Interior Hessian estimates for Hessian quotient equations in dimension three
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866914409631186944 |
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| author | Jiao, Heming Sui, Zhenan |
| author_facet | Jiao, Heming Sui, Zhenan |
| contents | In this paper, we establish the interior Hessian estimates for $2$-convex solutions to $\frac{σ_2}{σ_1} (D^2 u) = ψ(x,u)$ in dimension three. In higher dimensions ($n \geq 4$), we prove the interior Hessian estimates for semi-convex solutions. We provide a new method to prove the doubling inequality for smooth solutions in dimensions three and four. In higher dimensions ($n\geq 5$) the doubling inequality is proved under an additional dynamic semi-convexity condition which is the same to that in \cite{SY2025}. The method also applies to the equation $σ_2 (D^2 u) = ψ(x, u, \nabla u)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14064 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Interior Hessian estimates for Hessian quotient equations in dimension three Jiao, Heming Sui, Zhenan Analysis of PDEs In this paper, we establish the interior Hessian estimates for $2$-convex solutions to $\frac{σ_2}{σ_1} (D^2 u) = ψ(x,u)$ in dimension three. In higher dimensions ($n \geq 4$), we prove the interior Hessian estimates for semi-convex solutions. We provide a new method to prove the doubling inequality for smooth solutions in dimensions three and four. In higher dimensions ($n\geq 5$) the doubling inequality is proved under an additional dynamic semi-convexity condition which is the same to that in \cite{SY2025}. The method also applies to the equation $σ_2 (D^2 u) = ψ(x, u, \nabla u)$. |
| title | Interior Hessian estimates for Hessian quotient equations in dimension three |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.14064 |