Geometric origin and some properties of the arctangential heat equation

Abstract

We establish the geometric origin of the nonlinear heat equation with arctangential nonlinearity: ${\partial }_{t}D=\mathrm{\Delta }\left(arctanD\right)$ by deriving it, together and in duality 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 à la Otto and its relationship with the Born–Infeld theory of electromagnetism), we briefly discuss its possible use for image processing, once written in nonconservative form and properly discretized.