Boundary behavior of non-negative solutions to degenerate sub-elliptic equations

Abstract

Let X={X1,…,Xm} be a system of C∞ vector fields in Rn satisfying Hörmanderʼs finite rank condition and let Ω be a non-tangentially accessible domain with respect to the Carnot–Carathéodory distance d induced by X. We study the boundary behavior of non-negative solutions to the equationLu=∑i,j=1mXi⁎(aijXju)=∑i,j=1mXi⁎(x)(aij(x)Xj(x)u(x))=0 where Xi⁎ is the formal adjoint of Xi and x∈Ω. Concerning A(x)={aij(x)} we assume that A(x) is real, symmetric and thatβ−1λ(x)|ξ|2⩽∑i,j=1maij(x)ξiξj⩽βλ(x)|ξ|2for all x∈Rn,ξ∈Rm, for some constant β⩾1 and for some non-negative and real-valued function λ=λ(x). Concerning λ we assume that λ defines an A2-weight with respect to the metric introduced by the system of vector fields X={X1,…,Xm}. Our main results include a proof of the doubling property of the associated elliptic measure and the Hölder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. Our results generalize previous results of Fabes et al. (1982, 1983) [18–20] (m=n, {X1,…,Xm}={∂x1,…,∂xn}, λ is an A2-weight) and Capogna and Garofalo (1998) [6] (X={X1,…,Xm} satisfies Hörmanderʼs finite rank condition and λ(x)≡λ for some constant λ). One motivation for this study is the ambition to generalize, as far as possible, the results in Lewis and Nyström (2007, 2010, 2008) [35–38], Lewis et al. (2008) [34] concerning the boundary behavior of non-negative solutions to (Euclidean) quasi-linear equations of p-Laplace type, to non-negative solutions to certain sub-elliptic quasi-linear equations of p-Laplace type.