Partial Differential Equations

1. Front Matter

   Pages i-xiii

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2. Introduction: What Are Partial Differential Equations?

   • Jürgen Jost

   Pages 1-7

3. The Laplace Equation as the Prototype of an Elliptic Partial Differential Equation of Second Order

   • Jürgen Jost

   Pages 9-35

4. The Maximum Principle

   • Jürgen Jost

   Pages 37-57

5. Existence Techniques I: Methods Based on the Maximum Principle

   • Jürgen Jost

   Pages 59-83

6. Existence Techniques II: Parabolic Methods. The Heat Equation

   • Jürgen Jost

   Pages 85-126

7. Reaction–Diffusion Equations and Systems

   • Jürgen Jost

   Pages 127-148

8. Hyperbolic Equations

   • Jürgen Jost

   Pages 149-172

9. The Heat Equation, Semigroups, and Brownian Motion

   • Jürgen Jost

   Pages 173-206

10. Relationships Between Different Partial Differential Equations

    • Jürgen Jost

    Pages 207-213

11. The Dirichlet Principle. Variational Methods for the Solution of PDEs (Existence Techniques III)

    • Jürgen Jost

    Pages 215-253

12. Sobolev Spaces and L ² Regularity Theory

    • Jürgen Jost

    Pages 255-309

13. Strong Solutions

    • Jürgen Jost

    Pages 311-328

14. The Regularity Theory of Schauder and the Continuity Method (Existence Techniques IV)

    • Jürgen Jost

    Pages 329-351

15. The Moser Iteration Method and the Regularity Theorem of de Giorgi and Nash

    • Jürgen Jost

    Pages 353-392

16. Back Matter

    Pages 393-410

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This book offers an ideal graduate-level introduction to the theory of partial differential equations. The first part of the book describes the basic mathematical problems and structures associated with elliptic, parabolic, and hyperbolic partial differential equations, and explores the connections between these fundamental types. Aspects of  Brownian motion or pattern formation processes are also presented. The second part focuses on existence schemes and develops estimates for solutions of elliptic equations, such as Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. In particular, the reader will learn the basic techniques underlying current research in elliptic partial differential equations.

This revised and expanded third edition is enhanced with many additional examples that will help motivate the reader.  New features include a reorganized and extended chapter on hyperbolic equations, as well as a new chapter on the relations between different types of partial differential equations, including first-order hyperbolic systems, Langevin and Fokker-Planck equations, viscosity solutions for elliptic PDEs, and much more. Also, the new edition contains additional material on systems of elliptic partial differential equations, and it explains in more detail how the Harnack inequality can be used for the regularity of solutions.

• Brownian motion
• Harnack inequality
• Hilbert space methods
• Moser iteration
• Schauder estimates
• Sobolev spaces
• Turing mechanism
• eigenvalues
• harmonic functions
• heat equation
• maximum principle
• nonlinear partial differential equations
• pattern formation
• reaction-diffusion equations and systems
• semigroups
• wave equation
• partial differential equations

From the book reviews:

“This graduate-level book is an introduction to the modern theory of partial differential equations (PDEs) with an emphasis on elliptic PDEs. … The book is undoubtedly a success in the presentation of diverse methods in PDEs at such an introductory level. The reader has a great opportunity to learn basic techniques underlying current research in elliptic PDEs and be motivated for advanced theory of more general elliptic PDEs and nonlinear PDEs.” (Dhruba Adhikari, MAA Reviews, December, 2014)

“This revised version gives an introduction to the theory of partial differential equations. … Every chapter has at the end a very helpful summary and some exercises. This book is very useful for a PhD course.” (Vincenzo Vespri, Zentralblatt MATH, Vol. 1259, 2013)

"Because of the nice global presentation, I recommend this book to students and young researchers who need the now classical properties of these second-order partial differential equations. Teachers will also find in this textbook the basis of an introductory course on second-order partial differential equations."

- Alain Brillard, Mathematical Reviews

"Beautifully written and superbly well-organised, I strongly recommend this book to anyone seeking a stylish, balanced, up-to-date survey of this central area of mathematics."

- Nick Lord, The Mathematical Gazette

• für Mathematik in den Naturwissenschafte, Max Planck Institut für Mathematik in den Naturwissenschafte, Leipzig, Germany

  Jürgen Jost

Jürgen Jost is currently a codirector of the Max Planck Institute for Mathematics in the Sciences and an honorary professor of mathematics at the University of Leipzig.