Title: On maximizing the fundamental frequency of the complement of an obstacle,
Abstract: Let $\Omega \subset \mathbb{R}^n$ be a bounded domain satisfying a
Hayman-type asymmetry condition, and let $ D $ be an arbitrary bounded domain
referred to as "obstacle". We are interested in the behaviour of the first
Dirichlet eigenvalue $ \lambda_1(\Omega \setminus (x+D)) $. First, we prove an
upper bound on $ \lambda_1(\Omega \setminus (x+D)) $ in terms of the distance
of the set $ x+D $ to the set of maximum points $ x_0 $ of the first Dirichlet
ground state $ \phi_{\lambda_1} > 0 $ of $ \Omega $. In short, a direct
corollary is that if \begin{equation} \mu_\Omega := \max_{x}\lambda_1(\Omega
\setminus (x+D)) \end{equation} is large enough in terms of $ \lambda_1(\Omega)
$, then all maximizer sets $ x+D $ of $ \mu_\Omega $ are close to each maximum
point $ x_0 $ of $ \phi_{\lambda_1} $.
Second, we discuss the distribution of $ \phi_{\lambda_1(\Omega)} $ and the
possibility to inscribe wavelength balls at a given point in $ \Omega $.
Finally, we specify our observations to convex obstacles $ D $ and show that
if $ \mu_\Omega $ is sufficiently large with respect to $ \lambda_1(\Omega) $,
then all maximizers $ x+D $ of $ \mu_\Omega $ contain all maximum points $ x_0
$ of $ \phi_{\lambda_1(\Omega)} $.