Neumann Problems for the Stokes Equations in Convex Domains

TL;DR

This paper establishes improved boundary value problem estimates for the Stokes equations in convex domains, expanding the known range of $p$ values compared to Lipschitz domains, based on new $W^{2,2}$ estimates.

Contribution

It provides new $L^p$ and $W^{1,p}$ estimates for the Neumann problem in convex domains, extending the known results to larger $p$ ranges.

Findings

Enhanced $L^p$ estimates for convex domains

Broader $p$ ranges than Lipschitz domains

Key $W^{2,2}$ estimate for convex domains

Abstract

This paper studies the Neumann boundary value problems for the Stokes equations in a convex domain in $\mathbb{R}^d$. We obtain nontangential-maximal-function estimates in $L^p$ and $W^{1, p}$ estimates for $p$ in certain ranges depending on $d$. These ranges are larger than the known ranges for Lipschitz domains. The proof relies on a $W^{2, 2}$ estimate for the Stokes equations in convex domains.