Defective Laplacians and paradoxical phenomena in crowd motion modeling - Maury, Bertrand (Comptes Rendus. Mécanique)
In both continuous and discrete settings, Laplace operators appear quite commonly in the modeling of natural phenomena, in several context: diffusion, heat propagation, porous media, fluid flows through pipes, electricity.... In these contexts, the discrete Laplace operator enjoys all the properties of its continuous counterpart, in particular: self-adjointness, variational formulation, stochastic interpretation, mean value property, maximum principle, ...In a first part, we give a detailed description of the correspondence between these mathematical properties and modeling considerations, in contexts where the continuous and the discrete settings perfectly match. In a second part, we describe a pathological situation, in the context of granular crowd motion models. Accounting for the non-overlapping constraint between hard discs leads to a particular operator acting on a field of Lagrange multipliers, defined on the dual graph of the contact network. This operator is defective in a certain sense: although it is the microscopic counterpart of the macroscopic Laplace operator, this discrete operator indeed lacks some properties, in particular the maximum principle. We investigate here how this very defectivity may explain some paradoxical phenomena that are observed in crowd motions and granular materials, phenomena that are not reproduced by macroscopic models.