A relation between the Dirichlet and the Regularity problem for Parabolic equations

Abstract: We study the relationship between the Dirichlet and Regularity problem for parabolic operators of the form $ L = \mbox{div}(A\nabla\cdot) - \partial_t $ on cylindrical domains $ \Omega = \mathcal O \times \mathbb R $, where the base $ \mathcal O \subset \mathbb R^{n} $ is a $1$-sided chord arc domain (and for one result Lipschitz) in the spatial variables. In the paper we answer the question when the solvability of the $L^p$ Regularity problem for $L$ (denoted by $ (R_L){p} $) can be deduced from the solvability of the $ L^{p'} $ Dirichlet problem for the adjoint operator $L^*$ (denoted $ (D_L^*){p'} $). We show that this holds if for at least of $q\in(1,\infty)$ the problem $ (R_L)_{q} $ is solvable. That is, we establish a duality/dichotomy result: Dirichlet solvability implies Regularity solvability in the dual $L^p$ range, or the Regularity problem is not solvable in any $L^p$. Results like these were only known in the elliptic settings (Kenig-Pipher (1993) and Shen (2006)) but are new for parabolic PDEs. Our result is one of the key components needed for the recent advancement of Dindo\v{s}, Li and Pipher in understanding solvability of the Regularity problem for operators whose coefficients satisfy certain natural Carleson condition (called also DKP-condition in the elliptic case).