Boundary regularity for subelliptic equations in the Heisenberg group

TL;DR

This paper establishes boundary regularity results for a class of degenerate subelliptic equations in the Heisenberg group, including new second order expansion results near characteristic points.

Contribution

It proves boundary Hölder and Lipschitz regularity for degenerate elliptic equations in the Heisenberg group with measurable coefficients and source terms, extending known results to new settings.

Findings

Boundary Hölder and Lipschitz regularity established

Intrinsic second order expansion near characteristic points obtained

Results apply to equations with bounded measurable coefficients and source terms

Abstract

We prove boundary H\"older and Lipschitz regularity for a class of degenerate elliptic, second order, inhomogeneous equations in non-divergence form structured on the left-invariant vector fields of the Heisenberg group. Our focus is on the case of operators with bounded and measurable coefficients and bounded right-hand side; when necessary, we impose a dimensional restriction on the ellipticity ratio and a growth rate for the source term near characteristic points of the boundary. For solutions in the characteristic half-space $\{t>0\}$, we obtain an intrinsic second order expansion near the origin when the source term belongs to an appropriate weighted $L^{\infty}$ space; this is a new result even for the frequently studied sub-Laplacian.