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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 thecapacity of holes when $\mu\in L^\infty(\Omega).$ In this paper, wecharacterize $\mu$ as the limit of the density of the capacity of holes alsofor many $\mu\in W^{-1,\infty}(\Omega).$ We apply the result for the inversehomogenization problem, i.e. we construct holes corresponding to the givenpotential $\mu\in L^d(\Omega)+L^\infty(\delta_S)$ where $\delta_S$ is a surfacemeasure.

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