# Regularity of solutions to elliptic equations with Grushin's operator

## TL;DR

This paper investigates the regularity of solutions to elliptic equations involving Grushin's operator, a degenerate operator, using heat kernel analysis via the Feynman-Kac formula to establish optimal regularity results.

## Contribution

It introduces a novel approach to analyze the regularity of solutions with a degenerate operator using heat kernel properties and the Feynman-Kac formula.

## Key findings

- Derived the heat kernel expression for Grushin's operator
- Established optimal regularity of solutions
- Extended regularity results to degenerate elliptic operators

## Abstract

In this paper, we consider the regularity of solutions to elliptic equation with Grushin's operator. By using the Feynman-Kac formula, we first get the expression of heat kernel, and then by using the properties of heat kernel, the optimal regularity of solutions will be obtained. The novelty of this paper is that the Grushin's operator is a degenerate operator.