Geometric origin and some properties of the arctangential heat equation

Abstract

We establish the geometric origin ot the nonlinear heat equation with arct-angential nonlinearity: ∂ t D = ∆(arctan D) by deriving it, together and in du-ality with the mean curvature flow equation, from the minimal surface equation in Minkowski space-time, through a suitable quadratic change of time. After examining various properties of the arctangential heat equation (in particular through its optimal transport interpretation a la Otto and its relationship with the Born-Infeld theory of Electromagnetism), we shortly discuss its possible use for image processing, once written in non-conservative form and properly discretized.