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 ${\varphi }_{{\lambda }_1}>0$ of $\Omega$. In short, a direct corollary is that if \begin{equation} μ_Ω:= \max_{x}λ_1(Ω\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 ${\varphi }_{{\lambda }_1}$. Second, we discuss the distribution of ${\varphi }_{{\lambda }_1(\Omega )}$ 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 ${\varphi }_{{\lambda }_1(\Omega )}$.

6 pages, comments most welcome!