Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions (Q1359505)

Based on Boggio's formula the authors derive estimates for Green's functions of polyharmonic operators in balls \(B\). These estimates are delicate and require a subtle analysis. Different types of maximum principles are then derived. At first, boundary value problems of the type \[ ((-\triangle)^m+A)u=f \text{ in }B, \;D_mu=0 \text{ on } \partial B, \] where \(A\) is a perturbation of the polyharmonic operator, are considered. Conditions are given which insure that a positive \(f\) implies that the solution is also positive. The same question is discussed for systems of equations. The use of the Neumann series plays a crucial role.