Springer, 2019. — 576 p. — (Grundlehren der mathematischen Wissenschaften 353; A Series of Comprehensive Studies in Mathematics). — ISBN 978-3-030-15992-4.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. Table of contents Prolegomena A Toolbox for Carleman Inequalities Operators with Simple Characteristics: Calderón’s Theorems Pseudo-convexity: Hörmander’s Theorems Complex Coefficients and Principal Normality On the Edge of Pseudo-convexity Operators with Partially Analytic Coefficients Strong Unique Continuation Properties for Elliptic Operators Carleman Estimates via Brenner’s Theorem and Strichartz Estimates Elliptic Operators with Jumps; Conditional Pseudo-convexity Perspectives and Developments
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