Carleman Inequalities

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 books first five chapters contain a thorough exposition of the most classical results, such as Calderóns and Hörmanders 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 Holmgrens and Hörmanders uniqueness theorems, Wolffs modification of Carlemans 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.

1 Prolegomena.- 2 A Toolbox for Carleman Inequalities.- 3 Operators with Simple Characteristics: Calderons Theorems.- 4 Pseudo-convexity: Hormanders Theorems.- 5 Complex Coefficients and Principal Normality.- 6 On the Edge of Pseudo-convexity.- 7 Operators with Partially Analytic Coefficients.- 8 Strong Unique Continuation Properties for Elliptic Operators.- 9 Carleman Estimates via Brenners Theorem and Strichartz Estimates.- 10 Elliptic Operators with Jumps Conditional Pseudo-convexity.- 11 Perspectives and Developments.- A Elements of Fourier Analysis.- B Miscellanea.- References.- Index.