Extrapolation of solvability of the parabolic $L^p$ Neumann problem on bounded Lipschitz cylinders

TL;DR

This paper extends the extrapolation of solvability of the parabolic Neumann problem from unbounded Lipschitz graph domains to bounded Lipschitz cylinders, providing new methods for the bounded case.

Contribution

It introduces a novel approach to establish solvability extrapolation for the parabolic Neumann problem on bounded Lipschitz cylinders, filling a gap in previous research.

Findings

Established solvability extrapolation for bounded Lipschitz cylinders

Developed new techniques for bounded domain cases

Extended previous unbounded domain results to bounded settings

Abstract

A recent result of the first author with Li and Pipher has established the extrapolation of solvability of the $L^p$ parabolic Neumann problem on unbounded graph domains of the form $\Omega=\{(x',x_n):\,x_n>\varphi(x')\}\times\mathbb R$, where $\varphi:\mathbb R^{n-1}\to\mathbb R$ is a Lipschitz function. The result shows that under the assumptions that the $L^p$ parabolic Neumann problem for the equation $Lu=-\partial_t u+\mbox{div}(A\nabla u)=0$ in $\Omega$ and also the $L^{p'}$ parabolic Dirichlet problem for the adjoint equation $L^*u=\partial_t u+\mbox{div}(A\nabla u)=0$ in $\Omega$ are solvable, then also the $L^q$ parabolic Neumann problem for the equation $Lu=0$ in $\Omega$ is solvable for all $1<q<p$. However the mentioned paper does not answer the question whether the same claim is also true for domains of the form $\mathcal O\times\mathbb R$, where $\mathcal O$ is a bounded Lipschitz domain (in spatial variables) since this case does not follow from our argument for the unbounded case. Indeed, the bounded Lipschitz cylinder case requires a significantly different approach which we present in this article and establish an analogous result when $\mathcal O$ is a bounded Lipschitz domain.