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{\large\bf Preface}
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This book is an outgrowth of a course which we have given almost periodically
over the last eight years. It is addressed to beginning graduate students
of mathematics, engineering, and the physical sciences.
Thus, we have attempted to present it while presupposing a minimal
background: the reader is assumed to have some prior acquaintance with
the concepts of ``linear'' and ``continuous'' and also to believe $L^2$
is complete.
An undergraduate mathematics training through Lebesgue integration
is an ideal background but we dare not assume it without turning away
many of our best students.
The formal prerequisite consists of a good advanced calculus course and
a motivation to study partial differential equations.
A problem is called {\it well-posed\/} if for each set of data there
exists exactly one solution and this dependence of the solution on the
data is continuous. To make this precise we must indicate the space from
which the solution is obtained, the space from which the data may come,
and the corresponding notion of continuity. Our goal in this book is to
show that various types of problems are well-posed.
These include boundary value problems for (stationary) elliptic partial
differential equations and initial-boundary value problems for
(time-dependent) equations of parabolic, hyperbolic, and
pseudo-parabolic types. Also, we consider some nonlinear elliptic
boundary value problems, variational or uni-lateral problems, and some
methods of numerical approximation of solutions.
We briefly describe the contents of the various chapters.
Chapter~I presents all the elementary Hilbert space theory that is needed
for the book. The first half of Chapter~I is presented in a rather brief
fashion and is intended both as a review for some readers and as a study
guide for others. Non-standard items to note here are the spaces
$C^m (\bar G)$, $V^*$, and $V'$.
The first consists of restrictions to the closure of $G$ of functions on
$\RR^n$ and the last two consist of conjugate-linear functionals.
Chapter II is an introduction to distributions and Sobolev spaces.
The latter are the Hilbert spaces in which we shall show various problems
are well-posed.
We use a primitive (and non-standard) notion of distribution which is
adequate for our purposes.
Our distributions are conjugate-linear and have the pedagogical advantage
of being independent of any discussion of topological vector space theory.
Chapter III is an exposition of the theory of linear elliptic boundary value
problems in variational form.
(The meaning of ``variational form'' is explained in Chapter~VII.)
We present an abstract Green's theorem which permits the separation of the
abstract problem into a partial differential equation on the region and a
condition on the boundary.
This approach has the pedagogical advantage of making optional the
discussion of regularity theorems.
(We construct an operator $\partial$ which is an extension of the normal
derivative on the boundary, whereas the normal derivative makes sense only
for appropriately regular functions.)
Chapter IV is an exposition of the generation theory of linear semigroups
of contractions and its applications to solve initial-boundary value
problems for partial differential equations.
Chapters~V and VI provide the immediate extensions to cover evolution
equations of second order and of implicit type.
In addition to the classical heat and wave equations with standard
boundary conditions, the applications in these chapters include a
multitude of non-standard problems such as equations of
pseudo-parabolic, Sobolev, viscoelasticity, degenerate or mixed type;
boundary conditions of periodic or non-local type or with time-derivatives;
and certain interface or even global constraints on solutions.
We hope this variety of applications may arouse the interests even of
experts.
Chapter VII begins with some reflections on Chapter~III and develops into an
elementary alternative treatment of certain elliptic boundary value problems
by the classical Dirichlet principle.
Then we briefly discuss certain unilateral boundary value problems,
optimal control problems, and numerical approximation methods.
This chapter can be read immediately after Chapter~III and it serves as a
natural place to begin work on nonlinear problems.
There are a variety of ways this book can be used as a text.
In a year course for a well-prepared class, one may complete the entire
book and supplement it with some related topics from nonlinear functional
analysis. In a semester course for a class with varied backgrounds, one
may cover Chapters~I, II, III, and VII.
Similarly, with that same class one could cover in one semester the first
four chapters. In any abbreviated treatment one could omit I.6, II.4, II.5,
III.6, the last three sections of IV, V, and VI, and VII.4.
We have included over 40 examples in the exposition and there are about
200 exercises.
The exercises are placed at the ends of the chapters and each is numbered
so as to indicate the section for which it is appropriate.
Some suggestions for further study are arranged by chapter and precede the
Bibliography.
If the reader develops the interest to pursue some topic in one of these
references, then this book will have served its purpose.
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R. E. Showalter; Austin, Texas, January, 1977.
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