Linked e-resources
Details
Table of Contents
Intro; Preface; Acknowledgements; Contents; 1 Prolegomena; 1.1 Preliminaries; 1.2 Hyperbolicity, the Energy Method and Well-Posedness; 1.3 The Lax-Mizohata Theorems; 1.3.1 Strictly Hyperbolic Operators; 1.3.2 Ill-Posedness Examples; 1.4 Holmgren's Uniqueness Theorems; 1.5 Carleman's Method Displayed on a Simple Example; 1.5.1 The overline Equation; 1.5.2 The Laplace Equation; 2 A Toolbox for Carleman Inequalities; 2.1 Weighted Inequalities; 2.2 Conjugation; 2.3 Sobolev Spaces with Parameter; 2.4 The Symbol of the Conjugate; 2.5 Choice of the Weight
3 Operators with Simple Characteristics: Calderón's Theorems3.1 Introduction; 3.2 Inequalities for Symbols; 3.3 A Carleman Inequality; 3.4 Examples; 3.4.1 Second-Order Real Elliptic Operators; 3.4.2 Strictly Hyperbolic Operators; 3.4.3 Products; 3.4.4 Generalizations of Calderón's Theorems; 3.5 Cutting the Regularity Requirements; 4 Pseudo-convexity: Hörmander's Theorems; 4.1 Introduction; 4.2 Inequalities for Symbols; 4.3 Pseudo-convexity; 4.3.1 Carleman Inequality, Definition; 4.3.2 Invariance Properties of Strong Pseudo-convexity; 4.3.3 Unique Continuation; 4.4 Examples
4.4.1 Pseudoconvexity for Real Second-Order Operators4.4.2 The Tricomi Operator; 4.4.3 Constant Coefficients; 4.4.4 The Characteristic Case; 4.5 Remarks and Open Problems; 4.5.1 Stability Under Perturbations; 4.5.2 Higher Order Tangential Bicharacteristics; 4.5.3 A Direct Method for Proving Carleman Estimates?; 5 Complex Coefficients and Principal Normality; 5.1 Introduction; 5.1.1 Complex-Valued Symbols; 5.1.2 Principal Normality; 5.1.3 Our Strategy for the Proof; 5.2 Pseudo-convexity and Principal Normality; 5.2.1 Pseudo-Convexity for Principally Normal Operators
5.2.2 Inequalities for Symbols5.2.3 Inequalities for Elliptic Symbols; 5.3 Unique Continuation via Pseudo-convexity; 5.4 Unique Continuation for Complex Vector Fields; 5.4.1 Warm-Up: Studying a Simple Model; 5.4.2 Carleman Estimates in Two Dimensions; 5.4.3 Unique Continuation in Two Dimensions; 5.4.4 Unique Continuation Under Condition (P); 5.5 Counterexamples for Complex Vector Fields; 5.5.1 Main Result; 5.5.2 Explaining the Counterexample; 5.5.3 Comments; 6 On the Edge of Pseudo-convexity; 6.1 Preliminaries; 6.1.1 Real Geometrical Optics; 6.1.2 Complex Geometrical Optics
6.2 The Alinhac-Baouendi Non-uniqueness Result6.2.1 Statement of the Result; 6.2.2 Proof of Theorem6.6; 6.3 Non-uniqueness for Analytic Non-linear Systems; 6.3.1 Preliminaries; 6.3.2 Proof of Theorem6.27; 6.4 Compact Uniqueness Results; 6.4.1 Preliminaries; 6.4.2 The Result; 6.4.3 The Proof; 6.5 Remarks, Open Problems and Conjectures; 6.5.1 Finite Type Conditions for Actual Uniqueness; 6.5.2 Ill-Posed Problems with Real-Valued Solutions; 7 Operators with Partially Analytic Coefficients; 7.1 Preliminaries; 7.1.1 Motivations; 7.1.2 Between Holmgren's and Hörmander's Theorems
3 Operators with Simple Characteristics: Calderón's Theorems3.1 Introduction; 3.2 Inequalities for Symbols; 3.3 A Carleman Inequality; 3.4 Examples; 3.4.1 Second-Order Real Elliptic Operators; 3.4.2 Strictly Hyperbolic Operators; 3.4.3 Products; 3.4.4 Generalizations of Calderón's Theorems; 3.5 Cutting the Regularity Requirements; 4 Pseudo-convexity: Hörmander's Theorems; 4.1 Introduction; 4.2 Inequalities for Symbols; 4.3 Pseudo-convexity; 4.3.1 Carleman Inequality, Definition; 4.3.2 Invariance Properties of Strong Pseudo-convexity; 4.3.3 Unique Continuation; 4.4 Examples
4.4.1 Pseudoconvexity for Real Second-Order Operators4.4.2 The Tricomi Operator; 4.4.3 Constant Coefficients; 4.4.4 The Characteristic Case; 4.5 Remarks and Open Problems; 4.5.1 Stability Under Perturbations; 4.5.2 Higher Order Tangential Bicharacteristics; 4.5.3 A Direct Method for Proving Carleman Estimates?; 5 Complex Coefficients and Principal Normality; 5.1 Introduction; 5.1.1 Complex-Valued Symbols; 5.1.2 Principal Normality; 5.1.3 Our Strategy for the Proof; 5.2 Pseudo-convexity and Principal Normality; 5.2.1 Pseudo-Convexity for Principally Normal Operators
5.2.2 Inequalities for Symbols5.2.3 Inequalities for Elliptic Symbols; 5.3 Unique Continuation via Pseudo-convexity; 5.4 Unique Continuation for Complex Vector Fields; 5.4.1 Warm-Up: Studying a Simple Model; 5.4.2 Carleman Estimates in Two Dimensions; 5.4.3 Unique Continuation in Two Dimensions; 5.4.4 Unique Continuation Under Condition (P); 5.5 Counterexamples for Complex Vector Fields; 5.5.1 Main Result; 5.5.2 Explaining the Counterexample; 5.5.3 Comments; 6 On the Edge of Pseudo-convexity; 6.1 Preliminaries; 6.1.1 Real Geometrical Optics; 6.1.2 Complex Geometrical Optics
6.2 The Alinhac-Baouendi Non-uniqueness Result6.2.1 Statement of the Result; 6.2.2 Proof of Theorem6.6; 6.3 Non-uniqueness for Analytic Non-linear Systems; 6.3.1 Preliminaries; 6.3.2 Proof of Theorem6.27; 6.4 Compact Uniqueness Results; 6.4.1 Preliminaries; 6.4.2 The Result; 6.4.3 The Proof; 6.5 Remarks, Open Problems and Conjectures; 6.5.1 Finite Type Conditions for Actual Uniqueness; 6.5.2 Ill-Posed Problems with Real-Valued Solutions; 7 Operators with Partially Analytic Coefficients; 7.1 Preliminaries; 7.1.1 Motivations; 7.1.2 Between Holmgren's and Hörmander's Theorems