Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

[Ondes progressives pour le mouvement par courbure moyenne forcée en toute dimension d'espace]

Monneau, Régis ; Roquejoffre, Jean-Michel ; Roussier-Michon, Violaine

Annales scientifiques de l'École Normale Supérieure, Tome 46 (2013), p. 217-248 / Harvested from Numdam

Résumé

Nous construisons des ondes progressives sous la forme de graphes $z=-ct+\phi (x)$, $\phi : x \in \mathbb {R}^{N-1} \mapsto \phi (x)\in \mathbb {R}$, $N \ge 2$, solutions du mouvement par courbure moyenne forcée $V_n=-c_0+\kappa $ ($c\ge c_0$) en dimension $N$ d’espace et avec un comportement asymptotique prescrit. Pour toute solution de viscosité $\phi _{\infty }$, $1$-homogène en espace, de l’équation eikonale $|D\phi _{\infty }|=\sqrt{(c/c_0)^2-1}$, nous mettons en évidence une solution régulière et concave du mouvement par courbure moyenne forcée dont le comportement asymptotique est donné par $\phi _{\infty }$. Nous décrivons aussi $\phi _{\infty }$ en terme d’une mesure de probabilité sur la sphère $\mathbb{S}^{N-2}$.

Abstract

We construct travelling wave graphs of the form $z=-ct+\phi (x)$, $\phi : x \in \mathbb {R}^{N-1} \mapsto \phi (x)\in \mathbb {R}$, $N \ge 2$, solutions to the $N$-dimensional forced mean curvature motion $V_n=-c_0+\kappa $ ($c\ge c_0$) with prescribed asymptotics. For any $1$-homogeneous function $\phi _{\infty }$, viscosity solution to the eikonal equation $|D\phi _{\infty }|=\sqrt{(c/c_0)^2-1}$, we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by $\phi _{\infty }$. We also describe $\phi _{\infty }$ in terms of a probability measure on $\mathbb{S}^{N-2}$.

Détail

Publié le : 2013-01-01
DOI : https://doi.org/10.24033/asens.2188
Classification:  53C44,  35F21,  35C07,  35D40,  35K57
Mots clés: mouvement par courbure moyenne forcée, équation eikonale, équations de Hamilton-Jacobi, solutions de viscosité, équations de réaction diffusion, fronts progressifs

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