Non-divergence form parabolic equations associated with non-commuting vector fields: Boundary behavior of nonnegative solutions

TL;DR

This paper investigates the boundary behavior of nonnegative solutions to certain subelliptic parabolic equations with H"older continuous coefficients, establishing key inequalities and properties that extend classical results to a non-commuting vector fields setting.

Contribution

It introduces boundary regularity results for subelliptic parabolic equations with H"older continuous coefficients, generalizing classical Lipschitz cylinder results to a non-commuting vector fields framework.

Findings

Backward Harnack inequality for solutions vanishing on boundary

H"older continuity of quotient of solutions up to boundary

Doubling property of the associated parabolic measure

Abstract

In a cylinder $\Omega_T=\Omega\times (0,T)\subset \R^{n+1}_+$ we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form \[ Hu =\sum_{i,j=1}^ma_{ij}(x,t) X_iX_ju - \p_tu = 0, \ (x,t)\in\R^{n+1}_+, \] where $X=\{X_1,...,X_m\}$ is a system of $C^\infty$ vector fields in $\Rn$ satisfying H\"ormander's finite rank condition \eqref{frc}, and $\Omega$ is a non-tangentially accessible domain with respect to the Carnot-Carath\'eodory distance $d$ induced by $X$. Concerning the matrix-valued function $A=\{a_{ij}\}$, we assume that it be real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries $a_{ij}$ be H\"older continuous with respect to the parabolic distance associated with $d$. Our main results are: 1) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem \ref{T:back}); 2) the H\"older continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem \ref{T:quotients}); 3) the doubling property for the parabolic measure associated with the operator $H$ (Theorem \ref{T:doubling}). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [FSY] and [SY]. With one proviso: in those papers the authors assume that the coefficients $a_{ij}$ be only bounded and measurable, whereas we assume H\"older continuity with respect to the intrinsic parabolic distance.