The goal of this book is twofold: First, to give the student a basic and classical introduction to the subject, including first-order equations by the method of characteristics and the linear second-order equations which arise in mathematical physics: the wave equation, Laplace equation and the heat equation (Chapters 1-5). Second, to introduce the student to a wide variety of modern methods, especially the use of functional analysis, which has characterized much of the recent development of partial differential equations.
Functional analysis starts in Chapter 6 with a rapid survey of the basic definitions and tools needed to study linear operators on Banach and Hilbert spaces. One finds proofs only for the trivial results, for the other proofs one has to look somewhere else. In Chapter 6 there are also included weak solutions, Sobolev spaces and imbedding theorems.
Weak solutions appear again in Chapter 7, in the contex of differential calculus on Banach spaces. The variational method of finding a weak solution by optimizing a functional is applied to several problems, including the eigenvalues of the Laplacian. The issue of the regularity of weak solutions is taken up in Chapter 8, where the basic elliptic \(L^2\)-estimates are obtained by Fourier analysis. There is also the discussion of maximum principles for elliptic operators, and then the issues of uniqueness and solvability. Chapter 9 consists of two additional methods. The first of these, the Schauder fixed point theory, is represented and then illustrated with its application to the stationary Navier-Stokes equations. The second “additional method” is the use of semigroups of operators on a Banach space to describe the dynamics of evolutionary partial differential equations. Beginning in Chapter 10, the focus switches from methods to applications, and here are developed the theories of hyperbolic systems of conservation laws in one space dimension (Chapter 10), linear and nonlinear diffusion (Chapter 11), linear and nonlinear waves (Chapter 12), and nonlinear elliptic equations (Chapter 13). Providing some background on each topic, the interested student will be able to consult more detailed and comprehensive treatment.
The text is enriched by a large number of exercises.