Boundary estimates for solutions to linear degenerate parabolic equations

Let Omega subset of R '' be a bounded NTA-domain and let Omega(T) = Omega x (0, T) for some T > 0. We study the boundary behaviour of non-negative solutions to the equation Hu = partial derivative(t)u - partial derivative(xi) (a(ij) (x, t)partial derivative x(j)u)= 0, (x, t) epsilon Omega(T.) We assume that A (x, t)= {a(ij) (x, t)} is measurable, real, symmetric and that beta(-1)lambda(x)vertical bar xi vertical bar(2) <= Sigma(n)(i,j=1) a(ij) (x, t)xi(i)xi(j) <= beta lambda(x)vertical bar xi vertical bar(2) for all (x, t) is an element of Rn+1 is an element of R-n, for some constant beta >= 1 and for some non-negative and real-valued function lambda = lambda(x) belonging to the Muckenhoupt class A(1+2/n)(R-n). Our main results include the doubling property of the associated parabolic measure and the Holder 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, Kenig, Jerison, Serapioni, see [18-20], to a parabolic setting. (C) 2015 Elsevier Inc. All rights reserved.