Uniform estimates for Stokes equations in a domain with a small hole and applications in homogenization problems
Yong Lu · arXiv (Cornell University) · 2015
We consider the Dirichlet problem of the Stokes equations in a domain with a shrinking hole in $\mathbb{R}^d, \ d\geq 2$. A typical observation is that, the Lipschitz norm of the domain goes to infinity as the size of the hole goes to zero. Thus, if $p eq 2$, the classical results indicate that the $W^{1,p}$ estimate of the solution may go to infinity as the size of the hole tends to zero. In this paper, we give a complete description for the uniform $W^{1,p}$ estimates of the solution for all $1