Cancellation Properties and Pointwise Bounds for the Green's Functions for the Laplace Operator

We derive a cancellation property satisfied by the derivatives of the Green’s functions for the Laplace operator corresponding to Dirichlet and Neumann boundary conditions on bounded sets in ${\mathbb{R}}^n$. The main result is derived in a broader, self-contained exposition which includes construction of and basic pointwise bounds for the Green’s functions and their derivatives. We also give an application of the cancellation property to a problem in fluid mechanics.