G. C. Evans

On potentials of positive mass. I

A recent contribution to potential theory is characterized by the names of Lebesgue, Wiener, Kellogg, Vasilesco and Bouligand. Central features of this contribution are the notions of capacity and of regular boundary point, which are related to each other by Kelloggs hypothetical lemma.t A recent memoir by de la Vallee Poussin reinterprets these theories in the light of potentials of positive mass and the Poincare sweeping out process.: In the present memoir, the authors aim is to push on the development of these central problems of mass distribution, regularity, capacity, and approximation, and to answer definitely some of the questions which have become important. Fortunately there is already at hand, in the analysis of the general integral or linear functional of Radon, Daniell, and F. Riesz, the precise mathematical tool which is necessary.? 1. Integrals and potentials. Let F be a closed bounded point set, T the infinite domain lying in the complement of F, whose boundary t consists entirely of points of F. We consider an arbitrary distribution of positive mass f(e) on F, finite in total amount.II The potential of this mass, at a point M, is given by the generalized Stieltjes integral

Volterra’s integral equation of the second kind, with discontinuous kernel

In this equation the function K(w, t ) is called the kerllel; the desired function is u(z) . VVhen the functions K(w, {) and +(.z) are continuous there is no diSculty in finding a continuous 6(z) that shall satisfy equation (1). This general case of the equation (1), under the conditions solelythat +(z) be continuouswhen a-z-b and that X(x, t:) be continuous in the triangular region a-t:-z _ b, was first investigated by VOLTERRA, t who showed that there is one and only one continuous solution in the interval a _ z c b . His method applies without essential change to equations whose kernels are finite in the region a c t _ z _ b and have discontinuities, provided the discontinuities are regularly t distributed. Let us consider, however, a certain equation to which we are led by a hydrostatical problem. Suppose we are given a tube lying in a vertical plane along a curve of arbitrary shape, s t6(.z), where s is the distance along the curve and x the altitude. Let us fill this tube with a liquid of variable linear density v, and then regulate its height .z in the tube by allowing various amounts to flow through the bottom. Let us then legard zo as an analytic function of the depth in the liquid, i. e. zo zo ( Z t: ) . The average linear density is given by the formula

Continua of minimum capacity

1. Surfaces containing a given volume. In an endeavour to simplify a proof of Liapounoff [2], to the effect that in the problem of the forms of equilibrium of rotating liquids the sphere would be the only form for a liquid at rest, Poincaré [ l ] was led to the consideration of electric capacities of solids of given volume, and arrived at the result that among such bodies the sphere would have minimum capacity. The present paper originated in the question of the determination of the surface sheet, without volume, which would be bounded by a given closed curve in space, and, among all such surfaces, have minimum capacity. In the discussion of his problem, Poincaré assumes tacitly that there do exist one or more bodies of the given volume, with smooth boundaries, which furnish relative minima for the capacity with respect to neighboring forms; and his treatment amounts to a proof that among these the sphere furnishes the absolute minimum. Let the body F, with smooth exterior boundary S, be considered as a conductor on which a positive charge m is spread so as to be in equilibrium—that is, the charge lies entirely on the surface S with a surface density cr(P), and its potential V(M) has a constant value Vo within S, is continuous across 5, satisfies Laplaces equation VV=0 outside S and vanishes at infinity. The density on S is given by the equation

Modern methods of analysis in potential theory

I shall take the liberty of interpreting the title in terms of a short discussion of certain problems that have been of recent interest, particularly to younger mathematicians, namely, the discontinuous boundary value problems of the Dirichlet and Neumann types, the mixed problem, and finally the generalizations of the potential integral itself to others in which the integrand 1/r is replaced by l/r, 0 < a < 3 . In this way my review will serve as a continuation of that of the late Professor Kellogg, but will diverge from it in the sort of problem to be considered.!