Partial differential equations/Mathematical physics
On maximizing the fundamental frequency of the complement of an obstacle
Comptes Rendus. Mathématique, Volume 356 (2018) no. 4, pp. 406-411.

Let ΩRn 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 λ1(Ω(x+D)).

First, we prove an upper bound on λ1(Ω(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 Ω. In short, a direct corollary is that if

μΩ:=maxxλ1(Ω(x+D))(1)
is large enough in terms of λ1(Ω), then all maximizer sets x+D of μΩ are close to each maximum point x0 of ϕλ1.

Second, we discuss the distribution of ϕλ1(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 μΩ is sufficiently large with respect to λ1(Ω), then all maximizers x+D of μΩ contain all maximum points x0 of ϕλ1(Ω).

Soit ΩRn un domaine borné satisfaisant une condition de type Hayman asymétrique et soit D un domaine borné arbitraire, dénommé « obstacle ». Nous nous intéressons au comportement de la première valeur propre de Dirichlet λ1(Ω(x+D)).

Nous établissons, dans un premier temps, une borne supérieure pour cette valeur propre en termes de la distance de l'ensemble x+D à l'ensemble des points x0 où la fonction propre du premier état de base de Dirichlet ϕλ1>0 de Ω atteint son maximum. En bref, un corollaire immédiat est que, si

μΩ:=maxxλ1(Ω(x+D))
est suffisamment grand en fonction de λ1(Ω), alors tous les ensembles maximisant x+D de μΩ sont proches de chaque point x0ϕλ1 est maximum.

Ensuite, nous discutons la distribution de ϕλ1(Ω) et la possibilité d'inscrire des boules de longueur d'onde en un point donné de Ω.

Enfin, nous appliquons nos observations aux obstacles convexes D, et nous montrons que, si μΩ est suffisamment grand par rapport à λ1(Ω), alors tous les ensembles maximisant x+D de μΩ contiennent tous les points x0ϕλ1(Ω) est maximum.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.01.018
Georgiev, Bogdan 1; Mukherjee, Mayukh 2

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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Georgiev, Bogdan; Mukherjee, Mayukh. On maximizing the fundamental frequency of the complement of an obstacle. Comptes Rendus. Mathématique, Volume 356 (2018) no. 4, pp. 406-411. doi : 10.1016/j.crma.2018.01.018. http://www.numdam.org/articles/10.1016/j.crma.2018.01.018/

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