Optimization of the Principal Eigenvalue for Elliptic Operators

TL;DR

This paper investigates the optimization of the principal eigenvalue for elliptic operators, establishing properties of the eigenvalue map, and providing existence, relaxation, and optimality conditions for maximization and minimization problems.

Contribution

It introduces the principal eigen map for elliptic operators, analyzes its properties, and develops relaxed solutions and optimality conditions for eigenvalue optimization problems.

Findings

Eigenvalue map is continuous, concave, and differentiable.

Existence of optimal relaxed solutions via convexification and $H$-convergence.

Necessary optimality conditions derived for both maximization and minimization cases.

Abstract

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.