© 2019

Carleman Inequalities

An Introduction and More


Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 353)

Table of contents

  1. Front Matter
    Pages i-xxvii
  2. Nicolas Lerner
    Pages 1-25
  3. Nicolas Lerner
    Pages 27-46
  4. Nicolas Lerner
    Pages 69-92
  5. Nicolas Lerner
    Pages 93-136
  6. Nicolas Lerner
    Pages 137-193
  7. Nicolas Lerner
    Pages 195-236
  8. Nicolas Lerner
    Pages 415-442
  9. Back Matter
    Pages 443-557

About this book


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.


Carleman inequalities Carleman estimates Unique Continuation Strong Unique Continuation Pseudo-convexity Principal Normality Counterexamples to Uniqueness of the Cauchy Problem Operators with partially analytic coefficients Strichartz estimates Conditional pseudo-convexity

Authors and affiliations

  1. 1.Institut de Mathématiques de JussieuSorbonne UniversitéParisFrance

About the authors

Nicolas Lerner is professor at Sorbonne Université (formerly Université Paris VI). He has written several articles on Carleman estimates and a book on pseudo-differential operators. He was an invited section speaker at the 2002 ICM in Beijing.

Bibliographic information

  • Book Title Carleman Inequalities
  • Book Subtitle An Introduction and More
  • Authors Nicolas Lerner
  • Series Title Grundlehren der mathematischen Wissenschaften
  • Series Abbreviated Title Grundlehren math. Wissensch.
  • DOI
  • Copyright Information Springer Nature Switzerland AG 2019
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-030-15992-4
  • eBook ISBN 978-3-030-15993-1
  • Series ISSN 0072-7830
  • Series E-ISSN 2196-9701
  • Edition Number 1
  • Number of Pages XXVII, 557
  • Number of Illustrations 97 b/w illustrations, 10 illustrations in colour
  • Topics Operator Theory
    Partial Differential Equations
  • Buy this book on publisher's site