On Counterexamples to Interior $C^2$ Estimates for Monge-Ampère Type Equations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908843169021952 |
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| author | Fung, Cheuk Yan |
| author_facet | Fung, Cheuk Yan |
| contents | 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 $α= 2 - \frac{2}{n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18009 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Counterexamples to Interior $C^2$ Estimates for Monge-Ampère Type Equations Fung, Cheuk Yan Analysis of PDEs 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 $α= 2 - \frac{2}{n}$. |
| title | On Counterexamples to Interior $C^2$ Estimates for Monge-Ampère Type Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.18009 |