Optimal regularity for degenerate parabolic equations on a flat boundary
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917980525297664 |
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| author | Yun, Hyungsung |
| author_facet | Yun, Hyungsung |
| contents | We establish the optimal regularity of viscosity solutions to
\begin{equation*}
u_t - x_n^γΔu = f,
\end{equation*} which arises in the regularity theory for the porous medium equation. Specifically, we prove that under the zero Dirichlet boundary condition on $\{x_n=0\}$, the optimal regularity of $u$ up to the flat boundary $\{x_n=0\}$ is $C^{1,1-γ}$. Moreover, for the homogeneous equations, we establish that the optimal regularity of $u$ is $C^{2,1-γ}$ in the spatial variables, and that $x_n^{-γ}u$ is smooth in the variables $x'$ and $t$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04824 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal regularity for degenerate parabolic equations on a flat boundary Yun, Hyungsung Analysis of PDEs Primary 35B65, Secondary 35D40, 35K65 We establish the optimal regularity of viscosity solutions to \begin{equation*} u_t - x_n^γΔu = f, \end{equation*} which arises in the regularity theory for the porous medium equation. Specifically, we prove that under the zero Dirichlet boundary condition on $\{x_n=0\}$, the optimal regularity of $u$ up to the flat boundary $\{x_n=0\}$ is $C^{1,1-γ}$. Moreover, for the homogeneous equations, we establish that the optimal regularity of $u$ is $C^{2,1-γ}$ in the spatial variables, and that $x_n^{-γ}u$ is smooth in the variables $x'$ and $t$. |
| title | Optimal regularity for degenerate parabolic equations on a flat boundary |
| topic | Analysis of PDEs Primary 35B65, Secondary 35D40, 35K65 |
| url | https://arxiv.org/abs/2504.04824 |