On the level surfaces of potentials of masses with fixed center of gravity
1952; Mathematical Sciences Publishers; Volume: 2; Issue: 2 Linguagem: Inglês
10.2140/pjm.1952.2.147
ISSN1945-5844
Autores Tópico(s)Geotechnical and Geomechanical Engineering
Resumo1. Introduction.Let μ(e) be a nonnegative additive set function on the closed unit sphere K in three dimensions with μ(K) = 1.We shall regard μ as a distribution of unit positive mass on K. Furthermore, we shall require that the center of gravity of this distribution be at the origin; that is,Such μ will be called admissible distributions, and their Newtonian potentials u(P), admissible potentials.In 0. D. Kellogg's Foundations of Potential Theory [l], there occurs on page 144 an exercise which amounts to the following: Show that a level surface of an admissible potential which lies outside a concentric sphere of radius 10 varies in distance from the origin by less than 1.2 per cent.The figure 1.2 appears to be correct, as we shall show by an example, only when variation is interpreted to mean variation from some intermediate sphere.Then with this interpretation, Kellogg's figure is easily obtained by considering the expansion of the potential in a series of spherical harmonics, whose use the set in which this problem appears was designed to illustrate.However, since the estimates used are fairly crude, the figure is not attained by any level surface.In this note is solved the problem of exactly how much a level surface of an admissible potential can depart from being spherical, and what distributions give the extreme level surfaces.More precisely, we shall prove the following theorem: THEOREM 1.If a level surface of an admissible potential has minimum radius r and maximum radius R, with r > 1, then 2R < [(r 2 + I) 1 / 2 + (r 2 + 5) ι / 2 ]. This value is attained if and only if the distribution consists of two equal point
Referência(s)