Perturbation theory for the parabolic Regularity and Neumann problem
Martin Ulmer
TL;DR
This work develops a perturbation theory for parabolic boundary value problems with time-dependent coefficients on parabolic cylinders $\Omega=\mathcal{O}\times\mathbb{R}$, establishing both small and large Carleson perturbation results for the parabolic Regularity problem and a small Carleson perturbation result for the Neumann problem. Building on elliptic perturbation methods, the authors adapt Green’s function techniques, non-tangential maximal function estimates, and tent-space arguments to the time-dependent parabolic setting, overcoming the distinct scaling and decay in time. The Regularity theory transfers solvability from an unperturbed operator $L_0$ to a perturbed operator $L_1$ under a Carleson condition $\|\mu\|_{\mathcal{C}}$, with a small-norm case (S) and a finite-norm case (L); for Neumann problems an additional solvability assumption for the adjoint Dirichlet problem yields a parallel perturbation result. Extensions to rougher domain bases are discussed, including when $\omega_{L_1}^*$ lies in a BMO-type class, and connections to the parabolic DKP framework are highlighted as a broader context. The results fill a gap by providing perturbation theory for parabolic Regularity and Neumann problems, expanding the toolkit for solvability under coefficient perturbations in the parabolic regime.
Abstract
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 $Ω=\mathcal{O}\times\mathbb{R}$, where $\mathcal{O}$ is a Lipschitz domain.