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A generic model for the shape optimization problems we consider in this paperis the optimization of the Dirichlet eigenvalues of the Laplace operator with avolume constraint. We deal with an obstacle placement problem which can beformulated as the following eigenvalue optimization problem: Fix two positivereal numbers $r_1$ and $A$. We consider a disk $B\subset \mathbb{R}^2$ havingradius $r_1$. We want to place an obstacle $P$ of area $A$ within $B$ so as tomaximize or minimize the fundamental Dirichlet eigenvalue $\lambda_1$ for theLaplacian on $B\setminus P$. That is, we want to study the behavior of thefunction $\rho \mapsto \lambda_1(B\setminus\rho(P))$, where $\rho$ runs overthe 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-concentricobstacle placement problem. The extremal configurations correspond to the caseswhere an axis of symmetry of $P$ coincide with an axis of symmetry of $B$. Wealso characterize the maximizing and the minimizing configurations in our mainresult, viz., Theorem 4.1. Equation (6), Propositions 5.1 and 5.2 imply Theorem4.1. We give many different generalizations of our result. At the end, weprovide some numerical evidence to validate our main theorem for the case wherethe obstacle $P$ has $\mathbb{D}_4$ symmetry. For the $n$ odd case, we identifysome 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 thedifficulties faced in proving Proposition 5.2 for this case. We providenumerical evidence for $n=5$ and conjecture that Theorem 4.1 holds true for $n$odd too.

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