任意维空间中受迫中曲率运动的行波图
Travelling graphs for the forced mean curvature motion in an arbitrary space dimension
摘要：
我们构造了$z=-ct+ϕ(x)$，$ϕ: x \in \mathbb{R}^{N-1} \mapsto ϕ(x)\in \mathbb{R}$，$N \geq 2$，$N$ -维强迫中曲率运动$V_n=-c_0+κ$（$c\geq c_0$）的解的行波图.对于任一1-齐次函数$ϕ_{\infty}$，程函方程$|Dϕ_{\infty}|=\sqrt{(c/c_0)^2-1}$的粘性解，我们给出了受迫中曲率运动的光滑凹解，其渐近性由$ϕ_{\infty}$驱动.我们还描述了$ϕ_{\infty}$在$\mathbb{S}^{N-2}$上的概率测度。
英文摘要：
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 $\mathbb{S}^{N-2}$.
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