Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

TL;DR

This paper establishes mean value formulas for classical solutions to certain elliptic and subelliptic equations, including those on Carnot groups, using fundamental solutions and finite perimeter theory.

Contribution

It extends mean value formulas to degenerate elliptic equations and subelliptic operators on Carnot groups, incorporating the theory of finite perimeters.

Findings

Mean value formulas for elliptic equations with Hölder continuous coefficients

Extension of formulas to subelliptic operators on Carnot groups

Use of fundamental solutions supported on level sets

Abstract

We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with H\"{o}lder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relevant the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeters on stratified Lie groups.