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

Ahmad El Soufi (1) , Rola Kiwan (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

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 $\rho\mapsto \lambda_1(D\setminus \rho (B))$, where $\rho$ runs over the set of rigid motions such that $\rho (B)\subset D$. 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 $D$.

Mots clés: eigenvalues; Dirichlet Laplacian; Schrödinger operator; extremal eigenvalue; obstacle; dihedral group

Domaines: Théorie spectrale [math.SP]; Optimisation et contrôle [math.OC]

Ahmad El Soufi, Rola Kiwan. Extremal first Dirichlet eigenvalue of doubly connected plane domains and dihedral symmetry. SIAM Journal on Mathematical Analysis, 2007, 39 (4), pp.1112 --1119. ⟨hal-00145248⟩