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BL
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| Record Number
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861488
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| Main Entry
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Lerner, Nicolas,1953-
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| Title & Author
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Carleman inequalities : : an introduction and more /\ Nicolas Lerner.
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| Publication Statement
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Cham :: Springer,, [2019]
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, ©2019
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| Series Statement
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Grundlehren der mathematischen Wissenschaften ;; volume 353
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| Page. NO
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1 online resource
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| ISBN
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3030159922
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: 3030159930
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: 3030159949
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: 9783030159924
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: 9783030159931
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: 9783030159948
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9783030159924
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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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
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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
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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
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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
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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
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| Abstract
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Over the past 25 years, Carleman estimates have become an essential tool in several areas related to partial differential equations such as control theory, inverse problems, or fluid mechanics. This book provides a detailed exposition of the basic techniques of Carleman Inequalities, driven by applications to various questions of unique continuation. Beginning with an elementary introduction to the topic, including examples accessible to readers without prior knowledge of advanced mathematics, the book's first five chapters contain a thorough exposition of the most classical results, such as Calderón's and Hörmander's theorems. Later chapters explore a selection of results of the last four decades around the themes of continuation for elliptic equations, with the Jerison-Kenig estimates for strong unique continuation, counterexamples to Cauchy uniqueness of Cohen and Alinhac & Baouendi, operators with partially analytic coefficients with intermediate results between Holmgren's and Hörmander's uniqueness theorems, Wolff's modification of Carleman's method, conditional pseudo-convexity, and more. With examples and special cases motivating the general theory, as well as appendices on mathematical background, this monograph provides an accessible, self-contained basic reference on the subject, including a selection of the developments of the past thirty years in unique continuation.
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| Subject
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Carleman theorem.
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Inequalities (Mathematics)
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Carleman theorem.
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Inequalities (Mathematics)
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| Subject
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MATHEMATICS-- Calculus.
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| Subject
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MATHEMATICS-- Mathematical Analysis.
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| Dewey Classification
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515/.26
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| LC Classification
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QA295
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