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.

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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