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

Cheuk Yan Fung

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 $z_\varepsilon(x_1, \dots, x_n) = (1+x_1^2)(1+x_2^2)(\varepsilon^2 + η^2)^{α/2},$ where $η= \sqrt{x_3^2 + \dots + x_n^2}$ and $α= 2 - \frac{2}{n}$.

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 estimates for the equations in dimension . We construct a sequence of solutions with second derivatives blowing up at the origin as , while the corresponding right-hand sides admit uniform estimates. Specifically, the counterexamples are given by where and .
Paper Structure (4 sections, 8 theorems, 48 equations)

This paper contains 4 sections, 8 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Reduction of the Determinant
  3. Uniform $C^2$ Regularity of the Determinant
  4. Admissibility of the Example

Key Result

Theorem 1.1

Let $n\geq 3$. The interior $C^2$ estimate fails for the equations (duduequation). Specifically, there exist constants $\rho>0$ and $\varepsilon_0 > 0$ such that the sequence of smooth functions $\{z_\varepsilon\}$ defined on $B_\rho(0) \subset \mathbb{R}^n$ satisfies the following for all $\varepsi Specifically, the functions are given by where $\eta = \sqrt{x_3^2 + \dots + x_n^2}$ and $\alpha =

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1: Schur's formula
  • Lemma 2.2: Matrix determinant lemma
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • ...and 2 more