$L^p$ estimate for positive harmonic functions near singularities and B\^{o}cher type theorems

TL;DR

This paper establishes $L^p$ integrability estimates for positive harmonic functions near 1-dimensional singularities and characterizes their Laplacian behavior, extending classical results and methods to more general settings.

Contribution

It introduces $L^p$ estimates for solutions near singularities and provides a characterization of the Laplacian, with methods adaptable to higher dimensions.

Findings

Established $L^p$ integrability estimates near singularities.

Characterized the behavior of $- riangle u$ around the circle.

Method applicable to higher-dimensional cases.

Abstract

In this paper, positive solutions to the Laplace equation with 1-dimensional circular singularities are investigated. First, we establish $L^p$ integrability estimates for such solutions $u$ near the singularities, in comparison with classical $L^1$ estimates. Then we characterize $-\Delta u$ around the circle. The method we developed may also be adapted to higher dimensional as well as more generalized cases.