Carnot-Carath\'{e}odory metrics associated to degenerate elliptic operators in three dimensions

TL;DR

This paper extends geometric analysis of Carnot-Carathéodory metrics to a specific class of degenerate elliptic operators in three dimensions, providing explicit geodesic calculations and measure estimates for metric balls.

Contribution

It generalizes previous results to a 3x3 matrix setting with degeneracy, explicitly computing geodesics and metric ball measures in the associated Carnot-Carathéodory space.

Findings

Explicit geodesic formulas derived

Estimates on Lebesgue measures of metric balls

Extension of geometric results to degenerate elliptic operators

Abstract

This note is a companion paper to arXiv:1608.01630 [math.CA]. Here we generalize some of the geometric results of arXiv:1608.01630 [math.CA] to the case of a $3\times 3$ matrix function $A(x)\approx \mathrm{diag}\{1,f(x_1), g(x_1)\}$. More precisely, we make explicit calculations of the geodesics in the Carnot-Carath\'{e}odory space associated to $A$, and provide estimates on the Lebesgue measures of metric balls centered at the origin in that space.