On maximizing the fundamental frequency of the complement of an obstacle

Let Omega subset ℝn 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 λ1(Omega setminus (x+D)) . First, we prove an upper bound on λ1(Omega setminus (x+D)) in terms of the distance of the set x+D to the set of maximum points x0 of the first Dirichlet ground state φλ1 > 0 of Omega . In short, a direct corollary is that if beginequation μ_Omega := maxxλ1(Omega setminus (x+D)) endequation is large enough in terms of λ1(Omega) , then all maximizer sets x+D of μ_Omega are close to each maximum point x0 of φλ1 . Second, we discuss the distribution of φλ1(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 μ_Omega is sufficiently large with respect to λ1(Omega) , then all maximizers x+D of μ_Omega contain all maximum points x0 of φλ1(Omega) .