Boundary estimates for solutions to linear degenerate parabolic equations

Let Ω⊂RnΩ⊂Rn be a bounded NTA-domain and let ΩT=Ω×(0,T)ΩT=Ω×(0,T) for some T>0T>0. We study the boundary behaviour of non-negative solutions to the equationHu=∂tu−∂xi(aij(x,t)∂xju)=0,(x,t)∈ΩT. We assume that A(x,t)={aij(x,t)}A(x,t)={aij(x,t)} is measurable, real, symmetric and thatβ−1λ(x)|ξ|2≤∑i,j=1naij(x,t)ξiξj≤βλ(x)|ξ|2 for all (x,t)∈Rn+1,ξ∈Rn, for some constant β≥1β≥1 and for some non-negative and real-valued function λ=λ(x)λ=λ(x) belonging to the Muckenhoupt class A1+2/n(Rn)A1+2/n(Rn). Our main results include the doubling property of the associated parabolic 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, Kenig, Jerison, Serapioni, see [18], [19] and [20], to a parabolic setting.