SIAM Journal on Applied Mathematics, Volume 80, Issue 4, Page 1607-1628, January 2020.
We consider the problem of minimizing the first nonzero eigenvalue of an elliptic operator with Neumann boundary conditions with respect to the distribution of two conducting materials with a prescribed area ratio in a given domain. In one dimension, we show monotone properties of the first nonzero eigenvalue with respect to various parameters and find the optimal distribution of two conducting materials on an interval under the assumption that the region that has lower conductivity is simply connected. On a rectangular domain in two dimensions, we show that the strip configuration of two conducting materials can be a local minimizer. For general domains, we propose a rearrangement algorithm to find the optimal distribution numerically. Many results on various domains are shown to demonstrate the efficiency and robustness of the algorithms. Topological changes of the optimal configurations are discussed on circles, ellipses, annuli, and L-shaped domains.

中文翻译：

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具有Neumann边界条件的两相导体的第一个非零特征值问题的最小化

SIAM应用数学杂志，第80卷，第4期，第1607-1628页，2020年1月。
对于给定域中具有规定面积比的两种导电材料的分布，我们考虑使具有Neumann边界条件的椭圆算子的第一个非零特征值最小化的问题。在一个维度上，我们展示了相对于各种参数的第一个非零特征值的单调性质，并假设简单地连接了具有较低电导率的区域，从而找到了一种间隔上两种导电材料的最佳分布。在二维的矩形区域上，我们显示了两种导电材料的带状构造可以是局部最小化。对于一般领域，我们提出了一种重排算法以数字方式找到最佳分布。各种领域的许多结果显示出证明了算法的效率和鲁棒性。