On maximizing the fundamental frequency of the complement of an obstacle

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 $\phi_{\lambda_1(\Omega)}$.