Ebook: Hilbert Space Methods for Partial Differential Equations

After a basic introduction to partial differential equations (e.g. G. Hellwig, Partial Differential
Equations) there are a variety of directions in which to pursue the topic. One of the more popular,
and probably the most appropriate, is the use of Hilbert space methods in partial differential
equations. In this approach, the basic questions of existence and uniqueness are resolved through
the use and exploitation of the Riesz representation theorem and making precise the space from
which the solution is obtained, the space from which the data may come, and the corresponding
notion of continuity. The book under review is devoted to this topic, and in the reviewer’s opinion
is probably the best introduction to the subject available. The attraction of Showalter’s book is
that while it covers the basic material, and in fact touches on some of the more recent developments,
It refrains from pursuing long and tedious generalizations, which can often hide the main ideas from
the prospective student. Hence, only boundary value problems for second-order elliptic equations
are discussed, regularity of solutions to elliptic equations is shown only in the context that u € H¹(G)
implies u € H²(G), and results on distribution theory and semigroups are restricted to what is
actually essential in the development of the resulting theory. In this manner, Showalter is able to cover a considerable amount of material in the space of only 184 pages.

We now survey the main contents of the book. Chapter I presents all the elementary Hilbert
space theory that is needed for future developments, including the Hilbert-Schmidt theorem.
Chapter II is an introduction to distributions and Sobolev spaces, including Sobolev’s lemma,
trace theorems, and Rellich’s lemma. Chapter III is an exposition of the theory of linear elliptic
boundary value problems in variational form. An abstract Green’s theorem is presented 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 the
discussion of regularity theorems optional. Also included in this chapter is the problem of
eigenfunction expansions. Chapter IV is an exposition of the theory of semigroups and its application
to solve initial-boundary value problems for evolution equations. Chapters V and VI provide the
immediate extension of the results of Chapter IV to cover evolution equations of second order and
implicit type. This material is not often included in an introductory text, and represents an area
where the author has made substantial contributions of his own. Included here is a discussion of
such topics as pseudo-parabolic equations, Sobolev equations, and equations of degenerate or
mixed type. Chapter VII is entitled “Optimization and Approximation Topics” and includes
material on Dirichlet’s principle, variational inequalities, optimal control of boundary value
problems, and approximation of solutions to partial differential equations. Although there are
many detailed computations throughout the text, the reviewer noted only a few printing errors,
all of which can be easily corrected.

The author claims in the Preface that ‘‘the reader is assumed to have some prior acquaintance
with the concepts of ‘linear’ and ‘continuous’ and also to believe L² is complete” and the formal
prerequisite “consists of a good advanced calculus course and motivation to study partial
differential equations’’. As the opening lines of this review indicate, the reviewer is a little doubtful
if this is sufficient background to fully appreciate the material in the text. Even with the excellent
introduction to Hilbert space theory in Chapter I, a reader who has never been exposed to functional
analysis is liable to feel over his head quite quickly. The reviewer would furthermore consider it a
mistake to learn the “modern” theory of partial differential equations without first learning such
“classical” topics as the classification of partial differential equations as to type, the theory of
characteristics, maximum principles, and elementary methods for proving existence theorems.
This is in no way a criticism of Showalter’s book, which the reviewer considers to be excellent,
but merely to point out at which point in the learning process the reviewer considers this material
to be most beneficial to the prospective student.
-------------------------------------------------------------------------------------- D.L.COLTON