The kernel space of linear operator on a class of Grushin equation

TL;DR

This paper investigates the kernel of linear operators associated with a class of Grushin equations, providing explicit descriptions and invertibility results for singularly perturbed cases.

Contribution

It offers a general analysis of the kernel space for Grushin equations and derives an exact expression for a special case, including invertibility results.

Findings

Kernel space characterized for general Grushin equations

Exact kernel expression for a special Grushin equation

Invertibility established for singularly perturbed Grushin operator

Abstract

In this paper, we concern the kernel of linear operator for a class of Grushin equation. First, we study the kernel space of linear operator for a general Grushin equation. Then, we provide an exact expression for the kernel space of linear operator for a special Grushin equation. Finally, we prove the linear operator related to the singularly perturbed Grushin equation is invertible when restricted to the complement of its approximate kernel space.