特征值的多相形状最优化问题：定性研究和数值结果
A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results
摘要：
考虑多相形状最优化问题$$\min\Big\{\sum_{i=1}^hλ_1(Ω_i)+α|Ω_i|:\ Ω_i\ \hbox{open},\ Ω_i\subset D,\ Ω_i\capΩ_j=\emptyset\Big\},$$，其中$α>0$为给定常数，$ D\subset\Bbb{R}^2$为具有Lipschitz边界的有界开集.给出了最优集的定性性质和相应特征函数的正则性的一些新结果。我们还提供了最佳分区的数值结果。
英文摘要：
We consider the multiphase shape optimization problem $$\min\Big\{\sum_{i=1}^hλ_1(Ω_i)+α|Ω_i|:\ Ω_i\ \hbox{open},\ Ω_i\subset D,\ Ω_i\capΩ_j=\emptyset\Big\},$$ where $α>0$ is a given constant and $ D\subset\Bbb{R}^2$ is a bounded open set with Lipschitz boundary. We give some new results concerning the qualitative properties of the optimal sets and the regularity of the corresponding eigenfunctions. We also provide numerical results for the optimal partitions.
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