Boundary Behavior of Subelliptic Parabolic Equations on Time-Dependent Domains

TL;DR

This paper investigates the boundary behavior of solutions to subelliptic heat equations in time-dependent domains, establishing key estimates and properties like Dahlberg estimates, Harnack inequalities, and boundary regularity.

Contribution

It introduces new boundary estimates and regularity results for subelliptic heat equations in time-varying domains satisfying H"ormander's condition.

Findings

Dahlberg type estimate relating X-caloric measure and Green function

Backward Harnack inequality for solutions

Doubling property and boundary H"older continuity of solutions

Abstract

In this paper we study the boundary behavior of solutions of a divergence-form subelliptic heat equation in a time-varying domain \Omega in R^{n+1}, structured on a set of vector fields X = (X_1, ... X_m) with smooth coefficients satisfying H\"ormander's finite rank condition. Assuming that \Omega is an X-NTA domain, we first prove a Dahlberg type estimate comparing the X-caloric measure of \Omega and the Green function of the subelliptic heat operator. We then prove a backward Harnack inequality, the doubling property for the X-caloric measure of \Omega, the H\"older continuity at the boundary for quotients of solutions of H, and a Fatou theorem.