Some asymptotic properties of the polylaplacian operator

TY - JOUR

T1 - Some asymptotic properties of the polylaplacian operator

AU - Eveson, Simon

PY - 2012/1/15

Y1 - 2012/1/15

N2 - We consider the asymptotic behaviour as m¿8 of the polylaplacian m(-¿) on the unit ball in n dimensions with Dirichlet boundary conditions, and derive strikingly simple asymptotically correct formulae for the Green's 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.

AB - We consider the asymptotic behaviour as m¿8 of the polylaplacian m(-¿) on the unit ball in n dimensions with Dirichlet boundary conditions, and derive strikingly simple asymptotically correct formulae for the Green's 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.

KW - Polylaplacian;

KW - Green's functions;

KW - Integral operators

UR - http://www.scopus.com/inward/record.url?scp=80052139388&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2011.07.012

DO - 10.1016/j.jmaa.2011.07.012

M3 - Article

SN - 0022-247X

VL - 385

SP - 828

EP - 835

JO - Journal of mathematical analysis and applications

JF - Journal of mathematical analysis and applications

IS - 2

ER -