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

Abstract

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

36 pages, 6 figures