Optimization of the principal eigenvalue for elliptic operators

Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with Dirichlet boundary condition are considered. The principal eigen map such is introduced some basic properties this map, including continuity, concavity, differentiability respect to parameter in diffusibility matrix, established. For maximization problem, admissible control set convexified get existence an optimal relaxed solution. Whereas, relaxation problem under H-convergence H-relaxed solution certain interesting special cases. Some necessary optimality conditions presented both a couple illustrative examples as well.