Perturbation theory for the parabolic Regularity and Neumann problem

We show small and large Carleson perturbation results for the parabolic Regularity boundary value problem with boundary data in ${\dot{L}}_{1,1\mathrm{/}2}^p$ and small Carelson perturbation results for the Neumann problem with boundary data in $L^p$. The operator we consider is $L:={\mathrm{\partial }}_t-\mathrm{d}\mathrm{i}\mathrm{v}(A\mathrm{\nabla }\cdot )$ and the domains are parabolic cylinders $\mathrm{\Omega }=\mathcal{O}\times \mathbb{R}$, where $\mathcal{O}$ is a Lipschitz domain.