Carnot–Carathéodory metrics associated to degenerate elliptic operators in three dimensions

Abstract

In this note, we generalize some of the geometric results obtained by Korobenko, Sawyer, Rios, and Shen, who studied operators of the form \(\nabla \cdot A\nabla \) with \(A(x)\approx \{1,f^2(x_1)\}\) to the 3D case where \({A(x)\approx \textrm{diag}\{1,f^2(x_1), g^2(x_1)\}}\). More precisely, we make explicit calculations of the geodesics in the Carnot–Carathéodory space associated to A and provide estimates on the Lebesgue measures of metric balls centered at the origin in that space.