Weak Solutions of Partial Differential Equations

This book offers a comprehensive introduction to the study of solutions of linear and nonlinear partial differential equations, covering elliptic, parabolic and hyperbolic types. It places particular emphasis on the concept of weak solution, a fundamental framework for addressing well-posed problems in PDE theory. The book examines the existence and uniqueness of solutions for various types of PDEs, along with their key properties. Additionally, many of the methods introduced are also applicable for analyzing the convergence of numerical schemes used to approximate these equations. Based on courses taught by the authors, this book is primarily aimed at graduate students and contains numerous exercises and problems with detailed solutions.

List of contents

1 Sobolev Spaces.- 2 Linear Elliptic Problems.- 3 Quasi-Linear Elliptic Problems.- 4 Parabolic Problems.- 5 Hyperbolic Problems.

About the author

Thierry Gallouët
was a professor of mathematics at the University of Chambéry (1985–1993), the École Normale Supérieure de Lyon (1993–1997), and the University of Aix-Marseille (1997–2023). A specialist in functional analysis and partial differential equations, particularly in the L1 theory of elliptic and parabolic equations, he later became interested in the numerical resolution of partial differential equations with applications in petroleum and hydraulic engineering. Along with Robert Eymard, he pioneered a general theory of the finite volume method.

Raphaèle Herbin
is a professor of mathematics at the University of Aix-Marseille. She specializes in numerical analysis, particularly in the discretization of partial differential equations. She has contributed to the mathematical analysis of finite volume schemes and their application to various fields, such as electrochemistry and fluid mechanics.

Product details

• Authors: Thierry Gallouët, Raphaèle Herbin
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 17.11.2025
• EAN: 9783031989810
• ISBN: 978-3-0-3198981-0
• No. of pages: 465
• Illustrations: XI, 465 p.
• Series: Mathématiques et Applications
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Funktionalanalysis und Abwandlungen, Functional Analysis, Differential equations, heat equation, Sobolev spaces, Riemann problem, parabolic partial differential equations, Aubin-Simon theorem, Saint-Venant equation, entropy solutions, compactness methods for PDE, nonlinear PDE, Leray-Lions problem, existence and uniqueness of solutions of PDE, topological degree, Schauder fixed point, Stefan problem, elliptic partial differential equations