On maximizing the fundamental frequency of the complement of an obstacle

Let $\mathrm{\Omega }?{\mathbb{R}}^n$ be a bounded domain satisfying a Hayman-type asymmetry condition, and let D be an arbitrary bounded domain referred to as an “obstacle”. We are interested in the behavior of the first Dirichlet eigenvalue ${\lambda }_1(\mathrm{\Omega }?(x+D))$.First, we prove an upper bound on ${\lambda }_1(\mathrm{\Omega }?(x+D))$ in terms of the distance of the set $x+D$ to the set of maximum points $x_0$ of the first Dirichlet ground state ${?}_{{\lambda }_1}>0$ of Ω. In short, a direct corollary is that if

(1)${\mu }_{\mathrm{\Omega }}:=\underset{x}{\max }?{\lambda }_1(\mathrm{\Omega }?(x+D))$

is large enough in terms of ${\lambda }_1(\mathrm{\Omega })$, then all maximizer sets $x+D$ of ${\mu }_{\mathrm{\Omega }}$ are close to each maximum point $x_0$ of ${?}_{{\lambda }_1}$.Second, we discuss the distribution of ${?}_{{\lambda }_1(\mathrm{\Omega })}$ and the possibility to inscribe wavelength balls at a given point in Ω.Finally, we specify our observations to convex obstacles D and show that if ${\mu }_{\mathrm{\Omega }}$ is sufficiently large with respect to ${\lambda }_1(\mathrm{\Omega })$, then all maximizers $x+D$ of ${\mu }_{\mathrm{\Omega }}$ contain all maximum points $x_0$ of ${?}_{{\lambda }_1(\mathrm{\Omega })}$.