The fundamental gap of a kind of two dimensional sub-elliptic operator

Abstract

This paper is concerned at the minimization fundamental gap problem for a class of two-dimensional degenerate sub-elliptic operators. We establish existence results for weak solutions, Sobolev embedding theorem and spectral theory of sub-elliptic operators. We provide the existence and characterization theorems for extremizing potentials $V\left(x\right)$ when $V\left(x\right)$ is subject to $L^{\infty }$ norm constraint.