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Optimization of the Principal Eigenvalue for Elliptic Operators
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 -convergence is introduced to get an optimal -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.Comment: 38 page