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A course on partial differential equations. (English) Zbl 1415.35002

Graduate Studies in Mathematics 197. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4292-7/hbk; 978-1-4704-5057-1/ebook). ix, 205 p. (2018).
This book represents an introductory course in partial differential equations (PDEs) and it is dedicated, as the author say, to fourth year undergraduate or first year graduate mathematics students. The material focuses on three most important aspects of the subject:
\(\bullet\)
theory – the existence, uniqueness and continuous dependence on given data or parameters;
\(\bullet\)
phenomenology – the study of the properties of the solutions of PDEs;
\(\bullet\)
applications – to exercise good scientific taste in choosing the topics in this text, based on physical or geometrical applications.
The book is structured in 8 chapters, bibliography and an index of some concepts used. Each chapter contains some theoretical aspects, exercises and projects.
In Chapter 1, the author makes an overview of the subject and gives some examples of some common PDEs with a brief overview of the settings where they appear, either as a characterization of mathematical properties of a geometrical object or as a description of a physical phenomenon.
Chapter 2 is dedicated to a class of wave equations (transport equation) and the topics presented here are: the Fourier transformation, the method of characteristics, conservation laws, the d’Alembert formula, Duhamel’s principle, the method of images and separation of variables.
In Chapter 3, the author presents some aspects regarding the heat equation: the heat kernel, convolution operators, the maximum principle, initial and initial-boundary value problems, conservation laws and evolution of moments, the heat equation in \(\mathbb{R}^n\), entropy and gradient flow.
Chapter 4 is dedicated to Laplace’s equation. The topics presented here are: boundary value problems associated with Laplace’s equation (Dirichlet, Poisson and Neumann), Green’s identity, the fundamental solution, maximum principle, Green’s function and Dirichlet-Neumann operators, Poisson kernel in \(\mathbb{R}^n_+\), oscillation and attenuation estimates and Hadamard variational formula.
The properties of the Fourier transform in three of the most important cases (Hilbert spaces, Schwarz class and \(L^1\) integrability) are presented in Chapter 5.
The wave equation on \(\mathbb{R}^n\) is the subject of the Chapter 6. Here the topics are: wave propagator by Fourier synthesis, Lorentz transformation, method of spherical means, Huygens’ principle, Paley-Wiener theory, Lagrangians an Hamiltonians PDEs.
In Chapter 7, the author presents the phenomenon of dispersion for PDEs.
Chapter 8 is dedicated to conservation laws and shocks and the topics are: first-order quasilinear equation, the Riemann problem and Lax-Olenik solutions.

MSC:

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35K05 Heat equation
35L05 Wave equation
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
35L60 First-order nonlinear hyperbolic equations
35Q41 Time-dependent Schrödinger equations and Dirac equations
35B50 Maximum principles in context of PDEs
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