On Counterexamples to Interior $C^2$ Estimates for Monge-Amp\`ere Type Equations

TL;DR

This paper constructs counterexamples showing that second derivative estimates do not hold uniformly for certain Monge-Ampère type equations in higher dimensions, challenging previous assumptions of regularity.

Contribution

It modifies Pogorelov's method to produce solutions with unbounded second derivatives despite bounded right-hand sides, demonstrating the failure of a priori $C^2$ estimates.

Findings

Second derivatives blow up at the origin as epsilon approaches zero.

Right-hand sides remain uniformly bounded in $C^2$ norm.

Counterexamples exist in dimensions three and higher.

Abstract

We modify Pogorelov's classic construction to demonstrate the absence of a priori $C^2$ estimates for the equations $\det(D^2 u \pm Du \otimes Du) = f(x)$ in dimension $n \ge 3$. We construct a sequence of solutions $z_\varepsilon$ with second derivatives blowing up at the origin as $\varepsilon \rightarrow 0$, while the corresponding right-hand sides $f_\varepsilon$ admit uniform $C^2$ estimates. Specifically, the counterexamples are given by $z_\varepsilon(x_1, \dots, x_n) = (1+x_1^2)(1+x_2^2)(\varepsilon^2 + \eta^2)^{\alpha/2},$ where $\eta = \sqrt{x_3^2 + \dots + x_n^2}$ and $\alpha = 2 - \frac{2}{n}$.