Parabolic Regularity and Dirichlet boundary value problems

Abstract

We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form $Lu=\text{div}(A \nabla u)-u_t=0$ in Lip$(1,1/2)$ time-varying cylinders, where the coefficient matrix $A = \left[ a_{ij}(X,t)\right] $ is uniformly elliptic and bounded. We show that if the Regularity problem $(R)_p$ for the equation $Lu=0$ is solvable for some $1<p<\infty$ 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. In the parabolic settings in the special case of the heat equation in slightly smoother domains this has been established by Hofmann and Lewis and Nystr\"om 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.