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

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 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

• Toán học
• Discrete bilaplacian
• Finite differences
• Discrete Campanato spaces
• Membrane model
• Entropic repulsion
• Gaussian field
• Efficient Algorithms and Architectures
• The Multiplication algorithms
• Conventional GNB Multiplication
• Gaussian Normal Bases
• The Field Multiplication
• A more efficient algorithm