Optimization of the Principal Eigenvalue for Elliptic Operators

Hongwei Lou, Jiongmin Yong

Published 2018-09-29Version 1

Maximization and minimization problems of the principle eigenvalue for divergence form elliptic operators with Dirichlet boundary condition are considered. The principal eigen map of elliptic operator is introduced and the continuity as well as the differentiability of such a map, with respect to the parameter in the diffusibility matrix, is 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 used to get an optimal $H$-relaxed solution. Some necessary optimality conditions are presented for both problems and illustrative examples are presented as well.