So it is, then, that this wonderful book is imbued with a marvelous historical perspective so that the reader is taught some very beautiful mathematics fitted in the proper historical perspective. Moreover, as regards mathematics per se, the book is full of terrific hard (as opposed to soft) analysis focused on a general theme that is exemplified by the authorâ€™s astute and elegant choice of topics. Dacorognaâ€™s arsenal (amassed in Chapter 1) includes Hölder continuity, Sobolev spaces and convex analysis, in addition to more familiar fare; thereafter he goes after the following big game: the Euler-Lagrange and Hamilton-Jacobi equations, the Dirichlet problem, of course, and then minimal surfaces and the isoperimetric inequality.
Dacorognaâ€™s dominant focus on the Dirichlet problem is explicit from the very outset: already on p.2 he characterizes the problem as â€œthe most celebrated problem of the calculus of variationsâ€ and points towards Hilbertâ€™s famous solution of the problem (recounted so beautifully in Constance Reidâ€™s Hilbert). Lebesgue and Tonelli soon joined in and, collectively, their approaches sired what are today called â€œthe direct methods of the calculus of variations.â€ Wasting no time, on p.3 Dacorogna presents a very accessible model for the Dirichlet problem, serving as a prototype and more, consisting in minimizing the appropriate Dirichlet integral as a function of â€œadmissibleâ€ functions whence everything largely depends on what one posits as the indicated a priori hypotheses on the admissible functions (in this connection see p.79 for example). The model given on p.3 sets the tone (and the stage) for much of what follows and evolves magnificently through the ensuing pages, lending the book an impressive cohesion.
This having been said, it should be noted that while Dacorogna advertises his book as â€œa concise and broad introductionâ€ to the calculus of variations at an undergraduate and beginning graduate level, he does presuppose the reader to be able and willing to work hard and do battle with some serious analysis. There are a lot of (outstanding) exercises and these are critical for a deeper understanding of the material. Happily all of Chapter 7 is devoted to their solutions, and this increases the bookâ€™s already considerable value as a source for self-study.
Also, Dacorogna claims that even if his own focus in Introduction to the Calculus of Variations is on â€œmathematical applications,â€ the book is still accessible to folks from physics, engineering, biology, etc. Letâ€™s just reply â€œYeah, right â€¦â€ to this: itâ€™s hard-core mathematics, make no mistake! But as a more sophisticated introduction to the calculus of variations itâ€™s a very beautiful treatment, and will reward the diligent reader with a solid introduction to a great and grand subject and to a lot of beautiful hard analysis.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.