PDE is one of things I wish I knew a lot more about. In another order of Providence, if it were possible to get, let’s say, 72 hours out of every day, wouldn’t it have been wonderful to pursue hard analysis, in addition to so many other marvelous mathematical adventures? But choices must be made in this life, and I was never more than an analysis dilettante.
Indeed, it was my impression decades ago, during those formative university years, that hard analysts populated an exclusive world all their own, much like the logicians. There might be détente between algebraists, algebraic geometers, number theorists, and combinatorialists, for example, but hard analysts were a law unto themselves (modulo a couple of straying differential geometers and differential topologists, every so often). My outsider status notwithstanding, I had the good fortune to be in a position, throughout my school days, to float from seminar to seminar, often at the kind invitation of an ecumenically minded friend. As I had several friends among the analysts I found myself quasi-learning about such arcana as the Banach-Alaoglu Theorem, the duality between BMO and (I think) the first Hardy space (one of Charles Fefferman’s triumphs), and all sorts of stuff about the Corona Theorem: inequalities and estimates galore, weird constructions everywhere, and counterintuitive counterexamples to make your head spin.
I wonder if times have changed. After all, with Thurston’s geometrization program forever altering the face of geometry, with Yau’s earlier work bringing analysis into differential geometry, with Poincaré-3 taking it on the chin via applications of analysis to topology (if I may be allowed an oversimplification), and with Navier-Stokes stirring up all kinds of activity among all kinds of enthusiasts (geometers, global analysts, hard analysts, who knows who else), can it be that some sort of perestroika involving hard analysis’ place among the other mathematical subdisciplines is afoot? It certainly seems that PDE is everywhere these days!
Well, good. It is an exciting field, with lots of opportunities for researchers of all generations, and it is certainly a proper direction in which to guide talented fledgelings.
Against this backdrop, the book under review, the second edition of Emmanuele DiBenedetto’s 1995 Partial Differential Equations, now appearing in Birkhäuser’s “Cornerstones” series, is an example of excellent timing. DiBenedetto states that the material in this second edition is “essentially the same [as in the first] except for three new chapters,” namely, Ch. 8, on first-order non-linear (e.g. Hamilton-Jacobi) equations, Ch. 9, on “direct variational methods for linear and quasi-linear elliptic equations” (as well as weak formulations and Sobolev spaces), and Ch. 10 on local methods “in the framework of DeGiorgi classes.” Otherwise, the material in PDE is that of the first edition, albeit “revised and extended.” (Caveat: “Some elementary background material (Weierstrass’ Theorem, mollifiers, Ascoli-Arzelá… Jensen’s inequality…) has been removed.”)
So it is that DiBenedetto, whose philosophical position regarding PDE is unabashedly that “although a branch of mathematics, [it is] closely related to physical phenomena,” presents us with marvelous coverage of (in order), quasi-linearity and Cauchy-Kowalevski (shouldn’t it be Cauchy-Kovalevskaya?), Laplace, BVP’s by “double-layer potentials,” [and my favorite three chapters:] integral equations and the eigenvalue problem, the heat equation, and the wave equation. Then he returns to quasi-linearity (for first order equations), goes on to non-linearity, linear elliptic equations with measurable coefficients (just think of the alternative!), and, finally, as already indicated, DeGiorgi classes.
This itinerary is already sufficient to convey that even if the book is meant to be an introduction to the subject (says the author: “[The book] assume[s] only advanced differential calculus and some basic Lp theory”), its aim is to form able and enthusiastic converts to the PDE cause. And it does an outstanding job. PDE is beautifully written, in clear and concise prose, the mathematics is cogent and complete, and the presentation testifies both to DiBenedetto’s fine taste in the subject and his experience in teaching this difficult material (ca. fourteen years, at Indiana, Northwestern, Rome II, and, now, Vanderbilt). To the extent this is possible, or reasonable, he even succeeds in making the notorious hard-analytic intricacies of PDE appear almost innocuous, courtesy of an excellent presentation and attractive writing-style.
Make no mistake: the book is neither chatty nor discursive, but there’s something more or less ineffable about it, making it appear somehow less austere than other texts on PDE. Check it out.
DiBenedetto has also included a decent number of what he calls “Problems and Complements,” and, to be sure, these should capture the attention of the conscientious student or reader.
Thus, DiBenedetto’s PDE is indeed a cornerstone text in the subject. It looks like a rare gem to me.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.