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