On maximizing the fundamental frequency of the complement of an obstacle

Abstract

Let $Ω\subset \mathbb{R}^n$ 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 $ λ_1(Ω\setminus (x+D)) $. First, we prove an upper bound on $ λ_1(Ω\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 $ ϕ_{λ_1} > 0 $ of $ Ω$. 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 $ λ_1(Ω) $, then all maximizer sets $ x+D $ of $ μ_Ω$ are close to each maximum point $ x_0 $ of $ ϕ_{λ_1} $. Second, we discuss the distribution of $ ϕ_{λ_1(Ω)} $ 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 $ μ_Ω$ is sufficiently large with respect to $ λ_1(Ω) $, then all maximizers $ x+D $ of $ μ_Ω$ contain all maximum points $ x_0 $ of $ ϕ_{λ_1(Ω)} $.