从电边界测量得到的二维物体中两种材料体积分数的精确边界：平移法
Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: the translation method
摘要：
我们处理的问题估计的体积的夹杂物使用有限数量的边界测量电阻抗断层成像。我们推导出夹杂物的体积分数的上限和下限，或更一般的两相混合物，使用两个边界测量在两个维度。这些边界是最优的，因为它们是通过具有某些边界数据的某些配置来实现的。我们推导出使用的平移方法，使用经典的变分原理与零拉格朗日的界限。然后，我们得到必要的条件，达到的界限，并证明这些界限是通过包含内的字段是均匀的。当施加特殊的边界条件的界限减少到那些得到米尔顿和这些反过来又显示在这里减少到那些Capdeboscq-Vogelius在极限时的体积分数趋于零。本文的界限和米尔顿的界限适用于任意体积分数的包含。然后，我们进行了一些数值实验，以证明这些界限是多么好。
英文摘要：
We deal with the problem of estimating the volume of inclusions using a finite number of boundary measurements in electrical impedance tomography. We derive upper and lower bounds on the volume fractions of inclusions, or more generally two phase mixtures, using two boundary measurements in two dimensions. These bounds are optimal in the sense that they are attained by certain configurations with some boundary data. We derive the bounds using the translation method which uses classical variational principles with a null Lagrangian. We then obtain necessary conditions for the bounds to be attained and prove that these bounds are attained by inclusions inside which the field is uniform. When special boundary conditions are imposed the bounds reduce to those obtained by Milton and these in turn are shown here to reduce to those of Capdeboscq-Vogelius in the limit when the volume fraction tends to zero. The bounds of this paper, and those of Milton, work for inclusions of arbitrary volume fractions. We then perform some numerical experiments to demonstrate how good these bounds are.
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