The role of positive boundary data in generalized clamped plate equations

Abstract

Positivity phenomena in higher order elliptic Dirichlet problems as e.g. in the clamped plate equation are in general rather subtle. It depends on the domain and on the particular form of the operator whether there are comparison principles or not. Until now most papers concentrated on positivity with respect to the right-hand side, i.e. on positivity of the Green function itself. In the present paper we focus on the role of the boundary data, i.e. on positivity of certain Poisson kernels. While it is expected that the Poisson kernel of highest order behaves similarly as the Green function, it may be surprising that for Dirichlet problems of arbitrary order and in any dimension there is also a positivity result with respect to a second Poisson kernel. Furthermore a perturbation theory for this result is developed.