Regularity for convex viscosity solutions of $σ_2$ Equation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917552310976512 |
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| author | Chen, Ruosi Jian, Huaiyu Tu, Xushan Zhou, Xingchen |
| author_facet | Chen, Ruosi Jian, Huaiyu Tu, Xushan Zhou, Xingchen |
| contents | We prove interior $C^{2}$ regularity result for convex viscosity solutions of the quadratic Hessian equation $σ_2(D^2u) = f(x)$, under the assumption that $f\in C^{0,1}$ with $\inf f>0$. The result is almost sharp: if $f$ are merely continuous, there exist convex viscosity solutions that fail to be $C^{1,1}$. When $f\in C^α$ for some $α\in (0,1)$, the corresponding interior regularity remains open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30823 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Regularity for convex viscosity solutions of $σ_2$ Equation Chen, Ruosi Jian, Huaiyu Tu, Xushan Zhou, Xingchen Analysis of PDEs We prove interior $C^{2}$ regularity result for convex viscosity solutions of the quadratic Hessian equation $σ_2(D^2u) = f(x)$, under the assumption that $f\in C^{0,1}$ with $\inf f>0$. The result is almost sharp: if $f$ are merely continuous, there exist convex viscosity solutions that fail to be $C^{1,1}$. When $f\in C^α$ for some $α\in (0,1)$, the corresponding interior regularity remains open. |
| title | Regularity for convex viscosity solutions of $σ_2$ Equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.30823 |