EXTREMAL FIRST DIRICHLET EIGENVALUE OF DOUBLYC ONNECTED PLANE DOMAINS AND DIHEDRAL SYMMETRY

摘要

We deal with the following eigenvalue optimization problem: Given a bounded domain D R2, place an obstacle B of fixed shape within D so as to maximize or minimize the fundamental eigenvalue λ1 of the Dirichlet Laplacian on DB. This means that we want to extremize the function ρ → λ1(D ρ(B)), where ρ runs over the set of rigid motions such that ρ(B) D. We answer this problem in the case where both D and B are invariant under the action of a dihedral group Dn, n ≥ 2, 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.