Perturbation theory for the parabolic Regularity and Neumann problem

TL;DR

This paper establishes perturbation results for parabolic boundary value problems, demonstrating stability of solutions under small changes in the operator within Lipschitz domains, with implications for regularity and Neumann problems.

Contribution

It provides new small and large Carleson perturbation results for parabolic boundary value problems with specific boundary data spaces, extending stability theory.

Findings

Perturbation stability for the parabolic Regularity problem.

Small Carleson perturbation results for the Neumann problem.

Applicability to Lipschitz domains and specific boundary data spaces.

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 $\Omega=\mathcal{O}\times\mathbb{R}$, where $\mathcal{O}$ is a Lipschitz domain.