Boundary behavior of positive solutions of the heat equation on a stratified Lie group

TL;DR

This paper investigates the boundary behavior of positive solutions to the heat equation on stratified Lie groups, establishing conditions for the existence of parabolic limits and exploring measure-theoretic properties.

Contribution

It provides a necessary and sufficient condition linking parabolic limits of solutions to the strong derivative of boundary measures on stratified Lie groups.

Findings

Parabolic limit exists iff the strong derivative of boundary measure exists.

Parabolic limit and strong derivative are equal at boundary points.

Example on the Heisenberg group shows the set of strong derivative points exceeds Lebesgue points.

Abstract

In this article, we are concerned with a certain type of boundary behavior of positive solutions of the heat equation on a stratified Lie group at a given boundary point. We prove that a necessary and sufficient condition for the existence of the parabolic limit of a positive solution $u$ at a point on the boundary is the existence of the strong derivative of the boundary measure of $u$ at that point. Moreover, the parabolic limit and the strong derivative are equal. We also construct an example of a positive measure on the Heisenberg group to show that the set of all points where strong derivative exists is strictly larger than the set of Lebesgue points of the measure.