We prove the solvability of the parabolic L p Dirichlet boundary value problem for 1 < p ≤ ∞ for a PDE of the form ut = div(A∇u) + B • ∇u on time-varying domains where the coefficients A = [a ij (X, t)] and B = [b i ] satisfy a certain natural small Carleson condition. This result brings the state of affairs in the parabolic setting up to the elliptic standard.Furthermore, we establish that if the coefficients of the operator A, B satisfy a vanishing Carleson condition and the time-varying domain is of VMO type then the parabolic L p Dirichlet boundary value problem is solvable for all 1 < p ≤ ∞. This result is related to results in papers by Maz'ya, Mitrea and Shaposhnikova, and Hofmann, Mitrea and Taylor where the fact that boundary of domain has normal in VMO or near VMO implies invertibility of certain boundary operators in L p for all 1 < p ≤ ∞ which then (using the method of layer potentials) implies solvability of the L p boundary value problem in the same range for certain elliptic PDEs.Our result does not use the method of layer potentials since the coefficients we consider are too rough to use this technique, but remarkably we recover L p solvability in the full range of p's as the two papers mentioned above.

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We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form Lu = div(A∇u)− ut = 0 in Lip(1, 1/2) time-varying cylinders, where the coefficient matrix A = [a ij (X, t)] is uniformly elliptic and bounded.We show that if the Regularity problem (R)p for the equation Lu = 0 is solvable for some 1 < p < ∞ then the Dirichlet problem (D * ) p ′ for the adjoint equation L * v = 0 is also solvable, where p ′ = p/(p − 1). This result is an analogue of the result established in the elliptic case by Kenig and Pipher [KP93]. In the parabolic settings in the special case of the heat equation in slightly smoother domains this has been established by Hofmann and Lewis [HL96] and Nyström [Nys06] for scalar parabolic systems. In comparison, our result is abstract with no assumption on the coefficients beyond the ellipticity condition and is valid in more general class of domains.