Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions

Abstract

Higher order elliptic partial dierential equations with Dirichlet boundary conditions in general do not satisfy a maximum principle. Polyharmonic operators on balls are an exception. Here it is shown that in IR2 small perturbations of polyharmonic operators and of the domain preserve the maximum principle. Hence the Green function for the clamped plate equation on an ellipse with small eccentricity is positive.