Starshapedeness for fully-nonlinear equations in Carnot groups

TL;DR

This paper proves that the level sets of capacitary potentials for a broad class of fully-nonlinear equations in Carnot groups are starshaped, extending geometric properties in sub-Riemannian analysis.

Contribution

It introduces a natural notion of starshapedness and establishes the starshapedness of level sets for fully-nonlinear equations in Carnot groups, a significant geometric insight.

Findings

Level sets are starshaped under certain conditions

Introduces a new notion of starshapedness in Carnot groups

Extends geometric analysis of nonlinear PDEs in sub-Riemannian settings

Abstract

In this paper we establish the starshapedness of the level sets of the capacitary potential of a large class of fully-nonlinear equations for condensers in Carnot groups, once a natural notion of starshapedness has been introduced. Our main result is Theorem 1.2 below.