On $d$ and $M$ problems for Newtonian potentials in Euclidean $ n $ space

TL;DR

This paper explores a conjecture about Newtonian potentials in Euclidean space with mass on the unit sphere, focusing on extremal potentials characterized by bounds on their maximum and minimum values.

Contribution

It introduces a conjecture classifying Newtonian potentials based on their extremal properties and proves the existence of extremal potentials and related integral inequalities.

Findings

Existence of extremal Newtonian potentials established.

An integral inequality involving extremal potentials proved.

Support for the conjecture provided through theoretical results.

Abstract

In this paper we first make and discuss a conjecture concerning Newtonian potentials in Euclidean n space which have all their mass on the unit sphere about the origin, and are normalized to be one at the origin. The conjecture essentially divides these potentials into subclasses whose criteria for membership is that a given member have its maximum on the closed unit ball at most M and its minimum at least d. It then lists the extremal potential in each subclass which is conjectured to solve certain extremal problems. In Theorem 1.1 we show existence of these extremal potentials. In Theorem 1.2 we prove an integral inequality on spheres about the origin, involving so called extremal potentials, which lends credence to the conjecture.