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

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

2011-07-05

arxiv.org/abs/1107.0896v1

Mathematics

Analysis of PDEs

36 pages, 6 figures

Scientific paper

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 $\mathbb{S}^{N-2}$.

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