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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2505.24410 |
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| _version_ | 1866911108688773120 |
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| author | Ji, Meng |
| author_facet | Ji, Meng |
| contents | In this paper, we study the regularity of the solution for the obstacle problem associated with the linearized Monge-Ampère operator: \begin{align*}
\begin{cases}
&u\geqφ\text{\quad in } Ω
&L_{ w}u=\tr( W D^{2}u)\leq 0 \text{\quad in } Ω
&L_{ w}u= 0 \text{\quad in } \{u>φ\}
&u=0 \text{\quad on } \partialΩ,
\end{cases} \end{align*} where $ W=(\det D^{2} w) D^{2} w^{-1}$ is the matrix of cofactor of $D^{2} w$, $w$ satisfies $λ\leq \det D^{2} w \leq Λ$ and $ w=0$ on $\partial Ω$, $φ$ is the obstacle with at least $C^{2}(\barΩ)$ smoothness, $Ω$ is an open bounded convex domain. We show the existence and uniqueness of a viscosity solution by using Perron's method and the comparison principle. Our primary result is to prove that the solution exhibits local $C^{1,γ}$ regularity for any $γ\in (0,1)$, provided that it is a strong solution in $W^{2,n}_{\text{loc}}(Ω)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24410 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $C^{1,α}$ regularity of the solution for the obstacle problem for the linearized Monge-Ampère operator Ji, Meng Analysis of PDEs In this paper, we study the regularity of the solution for the obstacle problem associated with the linearized Monge-Ampère operator: \begin{align*} \begin{cases} &u\geqφ\text{\quad in } Ω &L_{ w}u=\tr( W D^{2}u)\leq 0 \text{\quad in } Ω &L_{ w}u= 0 \text{\quad in } \{u>φ\} &u=0 \text{\quad on } \partialΩ, \end{cases} \end{align*} where $ W=(\det D^{2} w) D^{2} w^{-1}$ is the matrix of cofactor of $D^{2} w$, $w$ satisfies $λ\leq \det D^{2} w \leq Λ$ and $ w=0$ on $\partial Ω$, $φ$ is the obstacle with at least $C^{2}(\barΩ)$ smoothness, $Ω$ is an open bounded convex domain. We show the existence and uniqueness of a viscosity solution by using Perron's method and the comparison principle. Our primary result is to prove that the solution exhibits local $C^{1,γ}$ regularity for any $γ\in (0,1)$, provided that it is a strong solution in $W^{2,n}_{\text{loc}}(Ω)$. |
| title | $C^{1,α}$ regularity of the solution for the obstacle problem for the linearized Monge-Ampère operator |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.24410 |