Placement of an Obstacle for Optimizing the Fundamental Eigenvalue of Divergence Form Elliptic Operators

Abstract

In this chapter, we consider an obstacle placement problem inside a disk, represented by a shape optimization problem to find the maximum or the minimum fundamental eigenvalue of a general divergence form elliptic operator. The obstacle is invariant under the action of a dihedral group, which is usually the case with regular polygons and ellipses. This is a generalization of a previous work by the same authors concerning the fundamental Dirichlet eigenvalue optimization. We show that when the order of symmetry of the obstacle is even, the extremal configurations for the fundamental eigenvalue, with respect to rotations of the obstacle, correspond to the cases where an axis of symmetry of the obstacle coincides with an axis of symmetry of the disk. For the case of odd order of symmetry, we provide conjectures about the extremal configurations. Furthermore, for both even and odd symmetry cases, we characterize the global extremal configurations with respect to rotations and translations of the obstacle. Finally, we provide results of several numerical experiments for obstacles with different orders of symmetries and two different types of elliptic operators, which validates our theoretical findings.