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'sand 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.
