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

TL;DR

This paper investigates the derivatives of Green's functions for the Laplace operator with boundary conditions, establishing cancellation properties, pointwise bounds, and applying these results to fluid mechanics problems.

Contribution

It introduces a new cancellation property for Green's function derivatives and provides comprehensive bounds, extending understanding of Laplace Green's functions in bounded domains.

Findings

Derived cancellation property for Green's function derivatives.

Established pointwise bounds for Green's functions and derivatives.

Applied the cancellation property to a fluid mechanics problem.

Abstract

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 $\R^n$. 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.