Potential Theory

A function \(u\) is harmonic if it has continuous second partial derivatives and its Laplacian \(\Delta u = \sum \partial^2\!u/\partial x_k^2\) is zero. Potential theory concerns itself primarily with the study of harmonic functions — their existence, uniqueness, and structure under various conditions. Of specific interest is the Dirichlet problem, which asks for a harmonic function on the interior of a region that agrees with a given continuous function on the boundary. The Laplacian arises naturally in physics in the study of potentials, hence the name of the subject. Lester Helms’ Potential Theory is an update of his earlier 1969 lecture notes and features a rigorous exposition of potential theory in arbitrary dimensions. (Potential theory in two dimensions is really the domain of complex analysis. It is covered in special cases of more general results, but is not a focus of this text.)

The author sets the goal of the book as getting the reader from real analysis to the front line of potential theory as quickly as possible. The narrative is thorough yet quickly paced and Helms does a nice job of proving theorems as generally as possible but providing specific examples where appropriate. There are no formal exercises, but the author makes good choices about which details to leave for the reader. Each chapter begins with some physical and historical context, but the bulk of the text concerns a careful development of results. Although the author reminds us of some of the main tools from analysis in the first chapter, the reader will want a solid grounding in real analysis.

The content is quite sensibly organized. Although the exposition can get dense, Helms never lets the reader forget how each topic fits into the overall goal of understanding Dirichlet problems (also Neumann problems, in which the normal derivative rather than the function value is specified at the boundary). Early chapters establish the fundamental tools of potential theory, including harmonic functions, Green’s Theorem, the mean value property of harmonic functions, and the Poisson Integral Formula. Green’s functions and their properties are developed very carefully and thoroughly as the primary device for constructing solutions, followed by an excellent exposition of subharmonic and superharmonic functions and the Perron method, the Riesz Decomposition Theorem, polar sets, capacity, and energy. Later chapters deal with transformations of regions and their effects on solutions to Dirichlet and Neumann problems, Newtonian kernels, and reflection. The book concludes with constructing solutions of more general elliptic equations beyond the Laplacian and discusses the oblique derivative problem, a generalization of the Neumann problem.

It is impressive how far the book is able to go without sacrificing rigor; though the book is far from chatty, a consistent narrative helps keep things motivated. The excellent index is also very helpful in navigating the material. Nearly all of the material is classical; the most recent references date to the mid-1980s, but the content is at the heart of much active work.

With the requisite background, Potential Theory would make a fine text for self-study, reference, or a graduate course. Researchers in the field will consider it a standard and those in an adjacent field such as differential equations, functional analysis, or mathematical physics will also find it a valuable reference.

Bill Wood is a Project NExT Sun Dot, board game enthusiast, and unimpressive disc golf player.