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Partial differential equations. 2nd ed. (English) Zbl 1188.35001

Cornerstones. Boston, MA: Birkhäuser (ISBN 978-0-8176-4551-9/hbk; 978-0-8176-4552-6/ebook). xx, 389 p. (2010).
This is revised and extended version of the author’s elementary introduction to partial differential equations [Basel: Birkhäuser (1995; Zbl 0818.35001)]. The material is essentially the same except for three new chapters. Precisely, Chapter 1 is focused on the Cauchy-Kowalewskaja problem and the notion of characteristic surfaces. Chapter 2 and 3 study the Laplace equation and connected elliptic theory. Chapter 4 presents the Fredholm theory of integral equations, which is then applied to the Neumann problem and to eigenvalue problems. Chapter 5 treats the heat equation and related parabolic theory. The wave equation in its basic aspects is discussed in Chapter 6. Chapter 7 is an introduction to conservation laws, the theory is complemented by an analysis of the asymptotic behaviour, following Hopf and Lax.
The first new chapter (Chapter 8) is about nonlinear equations of first 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. The Hopf variational approach to the Cauchy problem for Hamilton-Jacobi equations is one of the most incisive examples of such an interplay. Chapter 9 is an introduction to weak formulations, Sobolev spaces, and direct variational methods for linear and quasi-linear elliptic equations. While terse, the material on Sobolev spaces is reasonably complete, at least for a PDE user. It includes all the basic embedding theorems, including their proofs, 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. Some attention has been paid to the local behavior of these weak solutions, both for the Dirichlet and Neumann problems. While efficient in terms of existence theory, weak solutions provide limited information on their local behaviour. The starting point is a sup bound for the solutions and weak forms of the maximum principle. A further step is their local Hölder continuity. An introduction to these local methods is in Chapter 10 in the framework of DeGiorgi classes. The investigation of the local and boundary behavior of functions in these classes are locally bounded and locally Hölder continuous, and there are given conditions for the regularity to extend up to the boundary. Finally it is proved that non-negative functions on the DeGiorgi classes satisfy the Harnack inequality. The last two chapters provide a background on a spectrum of techniques in local behaviour of solutions of elliptic PDEs, and build toward research topics of current active investigations.
The presentation is more terse and streamlined than in the first edition. Some elementary background material (Weierstrass Theorem, mollifiers, Ascoli-Arzelà Theorem, Jensen’s inequality, etc.) has been removed.

MSC:

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35Axx General topics in partial differential equations
35P05 General topics in linear spectral theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35K05 Heat equation
35L05 Wave equation

Citations:

Zbl 0818.35001
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