Some identities of Green's function for the polyharmonic operator with the Navier boundary conditions and its applications

Abstract

We prove several integral identities for Green's function of the polyharmonic operator \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$(-\Delta )^p, p \in \mathbb {N}$\end{document}, under the Navier boundary conditions. As an application, we prove the nondegeneracy of the critical point of the Robin function associated to the Green function on some symmetric domains.