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

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

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 2, page 217-248
  • ISSN: 0012-9593

Abstract

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We construct travelling wave graphs of the form z = - c t + φ ( x ) , φ : x ∈ ℝ N - 1 ↦ φ ( x ) ∈ ℝ , N ≥ 2 , solutions to the N -dimensional forced mean curvature motion V n = - c 0 + κ ( c ≥ c 0 ) with prescribed asymptotics. For any 1 -homogeneous function φ ∞ , viscosity solution to the eikonal equation | D φ ∞ | = ( c / c 0 ) 2 - 1 , we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by  φ ∞ . We also describe φ ∞ in terms of a probability measure on  § N - 2 .

How to cite

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Monneau, Régis, Roquejoffre, Jean-Michel, and Roussier-Michon, Violaine. "Travelling graphs for the forced mean curvature motion in an arbitrary space dimension." Annales scientifiques de l'École Normale Supérieure 46.2 (2013): 217-248. <http://eudml.org/doc/272133>.

@article{Monneau2013,
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 $§^\{N-2\}$.},
author = {Monneau, Régis, Roquejoffre, Jean-Michel, Roussier-Michon, Violaine},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {forced mean curvature movement; eikonal equation; Hamilton-Jacobi equations; viscosity solution; reaction diffusion equations; travelling fronts},
language = {eng},
number = {2},
pages = {217-248},
publisher = {Société mathématique de France},
title = {Travelling graphs for the forced mean curvature motion in an arbitrary space dimension},
url = {http://eudml.org/doc/272133},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Monneau, Régis
AU - Roquejoffre, Jean-Michel
AU - Roussier-Michon, Violaine
TI - Travelling graphs for the forced mean curvature motion in an arbitrary space dimension
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 2
SP - 217
EP - 248
AB - 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 $§^{N-2}$.
LA - eng
KW - forced mean curvature movement; eikonal equation; Hamilton-Jacobi equations; viscosity solution; reaction diffusion equations; travelling fronts
UR - http://eudml.org/doc/272133
ER -

References

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