Geometric origin and few properties of the arctangential heat equation

Yann Brenier (1, 2)

1 CNRS - Centre National de la Recherche Scientifique
2 DMA - Département de Mathématiques et Applications - ENS Paris

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 à 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.

Mots clés: Nonlinear heat equations; minimal surface equations; mean curvature flow; optimal transport; nonlinear electromagnetism; image processing

Domaines: Equations aux dérivées partielles [math.AP]

Yann Brenier. Geometric origin and few properties of the arctangential heat equation. Tunisian Journal of Mathematics, 2019, 1 (4), pp.561-584. ⟨10.2140/tunis.2019.1.561⟩. ⟨hal-01740320⟩