Localization and interpolation of parabolic $L^p$ Neumann problems

TL;DR

This paper develops localization estimates for solutions to parabolic Neumann problems and extends solvability results to Hardy spaces, broadening the understanding of boundary value problems for parabolic equations.

Contribution

It introduces a localization estimate for parabolic Neumann problems and proves solvability in Hardy spaces, extending previous $L^p$ results to more general function spaces.

Findings

Established a localization estimate for solutions with zero Neumann data.

Proved solvability of the Neumann problem in atomic Hardy spaces.

Extended the extrapolation of $L^p$ solvability to broader function spaces.

Abstract

We show a localization estimate for local solutions to the parabolic equation $-\partial_t u+\mbox{div} (A\nabla u)=0$ with zero Neumann data, assuming that the $L^p$ Neumann problem and $L^{p'}$ Dirichlet problem for the adjoint operator are solvable in a Lipschitz cylinder for some $p\in(1,\infty)$. Using this result, we establish the solvability of the Neumann problem in the atomic Hardy space for parabolic operators with bounded, measurable, time-dependent coefficients, and hence obtain the extrapolation of solvability of the $L^p$ Neumann problem.