Vol. 1, No. 4, 2019

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Geometric origin and some properties of the arctangential heat equation

Yann Brenier

Vol. 1 (2019), No. 4, 561–584
Abstract

We establish the geometric origin of the nonlinear heat equation with arctangential nonlinearity: tD = Δ(arctanD) 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.

Keywords
Nonlinear heat equations, minimal surface equations, mean curvature flow, optimal transport, nonlinear electromagnetism, image processing
Mathematical Subject Classification 2010
Primary: 35K55, 35L65, 53C44
Milestones
Received: 21 March 2018
Revised: 26 July 2018
Accepted: 16 August 2018
Published: 14 December 2018
Authors
Yann Brenier
Departement de mathematiques et applications
CNRS, DMA (UMR 8553)
École Normale Superieure
45 rue d’Ulm
75005 Paris
France