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

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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 Caccioppoli (or reverse Poincare) inequality. | Vietnam Journal of Mathematics (2019) 47: 133–181 https://doi.org/10.1007/s10013-018-0325-8 Estimates for the Green’s Function of the Discrete Bilaplacian in Dimensions 2 and 3 1 · Florian Schweiger2 ¨ Stefan Muller Received: 5 December 2017 / Accepted: 25 May 2018 / Published online: 21 Novemb er 2018 © Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018 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 Caccioppoli (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. Keywords Discrete bilaplacian · Finite differences · Discrete Campanato spaces · Membrane model · Entropic repulsion · Gaussian field Mathematics Subject Classification (2010) 65N06 · 31B30 · 39A14 · 60K35 · 82B41 1 Introduction Let V = [−1, 1]n and VN = N V ∩Zn with n ∈ N+ and N ∈ N+ . Consider the Hamiltonian HN (ψ) = 12 x∈Zn | 1 ψx |2 , where 1 is the discrete Laplacian and ψ ∈ RVN is a function on VN , extended by 0 to all of Zn . The associated Gibbs measure 1 PN (dψ) = exp(−HN (ψ)) dψx δ0 (dψx ) ZN n x∈VN x∈Z \VN is then the distribution of a Gaussian random field on Zn with 0 boundary data, the so-called membrane model. Its covariance matrix is given by the Green’s function GN of the discrete Dedicated to the memory of Eberhard Zeidler who has been an inspiration in so many ways. Whoever had the good fortune to meet him will never forget him. Stefan M¨uller stefan.mueller@hcm.uni-bonn.de Florian Schweiger schweiger@iam.uni-bonn.de 1 Hausdorff Center for Mathematics, Universit¨at Bonn, Endenicher Allee 60, 53115 Bonn, .