A relation between the Dirichlet and the Regularity problem for Parabolic equations
Martin Dindoš, Erika Nyström
TL;DR
The paper establishes a parabolic duality/dichotomy: solvability of the L^p Regularity problem $(R_L)_p$ is determined by solvability of the adjoint Dirichlet problem $(D_{L^*})_{p'}$ together with solvability of the Regularity problem at some intermediate exponent. It develops a parabolic framework with half-time differentiation, Hardy–Sobolev spaces, and a parabolic Calderón–Zygmund theory, including reverse Hölder inequalities and almost-λ arguments, to bootstrap regularity from endpoint to the full range. By constructing atomic Hardy–Sobolev spaces that respect parabolic scaling and proving extrapolation via real interpolation, the authors obtain solvability across the $L^p$ spectrum for operators on CAD/Lipschitz cylinders, even under Carleson-type coefficient conditions. These results generalize and complete the elliptic duality theory to the parabolic setting, enabling applications to broader classes of parabolic operators and informing the study of Regularity under natural Carleson conditions.
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 $ Ω= \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 Dindoš, 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).