** ** Designed to prepare readers to better understand the current literature in research journals, this book explains the basics of classical PDEs and a wide variety of more modern methods—especially the use of functional analysis—which has characterized much of the recent development of PDEs. It gives equal treatment to elliptic, hyperbolic, and parabolic theory, and features an abundance of applications to equations that are important in physics and engineering—both on the basic and more advanced level. Provides worked, figures and illustrations, and extensive references to other literature. ** ** First-Order Equations. Principles for Higher-Order Equations. The Wave Equation. The Laplace Equation. The Heat Equation. Linear Functional Analysis. Differential Calculus Methods. Linear Elliptic Theory. Two Additional Methods. Systems of Conservation Laws. Linear and Nonlinear Diffusion. Linear and Nonlinear Waves. Nonlinear Elliptic Equations. Appendix on Physics. ** ** For anyone using PDEs in physics and engineering applications.

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Designed to bridge the gap between graduate-level texts in partial differential equations and the current literature in research journals, this text introduces students to a wide variety of more modern methods -- especially the use of functional analysis -- which has characterized much of the recent development of PDEs.

I am grateful that so many individuals and institutions have chosen to use *Partial Differential Equations: Methods & Applications* since it first appeared in 1996. I have been even more grateful to the many individuals who have contacted me with suggestions and corrections for the first edition. I hope that this new edition will be much improved because of their interest and contributions.

The book originally evolved from a two-term graduate course in partial differential equations that I taught many times at Northeastern University. At that time, I felt there was an absence of textbooks that covered both the classical results of partial differential equations and more modern methods, such as functional analysis, which are used heavily in the current literature. Since I began to write the book, however, several other textbooks have appeared that also aspire to bridge the same gap: *An Introduction to Partial Differential Equations* by Renardy and Rogers (Springer-Verlag, 1993) and *Partial Differential Equations* by Lawrence C. Evans (AXIS, 1998) are two good examples.

As with any book on such a broad and diverse subject as partial differential equations, I have had to make some difficult decisions concerning content and exposition. I make no apologies for these decisions, but I do acknowledge that other choices might have been made. For example, this text begins with the method of characteristics and first-order equations; although other texts often omit or slight this material in preference to the treatment of second-order equations, I have chosen to include it, and even emphasize its constructive aspects, because I feel it offers motivation and insights that are valuable in the study of higher-order equations. Indeed, the method of characteristics leads naturally to the Cauchy problem for higher-order equations, as well as the classification of second-order equations, which I treat in Chapter 2 (along with a discussion of generalized solutions). Following this momentum, I decided to treat the wave equation before Laplace's equation, even though this causes the use of eigenfunctions in a bounded domain to be delayed until the next chapter. Similarly, I have chosen to treat the heat equation after the Laplace equation for reasons of the maximum principle; of course, a bonus is that eigenfunction expansions are available for the heat equation in a bounded domain. Other texts treat these three second-order equations in different orders, and they all have their own reasons for doing so.

Exposure to the use of functional analysis begins in Chapter 6 with a rapid survey of the basic definitions and tools needed to study linear operators on Banach and Hilbert spaces. The Sobolev spaces are introduced as early as possible, as are their application to obtain weak solutions of the Dirichlet problems for the Poisson equation and the Stokes system, before encountering the more subtle issues of weak convergence, continuous imbeddings, compactness, unbounded operators, and spectral theory.

The theme of weak solutions is picked up again in Chapter 7, in the context of differential calculus on Banach spaces. The variational method of finding a weak solution by optimizing a functional, possibly with constraints, is applied to several problems, including the eigenvalues of the Laplacian. The forum of differential calculus also enables us to introduce, at this point, the contraction mapping principle, the inverse and implicit function theorems, a discussion of when they apply to Sobolev spaces, and an application to the prescribed mean curvature equation.

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 on a torus, and transplantation to open domains. It is also natural, at this point, to discuss maximum principles for elliptic operators; and then the issues of uniqueness and solvability for linear elliptic equations.

Chapter 9 consists of two additional methods. The first of these, the Schauder fixed point theory, is presented and then illustrated with its application to the stationary Navier-Stokes equations; this application returns us to our theme of weak solutions in Sobolev spaces, and also builds on the discussion of the Stokes system in Chapter 6. The second additional method is the use of semigroups of operators on a Banach space to describe the dynamics of evolutionary partial differential equations. We first discuss systems of ordinary differential equations as a finite-dimensional example; this helps to motivate the ensuing discussion for partial differential equations, which is well seasoned with examples. This treatment of semigroups is very brief but serves the purpose of setting the stage for the hyperbolic and parabolic equations and systems that are studied in Chapters 10, 11, and 12.

Although Chapters 6 through 9 emphasize the development of tools and methods, I have tried to provide sufficient applications to motivate and illustrate the theory as it unfolds. However, beginning in Chapter 10, the focus switches from methods to applications, and developing the theories of hyperbolic systems 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) as far as possible in this limited space. I have, of course, needed to severely "limit the budget" in each of these last four chapters, but I hope I have given the flavor and some background on each topic, enough to enable the interested student to consult more detailed and comprehensive treatments.

Although I have made certain choices for the order, I have tried to make the exposition flexible enough to allow for the individual instructor to make changes without too much difficulty. For example, to enable the introduction of the spherical mean in connection with the Laplace equation instead of the wave equation, I have made Section 3.2a self-contained. This means that it is possible to reorder the material following Chapter 2: the one-dimensional wave equation, then Laplace's equation (with Section 3.2a added to Section 4.1d), and then the *n*-dimensional wave equation. Similarly, although I felt the need to collect all of the linear functional analysis and Sobolev space theory in Chapter 6, it is possible to discuss only the results for *H* O 1,2(Ω) in order to study more quickly the Dirichlet problems in Chapters 7, 8, and 9. Another example would be to jump into Chapter 10 after only a minimal amount of Banach space theory and the contraction mapping principle.

I have tried to include a large number of exercises. Some of these exercises are fairly routine applications of the material covered in the text. Other exercises are designed to supply some steps that are omitted from the exposition in the text; this not only helps to streamline the exposition, but it also engages the student more actively in the learning experience. Still other exercises are intended to give the student a brief exposure to related topics that have been reluctantly omitted from the textual exposition, casualties of more hard choices of mine. When I teach this course, I usually assign many exercises, including some of each type. On the other hand, the instructor may choose to use lecture time to solve all omitted steps of proofs and/or pursue some of the omitted topics. In any case, hints and solutions of selected exercises are provided after Chapter 13; 1 hope the instructor and student find these useful.

Now let me list the major changes and additional topics that I have included in this second edition. To begin with, I have attempted to provide more details to some of the sketchier arguments in the first edition. Second, I have added sections with additional applications to Chapters 3, 4, and 5: respectively, applications to light and sound, applications to vector fields, and applications to fluid dynamics. In Chapter 6, I have added a section on unbounded operators and spectral theory that provides essential background for results in later chapters. I also have added an appendix on physics, in which the most important partial differential equations are derived from basic principles. Finally, I have made substantial changes to the Hints and Solutions for Selected Exercises.

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