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Partial Differential Equations 1: Foundations and Integral Representations

Friedrich Sauvigny
Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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Friedrich Sauvigny’s remarkable two-volume opus, Partial Differential Equations, 1,2, is the English translation of Partielle Differentialgleichungen der Geometrie und der Physik 1 (Grundlagen u. Integraldarstellungen), 2 (Funktionalanalytische Lösungsmethoden) — unter Berücksichtigung der Vorlesungen von E. Heinz, and constitutes the author’s attempt to treat the beautiful and difficult subject of PDEs in a thorough and instructive way, at, in his own words, the intermediate level (about which more presently). Sauvigny characterizes his presentation of the subject as being particularly tied to functional analysis, which renders these books a multiple threat, so to speak (though “treat” is more on the mark).

Partial Differential Equations, 1,2 (henceforth simply PDE 1,2), also evince the pronounced influence of the author’s highly-regarded teacher, his Doktor-Vater, E. Heinz, who is credited in the German subtitle. Moreover, in the Introduction to the first volume, amidst paying special tribute to his alma mater , Göttingen, Sauvigny singles out G. C. Lichtenberg’s pedagogical maxim, “Teach students how they think, not what they think,” as his marching orders.

It is therefore fair to say that PDE 1,2 is heavily steeped in the famous Göttingen mathematical tradition, both pedagogically and qua continuity, given the University’s famous twin (and intertwined) roots in mathematics and physics and its historical and defining role in mathematical and scientific education reform over the last three centuries. In this connection Constance Reid’s Hilbert and Courant in Göttingen and New York are without doubt the most evocative sources, particularly as regards how the Göttingen style, as identified with Klein and Hilbert and characterized by a dialectic between mathematics and physics, came to leave a legacy that thrives even today.

Sauvigny is the happy beneficiary of this wonderful legacy and PDE 1,2 is liberally peppered with what he terms “historical notices,” coming at the end of each of the twelve chapters and frequently capped off by nice photographs of the luminaries of the corresponding subject (my favorite occurs on p. 296 of PDE 1: Hermann Amandus Schwarz). These notices, which amplify the cultural experience of reading and using these books, are indeed marvelously instructive (while concise) and attest to the author’s broad scholarship.

Regarding Sauvigny’s description of his treatment as being “intermediate” one naturally needs to place the word in context: the reader should not be a novice in the field of hard analysis and should be prepared to do the serious work that facilitates the transition from student to scholar, specifically at the stage of a beginning researcher in partial differential equations. The preface to Part 1 explains that it should be sufficient for the reader to have a strong background in fundamental analysis as presented in “Blue Rudin,” i.e., Principles of Mathematical Analysis, and this seems to be right on the money. For the student accordingly prepared, PDE 1,2 goes on to provide not just a deep treatment of partial differential equations but a nigh on autonomous course in graduate analysis.

Volume 1 starts off with a sequence of superb (and long) chapters on foundational themes, starting with calculus on manifolds and functional analysis and proceeding through to spherical harmonics. In the first 350 pages or so Sauvigny serves up such lynch-pin theorems as Poincaré’s Lemma on the exactness of certain closed forms (complete with a proof attributed to André Weil), measure and integration à la Daniell and Lebesgue (and Riemann), L. E. J. Brouwer’s work on the degree of a mapping (with a proof of the product theorem due to Lipman Bers), and the Dirichlet Problem. And then Volume 1 is capped off by a beautiful discussion of linear PDE’s in n-space.

Next, Volume 2 starts off with two terrific chapters on operators in Banach spaces and linear operators in Hilbert spaces, comprising a short, but still very thorough, course in functional analysis. Thereupon, wasting no time, Sauvigny closes in on some big game: linear elliptic differential equations, their weak solutions, non-linear PDEs, and non-linear elliptic systems. Like those in Volume 1, these chapters are chock-full of serious mathematics fitted compactly into a historical framework, with credit scrupulously given where credit is due. It’s very, very impressive scholarship.

While unsuited for light reading (except for the “historical notices”) and while it presupposes sufficient maturity and experience in the reader to justify the absence of problems and exercises, PDE 1,2 is a fine place to learn PDEs with the goal of doing serious work in the field. As already mentioned, Friedrich Sauvigny’s scholarship is exemplary and thorough; at the same time his book is both broad and deep and is a pleasure to read. Of course reading such a tome is not for the timid: margins must be covered with comments and notebooks must be filled with reformulations of Sauvigny’s proofs (to give the nod to one particular learning-style). But working through PDE 1,2 will richly reward those who go this route: they’ll emerge well-prepared for exploring the frontiers of the subject.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.

 Differentiation and Integration on Manifolds.- Foudations of Functional Analysis.- Brouwer´s Degree of Mapping with Geometric Applications.- Generalized Analytic Functions.- Potential Theory and Spherical Harmonics.- Linear Partial Differential Equations in Rn.