Strict starshapedness of solutions to the horizontal p-laplacian in the Heisenberg group

TL;DR

This paper investigates the geometric properties of solutions to the horizontal p-Laplacian in the Heisenberg group, establishing conditions under which their level sets are strictly starshaped, contributing to geometric analysis in sub-Riemannian spaces.

Contribution

It provides sharp geometric conditions ensuring the strict starshapedness of level sets of p-capacitary potentials in the Heisenberg group, advancing understanding of horizontal p-harmonic functions.

Findings

Level sets are strictly starshaped under certain geometric conditions.

Established sharp conditions for starshapedness of solutions.

Contributed to the geometric analysis of sub-Riemannian PDEs.

Abstract

We examine the geometry of the level sets of particular horizontally $p$-harmonic functions in the Heisenberg group. We find sharp, natural geometric conditions ensuring that the level sets of the $p$-capacitary potential of a bounded annulus in the Heisenberg group are strictly starshaped.