Asymptotic average solutions to linear second order semi-elliptic PDEs: a Pizzetti-type Theorem

TL;DR

This paper introduces the concept of asymptotic average solutions for hypoelliptic PDEs, extending classical ideas to ensure pointwise solvability of Poisson equations with continuous data.

Contribution

It generalizes Pizzetti's classical approach to a broader class of hypoelliptic operators, providing a new framework for solving Poisson equations pointwise.

Findings

Defines asymptotic average solutions for hypoelliptic PDEs

Ensures pointwise solvability of Poisson equations with continuous data

Extends classical Pizzetti's theorem to semi-elliptic operators

Abstract

By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of {\it asymptotic average solutions} making pointwise solvable every Poisson equation $\mathcal{L} u(x)=-f(x)$ with continuous data $f$, where $\mathcal{L}$ is a hypoelliptic linear partial differential operator with positive semidefinite characteristic form.