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