How to Place an Obstacle Having a Dihedral Symmetry Inside a Disk so as to Optimize the Fundamental Dirichlet Eigenvalue

Abstract

In this paper, we deal with an obstacle placement problem inside a disk that can be formulated as an optimization problem for the fundamental Dirichlet eigenvalue with respect to rotations of the obstacle about its center. In this setup, the center of the obstacle, which satisfies the property of dihedral symmetry, and the center of the disk are non-concentric with each other. Such a problem finds important applications in the design of liquid crystal devices, musical instruments and optimal accelerator cavities. We show that the extremal configurations correspond to the cases, where an axis of symmetry of the obstacle coincides with an axis of symmetry of the disk. We also characterize the local and global maximizing and minimizing configurations for the case, when the obstacle has a dihedral symmetry of even order. For the case of odd order symmetry, we have partial results. We highlight the difficulties faced in characterizing the optimal configurations completely. We state our conjectures about such configurations. Finally, various numerical experiments validate the theoretical results obtained as well as the stated conjectures.