Proofs in Adapted Local Coordinates -- 3.5. Patching Estimates -- 3.6. Global Estimates with Observation Terms -- 3.6.1. A Global Estimate with an Inner Observation Term -- 3.6.2. A Global Estimate with a Boundary Observation Term -- 3.7. Alternative Approach -- 3.7.1. A Modified Carleman Estimate Derivation Away from Boundaries -- 3.7.2. A Modified Carleman Estimate Derivation at a Boundary -- 3.7.3. Alternative Derivation in the Case of Limited Smoothness -- 3.7.4. Valuable Aspects of the Different Approaches -- 3.8. Notes -- Appendices -- 3.A. Poisson Bracket and Weight Function -- 3.A.1. Smoothness of the Characteristic Set -- 3.A.2. Expression of the Poisson Bracket -- 3.A.3. Construction of a Weight Function -- 3.A.4. Local Extension of the Domain Where Sub-ellipticity Holds -- 3.B. Symbol Positivity -- 3.B.1. Symbol Positivity Away from a Boundary -- 3.B.2. Tangential Symbol Positivity Near a Boundary -- 3.B.3. Proof of Lemma 3.27 -- 3.B.4. Symbol Positivity in the Modified Approach -- 3.C. An Explicit Computation -- Chapter 4. Optimality Aspects of Carleman Estimates -- 4.1. On the Necessity of the Sub-ellipticity Property -- 4.1.1. Bracket Nonnegativity -- 4.1.2. Optimal Strength in the Large Parameter and Bracket Positivity -- 4.2. Limiting Weights and Limiting Carleman Estimates -- 4.2.1. Limiting Weights -- 4.2.2. Convexification.
4.2.3. Limiting Carleman Estimates Away from a Boundary -- 4.2.4. Global Limiting Carleman Estimates -- 4.3. Carleman Weight Behavior at a Boundary -- 4.4. Notes -- Appendix -- 4.A. Some Technical Results -- 4.A.1. A Linear Algebra Lemma -- 4.A.2. Sub-ellipticity for First-Order Operators with Linear Symbols -- 4.A.3. A Particular Class of Limiting Weights -- Part 2. Applications of Carleman Estimates -- Chapter 5. Unique Continuation -- 5.1. Introduction -- 5.2. Local and Global Unique Continuation -- 5.3. Quantification of Unique Continuation -- 5.3.1. Quantified Unique Continuation Away from a Boundary -- 5.3.2. Quantified Unique Continuation Up to a Boundary -- 5.4. Unique Continuation Initiated at the Boundary -- 5.5. Unique Continuation Without Any Prescribed Boundary Condition -- 5.6. Notes -- Appendix -- 5.A. A Hardy Inequality -- Chapter 6. Stabilization of the Wave Equation with an Inner Damping -- 6.1. Introduction and Setting -- 6.2. Preliminaries on the Damped Wave Equation -- 6.3. Stabilization and Resolvent Estimate -- 6.4. Remarks and Non-Quantified Stabilization Results -- 6.4.1. Comparison with Exponential Stability -- 6.4.2. Zero Eigenvalue -- 6.4.3. Non-Quantified Stabilization Results -- 6.5. Resolvent Estimate for the Damped Wave Generator -- 6.5.1. Estimations Through an Interpolation Inequality -- 6.5.2. Estimations Through the Derivation of a Global Carleman Estimate -- 6.6. Alternative Proof Scheme of the Resolvent Estimate -- 6.7. Notes -- Appendices -- 6.A. The Generator of the Damped-Wave Semigroup -- 6.B. Well-Posedness of the Damped Wave Equation -- 6.B.1. Proof of Well-Posedness -- 6.B.2. Other Formulations of Weak Solutions -- 6.C. From a Resolvent to a Semigroup Stabilization Estimate -- 6.D. Proofs of Non-Quantified Stabilization Results -- 6.D.1. Proof of Proposition 6.12 -- 6.D.2. Proof of Proposition 6.14.
6.D.3. Proof of Proposition 6.15 -- Chapter 7. Controllability of Parabolic Equations -- 7.1. Introduction and Setting -- 7.2. Exact Controllability for a Parabolic Equation -- 7.3. Null-Controllability for Semigroup Operators -- 7.4. Observability for the Semigroup Parabolic Equation -- 7.5. A Spectral Inequality -- 7.5.1. Spectral Inequality Through an Interpolation Inequality -- 7.5.2. Spectral Inequality Through the Derivation of a Global Carleman Estimate -- 7.5.3. Sharpness of the Spectral Inequality -- 7.6. Partial Observability and Partial Control -- 7.7. Construction of a Control Function for a Parabolic Equation -- 7.8. Dual Approach for Observability and Control Cost -- 7.9. Properties of the Reachable Set and Generalizations -- 7.10. Boundary Null-Controllability for Parabolic Equations -- 7.11. Notes -- Part 3. Background Material: Analysis and Evolution Equations -- Chapter 8. A Short Review of Distribution Theory -- 8.1. Distributions on an Open Set of Rd and on a Manifold -- 8.1.1. Test Functions -- 8.1.2. Definition of Distributions and Basic Properties -- 8.1.2.1. Localization and Support -- 8.1.2.2. Distributions with Compact Support -- 8.1.3. Composition by Diffeomorphisms, Distributions on aManifold -- 8.2. Temperate Distributions on Rd and Fourier Transformation -- 8.2.1. The Schwartz Space and Temperate Distributions -- 8.2.2. The Fourier Transformation on S(Rd), S'(Rd), and L2(Rd) -- 8.3. Distributions on a Product Space -- 8.3.1. Tensor Products of Functions -- 8.3.2. Tensor Products of Distributions -- 8.3.3. Convolution -- 8.3.4. The Kernel Theorem (Various Forms) -- 8.4. Notes -- Chapter 9. Invariance Under Change of Variables -- 9.1. A Review of the Actions of Change of Variables -- 9.1.1. Pullbacks and Push-Forwards -- 9.1.2. Action of a Change of Variables on a Differential Operator.
9.2. Action on Symplectic Structures -- 9.2.1. The Symplectic Two-Form -- 9.2.2. Hamiltonian Vector Fields -- 9.2.3. Poisson Bracket -- 9.3. Invariance of the Sub-ellipticity Condition -- 9.3.1. Action of a Change of Variables on the Conjugated Operator -- 9.3.2. The Sub-ellipticity Condition -- 9.4. Normal Geodesic Coordinates -- Chapter 10. Elliptic Operator with Dirichlet Data and Associated Semigroup -- 10.1. Resolvent and Spectral Properties of Elliptic Operators -- 10.1.1. Basic Properties of Second-Order Elliptic Operators -- 10.1.2. Spectral Properties -- 10.1.3. A Sobolev Scale and Operator Extensions -- 10.2. The Parabolic Semigroup -- 10.2.1. Spectral Representation of the Semigroup -- 10.2.2. Well-Posedness: An Elementary Proof -- 10.2.3. Additional Properties of the Parabolic Semigroup -- 10.2.4. Properties of the Parabolic Kernel -- 10.3. The Nonhomogeneous Parabolic Cauchy Problem -- 10.3.1. Properties of the Duhamel Term -- 10.3.2. Abstract Solutions of the Nonhomogeneous Semigroup Equations -- 10.3.3. Strong Solutions -- 10.3.4. Weak Solutions -- 10.4. Elementary Form of the Maximum Principle -- 10.5. The Dirichlet Lifting Map -- 10.6. Parabolic Equation with Dirichlet Boundary Data -- Chapter 11. Some Elements of Functional Analysis -- 11.1. Linear Operators in Banach Spaces -- 11.2. Continuous and Bounded Operators -- 11.3. Spectrum of a Linear Operator in a Banach Space -- 11.4. Adjoint Operator -- 11.5. Fredholm Operators -- 11.5.1. Characterization of Bounded Fredholm Operators -- 11.6. Linear Operators in Hilbert Spaces -- Chapter 12. Some Elements of Semigroup Theory -- 12.1. Strongly Continuous Semigroups -- 12.1.1. Definition and Basic Properties -- 12.1.2. The Hille-Yosida Theorem -- 12.1.3. The Lumer-Phillips Theorem -- 12.2. Differentiable and Analytic Semigroups -- 12.3. Mild Solution of the Inhomogeneous Cauchy Problem.
12.4. The Case of a Hilbert Space.
Record Nr. UNINA-9910556880603321
Le Rousseau Jérôme  
Cham, Switzerland : , : Birkhauser Verlag, , [2022]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
The hypoelliptic Laplacian and Ray-Singer metrics [[electronic resource] /] / Jean-Michel Bismut, Gilles Lebeau
The hypoelliptic Laplacian and Ray-Singer metrics [[electronic resource] /] / Jean-Michel Bismut, Gilles Lebeau
Autore Bismut Jean-Michel
Edizione [Course Book]
Pubbl/distr/stampa Princeton, : Princeton University Press, 2008
Descrizione fisica 1 online resource (378 p.)
Disciplina 515/.7242
Altri autori (Persone) LebeauGilles
Collana Annals of mathematics studies
Soggetto topico Differential equations, Hypoelliptic
Laplacian operator
Metric spaces
Soggetto genere / forma Electronic books.
ISBN 1-282-45837-X
9786612458378
1-4008-2906-2
Classificazione SK 620
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Frontmatter -- Contents -- Introduction -- Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles -- Chapter 2. The hypoelliptic Laplacian on the cotangent bundle -- Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel -- Chapter 4. Hypoelliptic Laplacians and odd Chern forms -- Chapter 5. The limit as t → +∞ and b → 0 of the superconnection forms -- Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics -- Chapter 7. The hypoelliptic torsion forms of a vector bundle -- Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula -- Chapter 9. A comparison formula for the Ray-Singer metrics -- Chapter 10. The harmonic forms for b → 0 and the formal Hodge theorem -- Chapter 11. A proof of equation (8.4.6) -- Chapter 12. A proof of equation (8.4.8) -- Chapter 13. A proof of equation (8.4.7) -- Chapter 14. The integration by parts formula -- Chapter 15. The hypoelliptic estimates -- Chapter 16. Harmonic oscillator and the J0 function -- Chapter 17. The limit of A'2φb,±H as b → 0 -- Bibliography -- Subject Index -- Index of Notation
Record Nr. UNINA-9910456689603321
Bismut Jean-Michel  
Princeton, : Princeton University Press, 2008
Materiale a stampa
Lo trovi qui: Univ. Federico II
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The hypoelliptic Laplacian and Ray-Singer metrics [[electronic resource] /] / Jean-Michel Bismut, Gilles Lebeau
The hypoelliptic Laplacian and Ray-Singer metrics [[electronic resource] /] / Jean-Michel Bismut, Gilles Lebeau
Autore Bismut Jean-Michel
Edizione [Course Book]
Pubbl/distr/stampa Princeton, : Princeton University Press, 2008
Descrizione fisica 1 online resource (378 p.)
Disciplina 515/.7242
Altri autori (Persone) LebeauGilles
Collana Annals of mathematics studies
Soggetto topico Differential equations, Hypoelliptic
Laplacian operator
Metric spaces
Soggetto non controllato Alexander Grothendieck
Analytic function
Asymptote
Asymptotic expansion
Berezin integral
Bijection
Brownian dynamics
Brownian motion
Chaos theory
Chern class
Classical Wiener space
Clifford algebra
Cohomology
Combination
Commutator
Computation
Connection form
Coordinate system
Cotangent bundle
Covariance matrix
Curvature tensor
Curvature
De Rham cohomology
Derivative
Determinant
Differentiable manifold
Differential operator
Dirac operator
Direct proof
Eigenform
Eigenvalues and eigenvectors
Ellipse
Embedding
Equation
Estimation
Euclidean space
Explicit formula
Explicit formulae (L-function)
Feynman–Kac formula
Fiber bundle
Fokker–Planck equation
Formal power series
Fourier series
Fourier transform
Fredholm determinant
Function space
Girsanov theorem
Ground state
Heat kernel
Hilbert space
Hodge theory
Holomorphic function
Holomorphic vector bundle
Hypoelliptic operator
Integration by parts
Invertible matrix
Logarithm
Malliavin calculus
Martingale (probability theory)
Matrix calculus
Mellin transform
Morse theory
Notation
Parameter
Parametrix
Parity (mathematics)
Polynomial
Principal bundle
Probabilistic method
Projection (linear algebra)
Rectangle
Resolvent set
Ricci curvature
Riemann–Roch theorem
Scientific notation
Self-adjoint operator
Self-adjoint
Sign convention
Smoothness
Sobolev space
Spectral theory
Square root
Stochastic calculus
Stochastic process
Summation
Supertrace
Symmetric space
Tangent space
Taylor series
Theorem
Theory
Torus
Trace class
Translational symmetry
Transversality (mathematics)
Uniform convergence
Variable (mathematics)
Vector bundle
Vector space
Wave equation
ISBN 1-282-45837-X
9786612458378
1-4008-2906-2
Classificazione SK 620
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Frontmatter -- Contents -- Introduction -- Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles -- Chapter 2. The hypoelliptic Laplacian on the cotangent bundle -- Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel -- Chapter 4. Hypoelliptic Laplacians and odd Chern forms -- Chapter 5. The limit as t → +∞ and b → 0 of the superconnection forms -- Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics -- Chapter 7. The hypoelliptic torsion forms of a vector bundle -- Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula -- Chapter 9. A comparison formula for the Ray-Singer metrics -- Chapter 10. The harmonic forms for b → 0 and the formal Hodge theorem -- Chapter 11. A proof of equation (8.4.6) -- Chapter 12. A proof of equation (8.4.8) -- Chapter 13. A proof of equation (8.4.7) -- Chapter 14. The integration by parts formula -- Chapter 15. The hypoelliptic estimates -- Chapter 16. Harmonic oscillator and the J0 function -- Chapter 17. The limit of A'2φb,±H as b → 0 -- Bibliography -- Subject Index -- Index of Notation
Record Nr. UNINA-9910781084803321
Bismut Jean-Michel  
Princeton, : Princeton University Press, 2008
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
The hypoelliptic Laplacian and Ray-Singer metrics / / Jean-Michel Bismut, Gilles Lebeau
The hypoelliptic Laplacian and Ray-Singer metrics / / Jean-Michel Bismut, Gilles Lebeau
Autore Bismut Jean-Michel
Edizione [Course Book]
Pubbl/distr/stampa Princeton, : Princeton University Press, 2008
Descrizione fisica 1 online resource (378 p.)
Disciplina 515/.7242
Altri autori (Persone) LebeauGilles
Collana Annals of mathematics studies
Soggetto topico Differential equations, Hypoelliptic
Laplacian operator
Metric spaces
ISBN 1-282-45837-X
9786612458378
1-4008-2906-2
Classificazione SK 620
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Frontmatter -- Contents -- Introduction -- Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles -- Chapter 2. The hypoelliptic Laplacian on the cotangent bundle -- Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel -- Chapter 4. Hypoelliptic Laplacians and odd Chern forms -- Chapter 5. The limit as t → +∞ and b → 0 of the superconnection forms -- Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics -- Chapter 7. The hypoelliptic torsion forms of a vector bundle -- Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula -- Chapter 9. A comparison formula for the Ray-Singer metrics -- Chapter 10. The harmonic forms for b → 0 and the formal Hodge theorem -- Chapter 11. A proof of equation (8.4.6) -- Chapter 12. A proof of equation (8.4.8) -- Chapter 13. A proof of equation (8.4.7) -- Chapter 14. The integration by parts formula -- Chapter 15. The hypoelliptic estimates -- Chapter 16. Harmonic oscillator and the J0 function -- Chapter 17. The limit of A'2φb,±H as b → 0 -- Bibliography -- Subject Index -- Index of Notation
Record Nr. UNINA-9910823892903321
Bismut Jean-Michel  
Princeton, : Princeton University Press, 2008
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui