Lectures on Cauchy's Problem in Linear PDEs by Jacques Hadamard
Lectures on Cauchy's Problem in Linear PDEs - Table of Contents
BOOK-I GENERAL PROPERTIES OF CAUCHY'S PROBLEM
1. Cauchy’s fundamental theorem. Characteristics
2. Discussion of Cauchy’s result
BOOK-II THE FUNDAMENTAL FORMULA AND THE ELEMENTARY SOLUTION
1. Classic cases and results
2. The fundamental formula
3. The elementary solution
BOOK-III THE EQUATIONS WITH AN ODD NUMBER OF INDEPENDENT VARIABLES
1. Introduction of a new kind of improper integral
2. The integration for an odd number of independent variables
3. Synthesis of the solution obtained
4. Applications to familiar equations
BOOK-IV THE EQUATIONS WITH AN EVEN NUMBER OF INDEPENDENT VARIABLES AND THE METHOD OF DESCENT
1. Integration of the equation in 2m variables
2. Other applications of the principle of descent
What You Will Learn in Lectures on Cauchy's Problem in Linear PDEs
"Lectures on Cauchy’s Problem in Linear Partial Differential Equations" by "Jacques Hadamard" is a foundational text that explores one of the most important questions in "partial differential equations": how solutions behave when initial conditions are specified. The book focuses on the "Cauchy problem", examining when solutions exist, whether they are unique, and how sensitive they are to small changes in data. Hadamard’s clear and rigorous approach makes complex ideas accessible to serious students of mathematics.
A central contribution of this work is the introduction of the concept of "well-posed problems", now a cornerstone of modern PDE theory. Hadamard shows that a mathematically meaningful problem must satisfy existence, uniqueness, and continuous dependence on initial data. Through careful analysis of "linear PDEs", the book demonstrates how violating these conditions leads to instability and non-physical results, especially in applications related to "mathematical physics".
Written in a lecture-based style, the book connects classical analysis with emerging ideas that later influenced functional analysis and applied mathematics. Despite its age, it remains highly relevant for graduate students and researchers interested in the theoretical foundations of PDEs, especially those studying "Cauchy problems", stability, and rigorous analytical methods.
Book Details & Specifications
Title:
Lectures on Cauchy's Problem in Linear PDEs by Jacques Hadamard
Publisher:
New Haven Yale University Press
Year:
1923
Pages:
334
Type:
PDF
Language:
English
ISBN-10 #:
0486495493
ISBN-13 #:
978-0486495491
License:
Public Domain Work
Amazon:
Amazon
About the Author: Jacques Hadamard
The author
Jacques Hadamard
(1865–1963) was a leading French mathematician whose work shaped modern "mathematical analysis" and "partial differential equations". He taught at major institutions such as the Collège de France and influenced generations of researchers through his clear, rigorous lecture style. Hadamard is especially known for his deep theoretical approach, connecting pure mathematics with real physical problems. He introduced the concept of "well-posed problems", now fundamental in PDE theory, and made major contributions to the study of the "Cauchy problem". Beyond PDEs, he is also celebrated for his proof of the "Prime Number Theorem", securing his lasting legacy in mathematics.
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