On the Dirichlet Problem for hypoelliptic evolution equations: Perron-Wiener solution and a cone-type criterion

TL;DR

This paper extends potential theory methods to solve the Dirichlet problem for a broad class of hypoelliptic evolution equations, introducing a Perron-Wiener solution and a generalized boundary regularity criterion.

Contribution

It develops a harmonic space approach for hypoelliptic PDEs, generalizing classical cone criteria for boundary regularity in the heat equation.

Findings

Constructed Perron-Wiener solutions for hypoelliptic PDEs

Provided a generalized cone-type regularity criterion

Extended potential theory methods to new classes of PDEs

Abstract

We show how to apply harmonic spaces potential theory in the study of the Dirichlet problem for a general class of evolution hypoelliptic partial differential equations of second order. We construct Perron-Wiener solution and we provide a sufficient condition for the regularity of the boundary points. Our criterion extends and generalizes the classical parabolic-cone criterion for the Heat equation due to Effros and Kazdan.