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

TL;DR

This work proves a localization estimate for parabolic Neumann problems with time-dependent, bounded, measurable coefficients on Lipschitz cylinders. The authors develop a Poisson–Neumann framework and employ reinforced weak/energy solutions, parabolic tent spaces, and duality to relate boundary data and interior gradients. They then derive an endpoint Hardy-space bound and interpolate to obtain $L^q$ solvability of the parabolic Neumann problem for all $1<q<p$, under the assumption that the adjoint problem satisfies the corresponding Dirichlet solvability. The results extend the parabolic boundary-value theory to general coefficients and provide a robust interpolation mechanism for $L^p$ solvability, with potential applications to classes of operators with Carleson-type coefficient conditions.

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 interpolation of solvability of the $L^p$ Neumann problem.

Theorems & Definitions (30)

• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Definition 2.1
• Definition 2.4
• Definition 2.6
• Lemma 2.9
• Lemma 2.12: Duality between $\widetilde{\mathcal{C}}$ and $\widetilde{N}$
• Definition 2.20: \ref{['Dq']}
• Definition 2.21: \ref{['Np']}