Second-order estimates for degenerate complex $k$-Hessian and Christoffel-Minkowski equations
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917301922562048 |
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| author | Lyu, Yasheng |
| author_facet | Lyu, Yasheng |
| contents | It is known that the complex $k$-Hessian equation admits almost $C^{1,1}$ regularity (i.e., $\supΔu<\infty$) and the Christoffel-Minkowski equation admits $C^{1,1}$ regularity under the sharp degenerate condition $f^{1/(k-1)}\in C^{1,1}$ for a nonnegative right-hand side $f$. Assuming instead the alternative sharp degenerate condition $f^{3/(2k-2)}\in C^{2,1}$, we prove almost $C^{1,1}$ regularity for the complex $k$-Hessian equation when $k\geq5$ and $C^{1,1}$ regularity for the Christoffel-Minkowski equation. The argument deeply exploits various concavity properties of the operators under the stronger regularity assumption on $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00561 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Second-order estimates for degenerate complex $k$-Hessian and Christoffel-Minkowski equations Lyu, Yasheng Analysis of PDEs It is known that the complex $k$-Hessian equation admits almost $C^{1,1}$ regularity (i.e., $\supΔu<\infty$) and the Christoffel-Minkowski equation admits $C^{1,1}$ regularity under the sharp degenerate condition $f^{1/(k-1)}\in C^{1,1}$ for a nonnegative right-hand side $f$. Assuming instead the alternative sharp degenerate condition $f^{3/(2k-2)}\in C^{2,1}$, we prove almost $C^{1,1}$ regularity for the complex $k$-Hessian equation when $k\geq5$ and $C^{1,1}$ regularity for the Christoffel-Minkowski equation. The argument deeply exploits various concavity properties of the operators under the stronger regularity assumption on $f$. |
| title | Second-order estimates for degenerate complex $k$-Hessian and Christoffel-Minkowski equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2603.00561 |