On the Dirichlet problem for the degenerate $k$-Hessian equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914199395893248 |
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| author | Lyu, Yasheng |
| author_facet | Lyu, Yasheng |
| contents | This paper investigates the existence of a global $C^{1,1}$ solution to the Dirichlet problem for the $k$-Hessian equation with a nonnegative right-hand side $f$, focusing on the required conditions for $f$. The conditions $f^{1/(k-1)}\in C^{1,1}(\overline{Ω_{0}})$ and $f^{3/(2k-2)}\in C^{2,1}(\overline{Ω_{0}})$, together with $f\geq0$ in a domain $Ω_{0}\SupsetΩ$, are optimal, as demonstrated by classical counterexamples. For the Monge-Ampère equation ($k=n$), we establish the existence under the optimal condition $f^{3/(2n-2)}\in C^{2,1}(\overline{Ω_{0}})$ together with $f\geq0$ in $Ω_{0}$. For the general $k$-Hessian equation, we establish the existence under the condition $f\geq0$ in $Ω_{0}$ together with one of the following three conditions: \begin{align*} &(1)\quad f^{1/(k-1)}\in C^{1,1}(\overline{Ω_{0}}),\ \ \inf_ΩΔu\geq1,\ \ 2\leq k\leq n-1;\\ &(2)\quad f^{3/(2k-2)}\in C^{2,1}(\overline{Ω_{0}}),\ \ \inf_ΩΔu\geq1,\ \ 5\leq k\leq n-1;\\ &(3)\quad f^{3/(2k)}\in C^{2,1}(\overline{Ω_{0}}),\ \ 2\leq k\leq n-1. \end{align*} |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_09205 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Dirichlet problem for the degenerate $k$-Hessian equation Lyu, Yasheng Analysis of PDEs This paper investigates the existence of a global $C^{1,1}$ solution to the Dirichlet problem for the $k$-Hessian equation with a nonnegative right-hand side $f$, focusing on the required conditions for $f$. The conditions $f^{1/(k-1)}\in C^{1,1}(\overline{Ω_{0}})$ and $f^{3/(2k-2)}\in C^{2,1}(\overline{Ω_{0}})$, together with $f\geq0$ in a domain $Ω_{0}\SupsetΩ$, are optimal, as demonstrated by classical counterexamples. For the Monge-Ampère equation ($k=n$), we establish the existence under the optimal condition $f^{3/(2n-2)}\in C^{2,1}(\overline{Ω_{0}})$ together with $f\geq0$ in $Ω_{0}$. For the general $k$-Hessian equation, we establish the existence under the condition $f\geq0$ in $Ω_{0}$ together with one of the following three conditions: \begin{align*} &(1)\quad f^{1/(k-1)}\in C^{1,1}(\overline{Ω_{0}}),\ \ \inf_ΩΔu\geq1,\ \ 2\leq k\leq n-1;\\ &(2)\quad f^{3/(2k-2)}\in C^{2,1}(\overline{Ω_{0}}),\ \ \inf_ΩΔu\geq1,\ \ 5\leq k\leq n-1;\\ &(3)\quad f^{3/(2k)}\in C^{2,1}(\overline{Ω_{0}}),\ \ 2\leq k\leq n-1. \end{align*} |
| title | On the Dirichlet problem for the degenerate $k$-Hessian equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.09205 |