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.

考虑了具有Dirichlet边界条件的二阶椭圆算子散度主特征值的最大化和最小化问题。介绍了这种椭圆算子的主本征图，并建立了该图的一些基本性质，包括关于扩散矩阵中参数的连续性、凹度和可微性。对于最大化问题，将容许控制集凸化以获得最优凸化松弛解的存在性。而对于最小化问题，引入H收敛下问题的松弛以获得最优H-针对某些有趣的特殊情况的轻松解决方案。为这两个问题提供了一些必要的优化条件，并提供了几个说明性示例。