Lectures on Elliptic Boundary Value Problems

In 1912 Hermann Weyl published a paper in the Mathematische Annalen carrying the title,”Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit Anwendung auf die Theorie der Hohlraumstrahlung).” Teutonic titles, be they of persons or of papers, are always dramatic and imposing, and, given their lengthy attention to detail, germane (if I may be forgiven a play on words) to the subject in question. The preceding Beispiel is no exception: “The asymptotic distribution of the eigenvalues of linear partial differential equations (with application to the theory of vacuum radiation).” With quantum mechanics firmly established as a twentieth century scientific and also a mathematical landmark, given that representation theory and functional analysis were transformed and extended so as to meet the challenged posed by the new physics, our current 20/20 hindsight just underscores the historical importance of Weyl’s paper. Schrödinger’s wave equation is a linear PDE, after all, and the eigenvalues corresponding to its time-independent formulation in the Hamiltonian formalism provide nothing less than the admissible (read measurable) energy levels for the particle, or quantum system, in question. To know what these energies do asymptotically is a dramatically meaningful question, therefore, not only on mathematical grounds but on physical ones.

But what does this have to do with Shmuel Agmon’s book, Lectures on Elliptic Boundary Value Problems, a brand new re-issue by (you gotta love ‘em!) AMS Chelsea of a 1965 classic? Well, on p.177 of the book we find the following fragment: “Following a classical paper by H. Weyl [loc.cit.], [a] formula such as (14.44) is referred to as Weyl’s law for the distribution of eigenvalues. Investigations on Weyl’s law, its generalizations and implications, form a very active field of research. As an example we mention the seminal work of L. Hörmander [nothing less than “The spectral function of an elliptic operator,” Acta Math 121 (1968)] dealing with estimates for the remainder in Weyl’s law, in a general set up.”

And this illustrates the tenor of the book under review: it’s hard analysis with a vengeance (so those who, as a matter of principle, find connections with physicist’s ways of doing things off-putting need not worry) and is in fact pitched at a rather high level. Of course the latter judgment on my part reflects the fact that no one would confuse me for a hard analyst: a monograph such as the present one, with inequalities seemingly populating every single page, is not my cup of tea. But this is a descriptive not a disparaging phrase: Lectures on Elliptic Boundary Value Problems is a wonderful and important book (indeed, a classic, as already noted), and analysts of the right disposition should rush to get their copy, if they don’t already have one (1965 being a long time ago, after all).

To wit, Agmon takes the reader from the foundations of the subject (calculus of L² derivatives, elliptic operators, local existence theory, and local regularity of solutions) to hard-core functional analysis (Garding, Aronszajn, Smith, Hilbert-Schmidt, &c.) and spectral theory , and then to eigenvalue problems for elliptic PDE, covering both the self-adjoint and non self-adjoint cases. The book closes with a chapter on the completeness of eigenfunctions.

Even to a non-analyst like me it is abundantly clear that this is a first rate exposition of beautiful material. It is proper that AMS Chelsea re-issue it now.

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