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

Abstract

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