The Green function for the Stokes system with measurable coefficients

Abstract

We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$. We construct the Green function in $\Omega$ under the condition $(A1)$ that weak solutions of the system enjoy interior Hölder continuity. We also prove that $(A1)$ holds, for example, when the coefficients are $\mathrm{VMO}$. Moreover, we obtain the global pointwise estimate for the Green function under the additional assumption $(A2)$ that weak solutions of Dirichlet problems are locally bounded up to the boundary of the domain. By proving a priori $L^q$-estimates for Stokes systems with $\mathrm{BMO}$ coefficients on a Reifenberg domain, we verify that $(A2)$ is satisfied when the coefficients are $\mathrm{VMO}$ and $\Omega$ is a bounded $C^1$ domain.

arXiv comment35 pages; part of Section 2.2 is revised; accepted in Communications on Pure and Applied Analysis