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

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 $\beta>\frac{1}{3}$.