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

Abstract

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

Figures (4)

• Figure 1: A domain perforated by periodic balls
• Figure 2: Definition of $\Omega_{ji}^\varepsilon~(j=1,2,3)$
• Figure 3: A domain perforated by holes
• Figure 4: Estimation for $\bigcup_{i~\mu(A_i^\varepsilon)>0}$ in \ref{['invgraph']}

Theorems & Definitions (48)

• Definition 1: capacity
• Definition 2
• Remark 1
• Proposition 1
• proof
• Theorem 1
• Proposition 2
• Theorem 2
• Lemma 1
• proof