Partial Differential Equations

1. Front Matter

   Pages 1-20

   PDF

2. Preliminaries

   • Emmanuele DiBenedetto

   Pages 1-16

3. Quasi-Linear Equations and the Cauchy–Kowalewski Theorem

   • Emmanuele DiBenedetto

   Pages 17-35

4. The Laplace Equation

   • Emmanuele DiBenedetto

   Pages 37-86

5. Boundary Value Problems by Double-Layer Potentials

   • Emmanuele DiBenedetto

   Pages 87-107

6. Integral Equations and Eigenvalue Problems

   • Emmanuele DiBenedetto

   Pages 109-134

7. The Heat Equation

   • Emmanuele DiBenedetto

   Pages 135-181

8. The Wave Equation

   • Emmanuele DiBenedetto

   Pages 183-224

9. Quasi-Linear Equations of First-Order

   • Emmanuele DiBenedetto

   Pages 225-263

10. Non-Linear Equations of First-Order

    • Emmanuele DiBenedetto

    Pages 265-295

11. Linear Elliptic Equations with Measurable Coefficients

    • Emmanuele DiBenedetto

    Pages 297-345

12. DeGiorgi Classes

    • Emmanuele DiBenedetto

    Pages 347-371

13. Back Matter

    Pages 1-17

    PDF

This is a revised and extended version of my 1995 elementary introduction to partial di?erential equations. The material is essentially the same except for three new chapters. The ?rst (Chapter 8) is about non-linear equations of ?rst order and in particular Hamilton–Jacobi equations. It builds on the continuing idea that PDEs, although a branch of mathematical analysis, are closely related to models of physical phenomena. Such underlying physics in turn provides ideas of solvability. The Hopf variational approach to the Cauchy problem for Hamilton–Jacobi equations is one of the clearest and most incisive examples of such an interplay. The method is a perfect blend of classical mechanics, through the role and properties of the Lagrangian and Hamiltonian, and calculus of variations. A delicate issue is that of identifying “uniqueness classes. ” An e?ort has been made to extract the geometrical conditions on the graph of solutions, such as quasi-concavity, for uniqueness to hold. Chapter 9 is an introduction to weak formulations, Sobolev spaces, and direct variationalmethods for linear and quasi-linearelliptic equations. While terse, the material on Sobolev spaces is reasonably complete, at least for a PDEuser. Itincludesallthebasicembeddingtheorems,includingtheirproofs, and the theory of traces. Weak formulations of the Dirichlet and Neumann problems build on this material. Related variational and Galerkin methods, as well as eigenvalue problems, are presented within their weak framework.

• Boundary value problem
• Cauchy--Kowalewski Theorem
• Conservation Laws
• Degiorgi Classes
• Elliptic Theory
• Green's Theorem
• Linear Elliptic Equations
• Partial Differential Equations
• Potential
• Viscosity Solutions
• partial differential equation
• wave equation

"The book under review, the second edition of Emmanuele DiBenedetto’s 1995 Partial Differential Equations, now appearing in Birkhäuser’s 'Cornerstones' series, is an example of excellent timing.  This is a well-written, self-contained, elementary introduction to linear, partial differential equations.

So it is that DiBenedetto, whose philosophical position regarding PDE is unabashedly that 'although a branch of mathematics, [it is] closely related to physical phenomena,' presents us with marvelous coverage of (in order), quasi-linearity and Cauchy-Kowalevski, Laplace, BVPs by 'double-layer potentials,' [and my favorite three chapters:] integral equations and the eigenvalue problem, the heat equation, and the wave equation. Then he returns to quasi-linearity (for first order equations), goes on to non-linearity, linear elliptic equations with measurable coefficients..., and, finally...DeGiorgi classes.

PDE is beautifully written, in clear and concise prose, the mathematics is cogent and complete, and the presentation testifies both to DiBenedetto’s fine taste in the subject and his experience in teaching this difficult material.

Make no mistake: the book is neither chatty nor discursive, but there’s something more or less ineffable about it, making it appear somehow less austere than other texts on PDE. Check it out.

DiBenedetto has also included a decent number of what he calls 'Problems and Complements,' and, to be sure, these should capture the attention of the conscientious student or reader.

Thus, DiBenedetto’s PDE is indeed a cornerstone text in the subject. It looks like a rare gem to me.

—MAA Reviews (Review of the Second Edition)

"The author's intent is to present an elementary introduction to pdes...In contrast to other elementary textbooks on pdes...much more material is presented on the three basic equations: Laplace's equation, the heat and wave equations...Thepresentation is clear and well organized...The text is complemented by numerous exercises and hints to proofs."

—Mathematical Reviews (Review of the First Edition)

"This is a well-written, self-contained, elementary introduction to linear, partial differential equations."

—Zentrallblatt MATH (Review of the First Edition)

"This book certainly can be recommended as an introduction to PDEs in mathematical faculties and technical universities."

—Applications of Mathematics (Review of the First Edition)

• Dept. Mathematics, Vanderbilt University, Nashville, U.S.A.

  Emmanuele DiBenedetto

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