The $L^p$ Dirichlet Problem for the Stokes System on Lipschitz Domains

TL;DR

This paper investigates the solvability of the $L^p$ Dirichlet problem for the Stokes system on Lipschitz domains, establishing conditions under which solutions exist for certain ranges of p and simplifying these conditions.

Contribution

It introduces a simplified condition that guarantees the solvability of the $L^p$ Dirichlet problem for the Stokes system on Lipschitz domains, extending results to higher dimensions.

Findings

A reverse H"{o}lder condition with exponent p suffices for solvability.

A simpler condition implying the reverse H"{o}lder condition is provided.

Solvability is established for $d extgreater 4$ and specific p ranges.

Abstract

We study the $L^p$ Dirichlet problem for the Stokes system on Lipschitz domains. For any fixed $p>2$, we show that a reverse H\"{o}lder condition with exponent $p$ is sufficient for the solvability of the Dirichlet problem with boundary data in $L^p_N(\partial\Omega,\rn{d})$. Then we obtain a much simpler condition which implies the reverse H\"{o}lder condition. Finally, we establish the solvability ofthe $L^p$ Dirichlet problem for $d\geq 4$ and $2-\varepsilon<p<\frac{2(d-1)}{d-3}+\varepsilon$.