Title: 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,
Abstract: 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.