11447nam 2200565 450 991055688060332120231110212625.03-030-88674-3(MiAaPQ)EBC6941389(Au-PeEL)EBL6941389(CKB)21435610200041(PPN)261518461(EXLCZ)992143561020004120221113d2022 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierElliptic Carleman estimates and applications to stabilization and controllabilityVolume 1 Dirichlet boundary conditions on Euclidean space /Jérôme Le Rousseau, Gilles Lebeau, and Luc RobbianoCham, Switzerland :Birkhauser Verlag,[2022]©20221 online resource (410 pages)Progress in Nonlinear Differential Equations and Their Applications ;v.97Print version: Le Rousseau, Jérôme Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I Cham : Springer International Publishing AG,c2022 9783030886738 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.Progress in Nonlinear Differential Equations and Their Applications Differential equations, PartialDifferential equations, PartialAsymptotic theoryEquacions en derivades parcialsthubLlibres electrònicsthubDifferential equations, Partial.Differential equations, PartialAsymptotic theory.Equacions en derivades parcials515.353Le Rousseau Jérôme1218695Lebeau GillesRobbiano LucMiAaPQMiAaPQMiAaPQBOOK9910556880603321Elliptic Carleman estimates and applications to stabilization and controllability2966965UNINA