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

Technical Summary: A Singular Profile for the Relativistic Heat Cost and the Special Lagrangian Curvature Equation

Problem Statement
This paper investigates the interior regularity of generalized solutions to the Monge–Ampère type equation governing optimal transportation under a relativistic cost function. The cost is defined as $c(x, y) = \sqrt{a^2 - |x-y|^2}$ for $|x-y| < a$ and $-\infty$ otherwise, where $a > 0$ represents a finite speed of transport. The associated equation, derived from the prescribed Jacobian of the $c$-normal map, is non-uniformly elliptic near infinite gradients.

The central problem is to determine the sharp regularity of solutions when the right-hand side of the equation is smooth (specifically constant). While classical optimal transport theory (under the Ma–Trudinger–Wang (MTW) condition) often yields $C^\infty$ regularity for smooth data, the relativistic cost violates the MTW condition. The paper seeks to construct explicit singular solutions to demonstrate that, despite smooth data, solutions may fail to be $C^{1,\beta}$ for certain $\beta$, and to identify the precise threshold of regularity.

Methodology
The author employs a constructive approach based on radial symmetry and dynamical systems analysis:

1. Radial Reduction: The author seeks a solution of the form $u(x', t) = \sqrt{r(t)^2 - |x'|^2}$ on a ball in $\mathbb{R}^n$. Substituting this ansatz into the Monge–Ampère equation reduces the problem to a second-order ordinary differential equation (ODE) for the meridian profile $r(t)$.
2. Singularity Analysis: By imposing initial conditions $r(0) = a$ and $\dot{r}(0) = 0$, the author identifies a degeneracy where the radial factor $A(t)$ vanishes at the origin. This forces the second derivative $\ddot{r}$ to blow up.
3. Autonomous System Reformulation: To rigorously handle the degeneracy at $t=0$, the second-order ODE is transformed into a first-order autonomous system in terms of the phase variable $s = \dot{r}$. The right-hand side of this system is smooth at $s=0$.
4. Existence and Asymptotics: Using the Picard–Lindelöf theorem, a smooth solution to the autonomous system is established. The author then inverts the time variable to derive the asymptotic behavior of the gradient $s(t)$ near the origin, showing $|s(t)| \sim |t|^{1/(2n-1)}$.
5. Verification: The constructed profile is verified to be a generalized (Aleksandrov) solution by proving $c$-convexity (using a reverse triangle inequality for the relativistic cost) and confirming the Monge–Ampère measure condition via slicing arguments.
6. Application to Special Lagrangian Curvature: In dimension $n=2$, the author utilizes a link established by Qiu and Zhou between the relativistic transport equation and the special Lagrangian curvature equation. By perturbing the initial conditions of the autonomous system to avoid the singularity, a sequence of smooth solutions is generated that converges uniformly to the singular profile.

Key Contributions and Results

• Sharp Regularity Threshold for Relativistic Cost: The paper constructs an explicit generalized solution $u$ on a ball in $\mathbb{R}^n$ ($n \ge 2$) that belongs to the class $C^{1, \frac{1}{2n-1}}$ but fails to belong to $C^{1,\beta}$ for any $\beta > \frac{1}{2n-1}$. This establishes that the exponent $\frac{1}{2n-1}$ is sharp for the relativistic cost, even with constant (smooth) right-hand side data.
• Comparison with MTW Theory: The result highlights a dichotomy in regularity theory. While the exponent $\frac{1}{2n-1}$ appears as a sharp threshold for optimal transport with bounded (rough) densities under the MTW condition (Liu [9]), this paper shows the same exponent arises from smooth data when the cost geometry violates the MTW condition. This implies that for the relativistic cost, no amount of regularity in the source term can improve the solution's regularity beyond this critical exponent.
• Non-Existence of Pure Interior Estimates for Special Lagrangian Curvature: Applying the construction to the two-dimensional special Lagrangian curvature equation (with phase $\Theta \in (0, \pi/2)$), the author produces a sequence of smooth graphical solutions converging uniformly to a limit in $C^{1, 1/3}$ but not $C^{1,\beta}$ for $\beta > 1/3$.
• Failure of Hessian Control: The constructed limit demonstrates that for the special Lagrangian curvature equation, the interior Hessian cannot be controlled solely by the $C^0$ norm of the solution and the data. This contrasts with the classical Monge–Ampère equation in the plane, where such pure interior second-derivative estimates are known to hold.

Significance
The paper demonstrates that the geometry of the cost function is a primary determinant of solution regularity, capable of inducing singularities even in the presence of smooth data. Specifically, it proves that the relativistic heat cost admits no dimension-free interior modulus of continuity for the gradient, as the Hölder exponent $\frac{1}{2n-1}$ tends to zero as the dimension $n$ increases.

Furthermore, the application to the special Lagrangian curvature equation resolves the question of pure interior estimates in two dimensions. By exhibiting a singular limit of smooth solutions, the paper establishes that the equation does not admit a pure interior $C^{1,\beta}$ estimate for any $\beta > 1/3$. This result underscores the delicacy of interior regularity for non-convex or geometrically constrained curvature equations, distinguishing them from the more robust regularity theory of the classical Monge–Ampère equation.