On the Shape Sensitivity of the First Dirichlet Eigenvalue for Two-Phase Problems
On the Shape Sensitivity of the First Dirichlet Eigenvalue for Two-Phase Problems
复制标题
关于两相问题的第一狄利克雷特征值的形状敏感性
DOI：
10.1007/s00245-010-9111-z
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发表时间：
2011
影响因子：
1.8
通讯作者：
D. Kateb
中科院分区：
文献类型：
--
作者：
M. Dambrine;D. Kateb
本文考虑导热系数中的两相问题：填充有导电率为σ 1的材料的夹杂物层压在导电率为σ2的物体中。当夹杂物和主体的边界都可以修改时，我们解决了与Dirichlet边界条件相关的第一特征值的形状敏感性。我们证明了一个可微性结果，并给出了一阶和二阶导数的表达式。我们将结果应用于绝缘体的优化设计。我们证明了稳定的最优设计感谢二阶分析。我们还继续研究了由孔卡等人发起的球中两相导体的极值本征值问题。（Appl. Math. Optim. 60（2）：173-184，2009），并在Conca et al.（CANUM 2008，ESAIM Proc.，第27卷，第311-321页，EDP Sci.，莱乌里斯，2009年）。
We consider a two-phase problem in thermal conductivity: inclusions filled with a material of conductivityσ1are layered in a body of conductivityσ2. We address the shape sensitivity of the first eigenvalue associated with Dirichlet boundary conditions when both the boundaries of the inclusions and the body can be modified. We prove a differentiability result and provide the expressions of the first and second order derivatives. We apply the results to the optimal design of an insulated body. We prove the stability of the optimal design thanks to a second order analysis. We also continue the study of an extremal eigenvalue problem for a two-phase conductor in a ball initiated by Conca et al. (Appl. Math. Optim. 60(2):173–184, 2009) and pursued in Conca et al. (CANUM 2008, ESAIM Proc., vol. 27, pp. 311–321, EDP Sci., Les Ulis, 2009).