Positivity for Perturbations of Polyharmonic Operators with Dirichlet BoundaryConditions in Two Dimensions

摘要

Maximum and comparison principles have proved to be a powerful tool in the theoryof second order elliptic differential equations. So, for a better understanding of higher order elliptic differential equations it is an obvious step to investigate the question whether similar results do exist there. That is, if omega is an appropriate bounded domain, does a function u that satisfies the higher order elliptic differential inequality Lu greater than or equal to 0, with zero Dirichlet boundary condition, have a fixed positive sign. Throughout this paper only two dimensional domains omega is contained in R(2) will be considered, if nothing different is stated. The authors have to leave the question open, whether the results of the present paper can be generalized to higher dimensions.