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

TL;DR

This paper constructs traveling wave solutions to the forced mean curvature motion in arbitrary dimensions, linking their asymptotics to solutions of the eikonal equation and probability measures on spheres.

Contribution

It introduces a method to construct smooth concave solutions with prescribed asymptotics for the N-dimensional forced mean curvature motion.

Findings

Constructed traveling wave graphs in arbitrary dimensions.

Linked asymptotics to solutions of the eikonal equation.

Described the asymptotics in terms of probability measures on spheres.

Abstract

We construct travelling wave graphs of the form $z=-ct+\phi(x)$, $\phi: x \in \mathbb{R}^{N-1} \mapsto \phi(x)\in \mathbb{R}$, $N \geq 2$, solutions to the $N$-dimensional forced mean curvature motion $V_n=-c_0+\kappa$ ($c\geq c_0$) 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 $\mathbb{S}^{N-2}$.