# Extremal first Dirichlet eigenvalue of doubly connected plane domains and dihedral symmetry

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1 LMPT - Laboratoire de Mathématiques et Physique Théorique

Abstract : We deal with the following eigenvalue optimization problem: Given a bounded domain $D\subset \R^2$, how to place an obstacle $B$ of fixed shape within $D$ so as to maximize or minimize the fundamental eigenvalue $\lambda 1$ of the Dirichlet Laplacian on $D\setminus B$. This means that we want to extremize the function $ho\mapsto \lambda 1D\setminus ho B$, where $ho$ runs over the set of rigid motions such that $ho B\subset D$. We answer this problem in the case where both $D$ and $B$ are invariant under the action of a dihedral group $\mathbb{D} n$, $n\ge2$, and where the distance from the origin to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of $B$ coincide with those of $D$.

keyword : eigenvalues Dirichlet Laplacian Schrödinger operator extremal eigenvalue obstacle dihedral group

Autor: Ahmad El Soufi - Rola Kiwan -

Fuente: https://hal.archives-ouvertes.fr/

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