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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.13230 |
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| _version_ | 1866909741477789696 |
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| author | Meng, Fanchen |
| author_facet | Meng, Fanchen |
| contents | We study the vanishing viscosity method for the eikonal equation $|Du|=V$ in $B(0,1)$ with homogeneous Dirichlet boundary value condition. By assuming $V$ is radially symmetric and restricting attention to radially symmetric solutions, we construct explicit formulas for both the viscous solution $u^ε$ and the limiting solution $u$. We prove $u^ε\rightarrow u$ as $ε\rightarrow 0^+$ qualitatively and quantitatively derive an $ε|\log ε|$ type local convergence rate. Finally, we discuss the uniqueness of viscosity solutions for the eikonal equation and give some examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13230 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The vanishing viscosity process for an eikonal equation in the radially symmetric setting Meng, Fanchen Analysis of PDEs We study the vanishing viscosity method for the eikonal equation $|Du|=V$ in $B(0,1)$ with homogeneous Dirichlet boundary value condition. By assuming $V$ is radially symmetric and restricting attention to radially symmetric solutions, we construct explicit formulas for both the viscous solution $u^ε$ and the limiting solution $u$. We prove $u^ε\rightarrow u$ as $ε\rightarrow 0^+$ qualitatively and quantitatively derive an $ε|\log ε|$ type local convergence rate. Finally, we discuss the uniqueness of viscosity solutions for the eikonal equation and give some examples. |
| title | The vanishing viscosity process for an eikonal equation in the radially symmetric setting |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.13230 |