2010/08/30 by M. Frentz, N. Garofalo, Frentz, M. +7
Mathematics · #31C05 #35C15 #65N99 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:31C05 #msc:35C15 #msc:65N99
paper · pdf · doi:10.48550/arxiv.1008.5082
arxiv created 2010/08/30 · arxiv updated 2010/08/31
In a cylinder ΩT=Ω× (0,T)⊂ \Rn+1+ we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form Hu =∑i,j=1maij(x,t) XiXju - \ptu = 0, (x,t)∈\Rn+1+, where X=\X1,...,Xm\ is a system of C^∞ vector fields in \Rn satisfying Hörmander's finite rank condition \eqreffrc, and Ω is a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance d induced by X. Concerning the matrix-valued function A=\aij\, we assume that it be real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries aij be Hölder 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 \refT:back); 2) the Hölder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem \refT:quotients); 3) the doubling property for the parabolic measure associated with the operator H (Theorem \refT: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 aij be only bounded and measurable, whereas we assume Hölder continuity with respect to the intrinsic parabolic distance.