Estimates for the Green's function of the discrete bilaplacian in dimensions 2 and 3

TL;DR

This paper establishes near-optimal estimates for the Green's function of the discrete bilaplacian in 2D and 3D, using novel discrete techniques, with applications to statistical physics models.

Contribution

It introduces a new discrete compactness argument and a discrete Cacciopoli inequality to transfer continuous estimates to the discrete setting.

Findings

Derived optimal estimates for the Green's function in squares and cubes.

Identified potential near-corner and near-edge deviations from optimality.

Provided tools applicable to the study of entropic repulsion in membrane models.

Abstract

We prove estimates for the Green's function of the discrete bilaplacian in squares and cubes in two and three dimensions which are optimal except possibly near the corners of the square and the edges and corners of the cube. The main idea is to transfer estimates for the continuous bilaplacian using a new discrete compactness argument and a discrete version of the Cacciopoli (or reverse Poincar\'e) inequality. One application that we have in mind is the study of entropic repulsion for the membrane model from statistical physics.