Optimization of the Principal Eigenvalue for Elliptic Operators
Optimization of the Principal Eigenvalue for Elliptic Operators
复制标题
椭圆算子主特征值的优化
DOI：
10.1007/s00526-021-02011-8
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Yong Jiongmin
中科院分区：
文献类型：
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作者：
Lou Hongwei;Yong Jiongmin
考虑具有狄利克雷边界条件的二阶椭圆型算子的发散型主本征值的最大化和最小化问题。引入了这类椭圆算子的主本征映射，并建立了该映射的一些基本性质，包括连续性、可微性和关于扩散矩阵中参数的可微性.对于最大化问题，对容许控制集进行了凸化，得到了最优凸化松弛解的存在性。而对于极小化问题，在H-收敛条件下引入了问题的松弛，得到了某些特殊情况下的最优H-松弛解.一些必要的最优性条件，这两个问题，并提出了一对夫妇的说明性例子。
Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced and some basic properties of this map, including continuity, concavity, and differentiability with respect to the parameter in the diffusibility matrix, are established. For maximization problem, the admissible control set is convexified to get the existence of an optimal convexified relaxed solution. Whereas, for minimization problem, the relaxation of the problem under $H$-convergence is introduced to get an optimal $H$-relaxed solution for certain interesting special cases. Some necessary optimality conditions are presented for both problems and a couple of illustrative examples are presented as well..