Cancellation properties and pointwise bounds for the Green's functions for the Laplace operator

Abstract

We derive a cancellation property satisfied by certain combinations of derivatives of the Green's functions for the Laplace operator corresponding to Dirichlet and Neumann boundary conditions. This cancellation property is expressed in terms of pointwise bounds independent of distances to the boundary and generalizes Newton's third law concerning equal and opposite forces. We give an application to fluid mechanics and we include a self-contained exposition of the construction of the Green's functions and the derivations of pointwise bounds for their general derivatives up to an order determined by the regularity of the domain.