Uniform estimates for polyharmonic Green functions in domains with small holes

TL;DR

This paper establishes uniform pointwise estimates for the Green's functions of polyharmonic operators in domains with small holes, extending techniques beyond the Laplacian where maximum principles apply.

Contribution

It introduces a novel method to obtain uniform Green's function estimates for polyharmonic operators in perforated domains, where traditional maximum principles are not available.

Findings

Uniform control of Green's functions as holes shrink

Extension of estimates to general polyharmonic operators

Techniques applicable without comparison principles

Abstract

We prove a pointwise control for the Green's function of polyharmonic operators with holes: this control is uniform while holes shrink. For the usual Laplacian, such a control is given by the maximum principle; the techniques developed here applies to general polyharmonic operators for which there is no comparison principle.