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
• Pubblicazione: Cham, Switzerland : , : Birkhauser Verlag, , [2022]; ©2022
• Descrizione fisica: 1 online resource (410 pages)
• Disciplina: 515.353
• Soggetto topico: Differential equations, Partial; Differential equations, Partial - Asymptotic theory; Equacions en derivades parcials
• Soggetto genere / forma: Llibres electrònics
• Persona (resp. second.): LebeauGilles; RobbianoLuc
• 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.
• Titolo autorizzato: Elliptic Carleman estimates and applications to stabilization and controllability
• ISBN: 3-030-88674-3
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910556880603321
• Lo trovi qui: Univ. Federico II
• Serie: Progress in Nonlinear Differential Equations and Their Applications