Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains

In this paper we consider general second order, symmetric and strongly elliptic parabolic systems with real valued and constant coefficients in the setting of a class of time-varying, non-smooth infinite cylinders Omega = {(x(0), x, t) is an element of R x Rn-1 x R : x(0) > A(x, t)}. We prove solvability of Dirichlet, Neumann as well as regularity type problems with data in L-p and L-1,1/2(p) (the parabolic Sobolev space having tangential (spatial) gradients and half a time derivative in L-p) for p is an element of (2 - epsilon, 2 + epsilon) assuming that A(x, .) is uniformly Lipschitz with respect to the time variable and that parallel to D(1/2)(t)A parallel to(*) <= epsilon(0) < infinity for epsilon(0) small enough (parallel to D(1/2)(t)A parallel to(*) is the parabolic BMO-norm of a half-derivative in time). We also prove a general structural theorem (duality theorem between Dirichlet and regularity problems) stating that if the Dirichlet problem is solvable in LP with the relevant bound on the parabolic non-tangential maximal function then the regularity problem can be solved with data in L-1,1/2(q)(partial derivative ohm) with q(-1) + p(-1) = 1. As a technical tool, which also is of independent interest, we prove certain square function estimates for solutions to the system.