How to place an obstacle having a dihedral symmetry centered at a given point inside a disk so as to optimize the fundamental Dirichlet eigenvalue

A generic model for the shape optimization problems we consider in this paper is the optimization of the Dirichlet eigenvalues of the Laplace operator with a volume constraint. We deal with an obstacle placement problem which can be formulated as the following eigenvalue optimization problem: Fix two positive real numbers $r_1$ and $A$. We consider a disk $B\subset \mathbb{R}^2$ having radius $r_1$. We want to place an obstacle $P$ of area $A$ within $B$ so as to maximize or minimize the fundamental Dirichlet eigenvalue $\lambda_1$ for the Laplacian on $B\setminus P$. That is, we want to study the behavior of the function $\rho \mapsto \lambda_1(B\setminus\rho(P))$, where $\rho$ runs over the set of all rigid motions of the plane fixing the center of mass for $P$ such that $\rho(P)\subset B$. In this paper, we consider a non-concentric obstacle placement problem. The extremal configurations correspond to the cases where an axis of symmetry of $P$ coincide with an axis of symmetry of $B$. We also characterize the maximizing and the minimizing configurations in our main result, viz., Theorem 4.1. Equation (6), Propositions 5.1 and 5.2 imply Theorem 4.1. We give many different generalizations of our result. At the end, we provide some numerical evidence to validate our main theorem for the case where the obstacle $P$ has $\mathbb{D}_4$ symmetry. For the $n$ odd case, we identify some of the extremal configuration for $\lambda_1$. We prove that equation (6) and Proposition 5.1 hold true for $n$ odd too. We highlight some of the difficulties faced in proving Proposition 5.2 for this case. We provide numerical evidence for $n=5$ and conjecture that Theorem 4.1 holds true for $n$ odd too.