On maximizing the fundamental frequency of the complement of an obstacle

Georgiev, Bogdan; Mukherjee, Mayukh

doi:10.1016/j.crma.2018.01.018

Partial differential equations/Mathematical physics

On maximizing the fundamental frequency of the complement of an obstacle
[Sur la maximisation de la fréquence fondamentale du complément d'un obstacle]

Présenté par : the Editorial Board

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

Comptes Rendus. Mathématique, Tome 356 (2018) no. 4, pp. 406-411.

Résumé
Abstract

Soit $Ω \subset R^{n}$ 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 $λ_{1} (Ω ∖ (x + D))$ .

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 $x_{0}$ où la fonction propre du premier état de base de Dirichlet $ϕ_{λ_{1}} > 0$ de Ω atteint son maximum. En bref, un corollaire immédiat est que, si

μ_{Ω} : = \max_{x} λ_{1} (Ω ∖ (x + D))

est suffisamment grand en fonction de

λ_{1} (Ω)

, alors tous les ensembles maximisant

x + D

μ_{Ω}

sont proches de chaque point

x_{0}

où

ϕ_{λ_{1}}

est maximum.

Ensuite, nous discutons la distribution de $ϕ_{λ_{1} (Ω)}$ 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 $μ_{Ω}$ est suffisamment grand par rapport à $λ_{1} (Ω)$ , alors tous les ensembles maximisant $x + D$ de $μ_{Ω}$ contiennent tous les points $x_{0}$ où $ϕ_{λ_{1} (Ω)}$ est maximum.

Let $Ω \subset R^{n}$ 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 $λ_{1} (Ω ∖ (x + D))$ .

First, we prove an upper bound on $λ_{1} (Ω ∖ (x + D))$ 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

μ_{Ω} : = \max_{x} λ_{1} (Ω ∖ (x + D))

(1)

is large enough in terms of

λ_{1} (Ω)

, then all maximizer sets

x + D

μ_{Ω}

are close to each maximum point

x_{0}

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 $ϕ_{λ_{1} (Ω)}$ .

Reçu le : 2017-02-20
Accepté le : 2018-01-29
Publié le : 2018-03-01

DOI : 10.1016/j.crma.2018.01.018

Affiliations des auteurs :

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, Tome 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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