Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions

when ε ≥ 0 is small. In particular, ∆2v + εv ≥ 0 in Ω, with v = ∆v = 0 on ∂Ω, implies v ≥ 0 for ε small. In numerical experiments ([14]) for one dimension a similar behaviour was observed under Dirichlet boundary conditions v = ∂ ∂nv = 0. In this paper we will derive a 3-G type theorem as in (1) but with G1,n replaced by the Green function Gm,n for the m-polyharmonic operator with Dirichlet boundary conditions and with Ω replaced by the unit ball B in Rn withn ≥ 1: { (−∆) u = f in B, Dmu = 0 on ∂B, (2)