Asymptotic Average Solutions to Linear Second Order Semi-Elliptic PDEs: A Pizzetti-Type Theorem

Abstract

By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of 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.