Partial Differential Equations: A Unified Hilbert Space Approach

This interesting and unusual book presents its case in a very well-written and informative Preface in which the authors give their rationale for opting to restrict their aims in what at first glance might seem rather restrictive ways: they work not in Banach spaces but on Hilbert spaces, and their focus falls on linear PDEs. They explain:

… our focus on a Hilbert space setting is not a constraint, but rather a highly suitable approach for providing a more transparent and even fairly elementary framework for presenting the main issues in the discussion of a solution theory for partial differential equations.

This is particularly fitting and pedagogically sound in light of their earlier observation that

the whole approach … [is] (with some additional work) extendable to a more general Banach space setting … [which is] sometimes more appropriate, but usually the core results rely nevertheless on a Hilbert space solution theory, a fact sometimes only tacitly acknowledged…

And then, as regards their restriction, by and large, to linear PDE, they suggest that

by a rule of thumb non-linear problems, if they are at all well-posed, are frequently solvable by using a priori estimates and a fixed point argument based on perturbations of the linear theory, [whence] the restriction to linear partial differential equations [should be seen] as foundation laying rather than an exclusion of non-linear issues.

This having been said, the book under review is evidently offered as a pointed and practical introduction to the field, oriented toward getting at the hard-core methodology of solving PDEs with sound and accessible methods, amenable to generalization but fascinating and important in their own right — and it all starts with functional analysis. To wit:

Accordingly the book is divided into several natural parts. In Chapter 1 we supply some additional material on functional analysis in Hilbert space which may be difficult to find elsewhere.

Picard and McGhee note that “the reader [should be] familiar with basic functional analysis in Hilbert space.” So: read Halmos.

Thereafter the trajectory of the book is thus: Sobolev lattices; PDEs with constant coefficients in real space of dimension n+1, presently extended to tempered distributions; generalization to abstract evolution equations; then “[i]n Chapter 5 this general setting is exemplified by applications to a variety of initial-boundary value problems from mathematical physics”; and finally, “Chapter 6 offers a new approach to initial boundary value problems by expanding on the ideas and problems presented.”

Thus, Partial Differential Equations: a Unified Hilbert Space Approach presents a thoroughly thought out and carefully crafted deviation from the usual approach to hard analysis in the shape of PDEs, with a very practical subtext, viz. to get to the difficult business of solving actual PDEs in an effective way, as sketched above. This emphasis on problem solving is in evidence even before we get to the first chapter of the book, what with a list of bullet points on p.viii (in the Preface), starting with “Hadamard’s celebrated criteria for well-posedness.”

But, as already indicated, the book is something of a departure:

The systematic approach presented here will shed a different and hopefully more illuminating light on [e.g., regularity considerations] and other issues by proposing a different classification scheme, advocating the construction of “tailor-made” distribution spaces adapted to the particular equation at hand, [and] showing that from our point of view parabolic and hyperbolic partial differential equations are, in a way, “easier” than elliptic partial differential equations.

The book is pitched at the level of “introductory and advanced graduate level courses on partial differential equations and functional analysis”; it is very clearly written with complete proofs given (of course), and comes with plenty of remarks and useful examples. One final quote from the authors:

Apart from the novel approach the material presented in this monograph may in many ways be considered elementary, however, researchers will nevertheless find new results for particular evolutionary systems from mathematical physics in later parts of this monograph as well as a very different perspective on seemingly familiar evolutionary problems.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.