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Kernel Functions and Elliptic Differential Equations in Mathematical Physics

Stefan Bergman and Menahem Schiffer
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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Stefan Bergman is one of those few mathematicians whose name is inextricably attached to a specific mathematical object. Just as we have, for example, Hilbert, Banach, and Sobololev spaces, the Möbius strip, the Klein bottle, the Ricci flow, the Weil representation, and Galois fields, we also have the Bergman kernel. When I was a student at UCLA in the 1970s this latter object was on the lips of just about every analyst you’d run into (and there were many). It actually ran a close second to the corona theorem, it seemed, but it was pretty popular. So, it’s a very big deal and we can’t help but ask, right off: what is this kernel? Well, the ever-taunting Wikipedia option yields the following definition: “In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is a reproducing kernel for the Hilbert space of all square integrable holomorphic functions on a domain \(D\) in \(\mathbb{C}^n\).” Fair enough. Evidently it is a mainstay in the study of several complex variables, or presumably even just one, and this certainly underscores the quality of the credentials of the one of the authors of the book under review.

The other author, Max Schiffer, is also a very wonderful scholar, eloquently eulogized in the Notices of the AMS in the following terms: “… one of the most distinguished mathematicians and scholars of his time … always writing, by his own testimony, with the reader in mind … [his] passing marked the end of an era, in which celebrated names from the ‘old world,’ including Bergman, Loewner, Pólya, Schiffer, and Szegő, created at Stanford University one of the great world centers for classical analysis…” Thus, the book under review is part of a wonderful tradition of analysis and is possessed, to boot, of a certain evocative historical quality. (By the way, regarding Schiffer, I had occasion to review another of his collaborations on the order of a year ago.)

Kernel Functions and Elliptic Differential Equations in Mathematical Physics is obviously situated at the interface of analysis and mathematical physics, which is altogether natural, really, given the role played by integrating kernels in modern physics, in the wake of, for instance, Methoden der Mathematische Physik I,II, by Hilbert and Courant, the classic set of texts famously providing the mathematics needed for quantum mechanics. In point of fact, Hilbert-Courant is mentioned by the authors in the present book’s preface, in the context of a caveat to the effect that while “[t]he present book is not to be considered a textbook on partial differential equations … [i]t assumes a fair acquaintance with the standard methods of analysis provided … by … Courant-Hilbert, Frank-Mises, and Jeffreys.” Regarding these references, it looks like Frank-Mises is nothing less than Die Differential- und Integralgleichungen der Mechanik und Physik, by Phillip Frank and Richard von Mises, with additional author’s credit given to none other than Heinrich Weber and (yes) Bernhard Riemann; the other text is the 718 page tome by Harold and Bertha Jeffreys, Methods of Mathematical Physics (cf. ), another acknowledged classic.

Thus, in the book by Bergman and Schiffer we’re dealing with some old-fashioned scholarship and pedagogy, so to speak, in the sense that the student, or the reader, was (and is) expected to have some very serious mathematics under control already, having read some correspondingly serious books. Bergman and Schiffer indeed go on to say that “… an engineer or a physicist with the conventional mathematical training may possibly find some parts [of their book] difficult to read …,” and so be it: the authors focus on PDEs as mathematical objects, despite their prevalence and utility in physics and engineering, and it is only proper that the indicated prerequisites are required.

That said, however, the book does start with the themes of heat conduction and fluid dynamics, and then goes on to “electro- and magnetostatics” and elasticity. These four chapters, adding up to about 250 pages, constitute the book’s “Part A,” dealing with PDEs of elliptic type and the according boundary value problems: here, clearly, we encounter the explicit tenor of hard analysis (as opposed to the soft kind: we’re not talking about difficult vs. easy here — those days are long gone), and we get a good deal of airplay given to the Dirichlet principle, Green’s functions, spherical harmonics, orthogonal harmonic functions, and so forth. This is still pretty unforgiving material for the uninitiated or the easily cowed.

Then, in Part B, we hit the pay load: “Kernel function methods in the theory of boundary value problems.” It is remarkable indeed to see what a wealth of deep analysis is covered here, including existence results (for certain types of solutions of elliptic PDEs (cf. p. 258)), Dirichlet integrals, as well as a good deal of coverage of integral equations (not just regarding kernels, but also the corresponding eigenvalue analysis). A propos, for a number theorist like me it is striking that Dirichlet’s name appears with such great frequency in a book with such close connections to physics: it is a reminder that the same wonderful mathematician who gave us the gorgeous theorem on primes in arithmetic progressions and the unit theorem for number fields also coined the Dirichlet principle central to physics.

And that brings me to Riemann: in his hands the latter principle was taken to a very high level indeed (and it was Hilbert who finally presented it in a form consonant with rigorous mathematics. Riemann was breaking new ground, and there were intuitions to be justified — I guess Reid’s Hilbert best describes this state of affairs and Hilbert’s definitive work on this theme). Riemann is possibly the ne plus ultra as far as mathematical breadth is concerned, seeing that his applied analytic work, and his physics (and differential geometry), was offset by the single most remarkable and influential article ever written on analytic number theory, giving us both the prime number theorem and the Riemann hypothesis. But I digress — and how. Getting back to the book under review, the salient point is that Kernel Functions and Elliptic Differential Equations in Mathematical Physics is a historically and (still) mathematically significant work, and it sits squarely in a long tradition of interplay and cross fertilization between mathematics and physics. Even in this day and age, over 60 years after its first appearance, this book is a very valuable contribution indeed.

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

Note to the Reader
A. Boundary Value Problems for Partial Differential Equations of Elliptic Type
1. Theory of Heat Conduction
2. Fluid Dynamics
3. Electro- and Magnetostatics
4. Elasticity
B. Kernel Function Methods in the Theory of Boundary Value Problems
1. Properties of Solutions
2. The Kernel Functions and Their Properties
3. Variational and Comparison Theory
4. Existence Theory
5. Dependence of Kernels on Boundary Conditions and the Differential Equation
6. Generalizations
List of Symbols Used in Part B
Author Index
Subject Index