Inverse homogenization problem for the Drichlet problem for Poisson equation for $W^{-1,\mathrm{\infty }}$ potential

We consider Poisson problems $-\mathrm{\Delta }{u}^{\epsilon }=f$ on perforated domains, and characterize the limit of $u^{\epsilon }$ as the solution to $(-\mathrm{\Delta }+\mu )u=f$ on domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with some potential $\mu \in W^{-1,\mathrm{\infty }}(\mathrm{\Omega })\mathrm{.}$ It is known that $\mu$ is related to the capacity of holes when $\mu \in L^{\mathrm{\infty }}(\mathrm{\Omega })\mathrm{.}$ In this paper, we characterize $\mu$ as the limit of the density of the capacity of holes also for many $\mu \in W^{-1,\mathrm{\infty }}(\mathrm{\Omega })\mathrm{.}$ We apply the result for the inverse homogenization problem, i.e. we construct holes corresponding to the given potential $\mu \in L^d(\mathrm{\Omega })+L^{\mathrm{\infty }}({\delta }_S)$ where ${\delta }_S$ is a surface measure.