将低体积分数边界电压扰动的表示公式扩展到对比度很大的电导率不均匀性
Extending Representation Formulae for Boundary Voltage Perturbations of Low Volume Fraction to Very Contrasted Conductivity Inhomogeneities
摘要：
在光滑有界区域$Ω$的边界上施加Dirichlet或Neumann边界条件，研究了电导率在一个小测度集上改变时电势引起的扰动.我们认为$\left(γ_{n}\right)_{n\in\mathbb{N}}$，一个序列的扰动电导率矩阵不同于一个光滑的$γ_{0}$背景电导率矩阵上的一个可可测集以及在域中，我们假设$\left(γ_{n}-γ_{0}\right)γ_{n}^{-1}\left(γ_{n}-γ_{0}\right)\to0$在$L^{1}(Ω)$。采用极限测度，我们证明了在\citep{capdeboscq-vogelius-03a}中引入的有界对比度的一般表示公式可以推广到无界的矩阵值电导率序列.
英文摘要：
Imposing either Dirichlet or Neumann boundary conditions on the boundary of a smooth bounded domain $Ω$, we study the perturbation incurred by the voltage potential when the conductivity is modified in a set of small measure. We consider $\left(γ_{n}\right)_{n\in\mathbb{N}}$, a sequence of perturbed conductivity matrices differing from a smooth $γ_{0}$ background conductivity matrix on a measurable set well within the domain, and we assume $\left(γ_{n}-γ_{0}\right)γ_{n}^{-1}\left(γ_{n}-γ_{0}\right)\to0$ in $L^{1}(Ω)$. Adapting the limit measure, we show that the general representation formula introduced for bounded contrasts in \citep{capdeboscq-vogelius-03a} can be extended to unbounded sequencesof matrix valued conductivities.
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