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

John Lewis

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.

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

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.

Paper Structure

This paper contains 5 sections, 6 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Notation, Definitions, and Basic Lemmas
  3. Proof of Theorem 1.1
  4. Proof of Proposition 4.1
  5. Proof of Theorem 1.2

Key Result

Theorem 1.1

If $n \geq 3,$ there is a 1-1 map from for which there exists a potential $P = P ( \cdot, d, M ) \in \mathcal{F}$ satisfying 1.1.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • proof
  • Remark 3.1
  • ...and 8 more