A singular profile for the relativistic heat cost and the special Lagrangian curvature equation

Abstract

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 $C^{1,\beta }$ for every $\beta >\frac{1}{2n-1}$. 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 $\Theta \in (0,\pi /2)$ 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 $C^{1,\beta }$ estimate for any $\beta >\frac{1}{3}$.