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

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

2007-05-09

arxiv.org/abs/0705.1262v1

SIAM Journal on Mathematical Analysis / SIAM Journal of Mathematical Analysis 39, 4 (2007) 1112 --1119

Mathematics

Spectral Theory

To appear in SIAM Journal on Mathematical Analysis

Scientific paper

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 $D$.

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