# On maximizing the fundamental frequency of the complement of an obstacle

arXiv: https://arxiv.org/abs/1706.02138 · HTML: https://mlsanity.com/paper/1706.02138/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2017.

## Built on by (most cited first, 1 of 1)

- [Heating up decision boundaries: isocapacitory saturation, adversarial scenarios and generalization bounds](https://mlsanity.com/paper/2101.06061/) (2021)

## Abstract (arXiv)

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) .