Geometric estimation of a potential and cone conditions of a body

TL;DR

This paper develops geometric estimates for a potential function derived from a body with exterior cones, aiding in localizing potential maximizers and applying these results to the Poisson integral in the upper half space.

Contribution

It introduces a method to estimate potentials using cone conditions, improving the understanding of potential maximizers within bodies.

Findings

Potential maximizers are confined to smaller regions within the body.

The estimates are applicable to the Poisson integral in the upper half space.

Abstract

We investigate a potential obtained as the convolution of a radially symmetric function and the characteristic function of a body (the closure of a bonded open set) with exterior cones. In order to restrict the location of a maximizer of the potential into a smaller closed region contained in the interior of the body, we give an estimate of the potential using the exterior cones of the body. Moreover, we apply the result to the Poisson integral for the upper half space.