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

TL;DR

This paper establishes a duality relationship between the Dirichlet and Regularity problems for parabolic equations on cylindrical domains, extending known elliptic results to the parabolic setting and aiding in understanding solvability under Carleson conditions.

Contribution

It proves that Regularity solvability can be deduced from Dirichlet solvability for parabolic operators, a novel result not previously known for parabolic PDEs.

Findings

Duality between Dirichlet and Regularity problems established

Regularity solvability implied by Dirichlet solvability in dual L^p range

Results extend elliptic duality to 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).