Some asymptotic properties of the polylaplacian operator

Abstract

We consider the asymptotic behaviour as m → ∞ of the polylaplacian ( − Δ ) m on the unit ball in n dimensions with Dirichlet boundary conditions, and derive strikingly simple asymptotically correct formulae for the Greenʼs function, for the minimal eigenvalue, and for the associated eigenvector. The principal feature observed is that, for large m, the solution operators can be well approximated in all Schatten norms by operators of rank 1. We also show that a fixed lower-order perturbation term has no effect on the asymptotic behaviour of the solution operator.