On maximizing the fundamental frequency of the complement of an obstacle |
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Authors: | Bogdan Georgiev Mayukh Mukherjee |
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Affiliation: | 1. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany;2. Mathematics Department, Technion – I.I.T., Haifa 32000, Israel |
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Abstract: | Let 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 .First, we prove an upper bound on in terms of the distance of the set to the set of maximum points of the first Dirichlet ground state of Ω. In short, a direct corollary is that if (1) is large enough in terms of , then all maximizer sets of are close to each maximum point of .Second, we discuss the distribution of 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 is sufficiently large with respect to , then all maximizers of contain all maximum points of . |
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