AI 中文总结
该研究分析了相对论成本对应的Monge–Ampère型方程广义解的正则性，构造出特定奇异正则性的解，并将其推广到二维特殊拉格朗日曲率方程，证明其不存在更高阶的内部正则估计。
AI 中文摘要
我们研究了$\boldsymbol{\text{R}}^n$上相对论成本$c(x,y)=\boldsymbol{\text{\textit{a}}}^2-|x-y|^2$的最优运输对应的Monge–Ampère型方程广义解的内部正则性。我们在球上构造了显式的单参数径向结构广义解族，并在其中找到一个属于$C^{1,\frac{1}{2n-1}}$但不属于更高Hölder类的解：它对任意$\beta>\frac{1}{2n-1}$都不属于$C^{1,\beta}$。该构造将方程简化为平面自治系统，其相位变量$s=\boldsymbol{\text{\textit{r}}}$在基变量中消失的阶为$(2n-1)$。作为应用，在二维情形下我们将该构造推广到特殊拉格朗日曲率方程：对每个相位$\boldsymbol{\text{\textit{Θ}}}\boldsymbol{\text{\textit{∈}}}(0,\boldsymbol{\text{\textit{π}}}/2)$，我们生成一列光滑图解，一致收敛到恰好属于$C^{1,1/3}$的极限。因此二维特殊拉格朗日曲率方程对任意$\beta>\frac{1}{3}$都不存在纯内部$C^{1,\beta}$估计。
英文摘要
We study the interior regularity of generalized solutions to the Monge--Ampère type equation governing optimal transportation for the relativistic cost $c(x,y)=\sqrt{a^2-|x-y|^2}$ on $\mathbb{R}^n$. We construct an explicit one-parameter family of radially structured generalized solutions on a ball and exhibit, among them, a solution that is of class $C^{1,\frac{1}{2n-1}}$ but of no better Hölder class: it fails to belong to $C^{1,β}$ for every $β>\frac{1}{2n-1}$. The construction reduces the equation to a planar autonomous system whose phase variable $s=\dot r$ vanishes to order $(2n-1)$ in the base variable. As an application, in dimension two we transfer the construction to the special Lagrangian curvature equation: for every phase $Θ\in(0,π/2)$ we produce a sequence of smooth graphical solutions converging uniformly to a limit of class exactly $C^{1,1/3}$. Consequently the two-dimensional special Lagrangian curvature equation admits no pure interior $C^{1,β}$ estimate for any $β>\frac{1}{3}$.
