This book offers a self-contained introduction to partial differential equations (PDEs), primarily focusing on linear equations, and also providing perspective on nonlinear equations. The treatment is mathematically rigorous with a generally theoretical layout, with indications to some of the physical origins of PDEs. The Second Edition is rewritten to incorporate years of classroom feedback, to correct errors and to improve clarity. The exposition offers many examples, problems and solutions to enhance understanding. Requiring only advanced differential calculus and some basic Lp theory, the book will appeal to advanced undergraduates and graduate students, and to applied mathematicians and mathematical physicists.
weitere Ausgaben werden ermittelt
Preliminaries.- Quasi-Linear Equations and the Cauchy#x2013;Kowalewski Theorem.- The Laplace Equation.- Boundary Value Problems by Double-Layer Potentials.- Integral Equations and Eigenvalue Problems.- The Heat Equation.- The Wave Equation.- Quasi-Linear Equations of First-Order.- Non-Linear Equations of First-Order.- Linear Elliptic Equations with Measurable Coefficients.- DeGiorgi Classes.
"The book under review, the second edition of Emmanuele DiBenedetto's 1995 Partial Differential Equations, now appearing in Birkhauser'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...The presentation 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)
This self-contained textbook offers an elementary introduction to partial differential equations (PDEs), primarily focusing on linear equations, but also providing a perspective on nonlinear equations, through Hamilton--Jacobi equations, elliptic equations with measurable coefficients and DeGiorgi classes. The exposition is complemented by examples, problems, and solutions that enhance understanding and explore related directions.
Large parts of this revised second edition have been streamlined and rewritten to incorporate years of classroom feedback, correct misprints, and improve clarity. The work can serve as a text for advanced undergraduates and graduate students in mathematics, physics, engineering, and the natural sciences, as well as an excellent reference for applied mathematicians and mathematical physicists.
The newly added three last chapters, on first order non-linear PDEs (Chapter 8), quasilinear elliptic equations with measurable coefficients (Chapter 9) and DeGiorgi classes (Chapter 10), point to issues and directions at the forefront of current investigations.
Reviews of the first 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. The presentation is clear and well organized. The text is complemented by numerous exercises and hints to proofs.
This is a well-written, self-contained, elementary introduction to linear, partial differential equations.
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