We derive local $C^2$ estimates for complete noncompact translating solitons of the Gauss curvature flow in ${ℝ}^3$ which are graphs over a convex domain $\Omega$. This is closely is related to deriving local $C^{1,1}$ estimates for the degenerate Monge–Ampère equation. As a result, given a weakly convex bounded domain $\Omega$, we establish the existence of a $C_{loc}^{1,1}$ translating soliton. In particular, when the boundary $\partial \Omega$ has line segments, we show the existence of flat sides of the translator from a local a priori nondegeneracy estimate near the free boundary.

中文翻译：

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平侧高斯曲率流的平移解

我们推导本地 $C^{\mathrm{2个}}$ 高斯曲率流的完全非紧平移孤立子的估计。 ${ℝ}^3$ 是凸域上的图 $\Omega$。这与派生本地关系密切$C^{\mathrm{1个}，\mathrm{1个}}$对退化的蒙格-安培方程的估计。结果，给定一个弱凸的有界域$\Omega$，我们确定一个 $C_{位置}^{\mathrm{1个}，\mathrm{1个}}$翻译孤子。特别是当边界$\partial \Omega$ 有线段，我们通过自由边界附近的局部先验非退化估计来显示转换器的平坦面。