论文标题
拉格朗日平均曲率类型方程的最佳规律性
Optimal regularity for Lagrangian mean curvature type equations
论文作者
论文摘要
我们将拉格朗日平均曲率方程式的规律性分类，其中包括规定的拉格朗日平均曲率的潜在方程，以及拉格朗日平均曲率流动自我缩减器和扩展器，翻译孤子和旋转孤子的平均值曲率曲率。某些此类方程的第二个边界价值问题的凸解是由Brendle-Warren 2010，Huang 2015和Wang-Huang-Bao 2023构建的。我们首先表明凸粘度解决方案是常规的，只要Lagrangian角度或相位是$ C^2 $，并且在梯度可变性中均可进行。接下来，我们表明，对于仅Hölder连续阶段而言，如果凸溶液为$ c^{1，β} $，对于足够大的$β$，则是规律的。给出了奇异的解决方案，以表明每种条件都是最佳的，并且Hölder指数很清晰。在此过程中，我们将Bian和Guan的恒定等级定理推广到包括对Legendre变换的任意依赖。
We classify regularity for Lagrangian mean curvature type equations, which include the potential equation for prescribed Lagrangian mean curvature and those for Lagrangian mean curvature flow self-shrinkers and expanders, translating solitons, and rotating solitons. Convex solutions of the second boundary value problem for certain such equations were constructed by Brendle-Warren 2010, Huang 2015, and Wang-Huang-Bao 2023. We first show that convex viscosity solutions are regular provided the Lagrangian angle or phase is $C^2$ and convex in the gradient variable. We next show that for merely Hölder continuous phases, convex solutions are regular if they are $C^{1,β}$ for sufficiently large $β$. Singular solutions are given to show that each condition is optimal and that the Hölder exponent is sharp. Along the way, we generalize the constant rank theorem of Bian and Guan to include arbitrary dependence on the Legendre transform.
