Explicit positive solutions to $G$-heat equations and the application to $G$-capacities

TL;DR

This paper constructs explicit positive solutions to $G$-heat equations via nonlinear ODEs and applies these solutions to determine the precise behavior of $G$-capacities as intervals shrink.

Contribution

It introduces a method to find explicit positive solutions to $G$-heat equations and uses them to analyze the asymptotic order of $G$-capacities.

Findings

Derived explicit positive solutions to $G$-heat equations.

Established the sharp order of $G$-capacity as interval size approaches zero.

Abstract

In this paper, we study a class of explicit positive solutions to $G$-heat equations by solving second order nonlinear ordinary differential equations. Based on the positive solutions, we give the sharp order of $G$-capacity $c_{\sigma}\left( \left[ -\varepsilon,\varepsilon \right] \right) $ when $\varepsilon \rightarrow0$.