On Counterexamples to Interior $C^2$ Estimates for Monge-Ampère Type Equations

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 $z_{\epsilon }$ with second derivatives blowing up at the origin as $\epsilon \to 0$, while the corresponding right-hand sides $f_{\epsilon }$ admit uniform $C^2$ estimates. Specifically, the counterexamples are given by $z_{\epsilon }(x_1,\dots ,x_n)=(1+x_1^2)(1+x_2^2)({\epsilon }^2+{\eta }^2{)}^{\alpha /2},$ where $\eta =\sqrt{x_3^2+\cdots +x_n^2}$ and $\alpha =2-\frac{2}{n}$.

13 pages