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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.20933 |
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| _version_ | 1866910921871327232 |
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| author | Lamy, Xavier Lorent, Andrew Peng, Guanying |
| author_facet | Lamy, Xavier Lorent, Andrew Peng, Guanying |
| contents | A weak solution of the two-dimensional eikonal equation amounts to a vector field $m\colonΩ\subset\mathbb R^2\to\mathbb R^2$ such that $|m|=1$ a.e. and $\mathrm{div}\,m=0$ in $\mathcal D'(Ω)$. It is known that, if $m$ has some low regularity, e.g., continuous or $W^{1/3,3}$, then $m$ is automatically more regular: locally Lipschitz outside a locally finite set. A long-standing conjecture by Aviles and Giga, if true, would imply the same regularizing effect under the Besov regularity assumption $m\in B^{1/3}_{p,\infty}$ for $p>3$. In this note we establish that regularizing effect in the borderline case $p=6$, above which the Besov regularity assumption implies continuity. If the domain is a disk and $m$ satisfies tangent boundary conditions, we also prove this for $p$ slightly below $6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20933 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Another regularizing property of the 2D eikonal equation Lamy, Xavier Lorent, Andrew Peng, Guanying Analysis of PDEs A weak solution of the two-dimensional eikonal equation amounts to a vector field $m\colonΩ\subset\mathbb R^2\to\mathbb R^2$ such that $|m|=1$ a.e. and $\mathrm{div}\,m=0$ in $\mathcal D'(Ω)$. It is known that, if $m$ has some low regularity, e.g., continuous or $W^{1/3,3}$, then $m$ is automatically more regular: locally Lipschitz outside a locally finite set. A long-standing conjecture by Aviles and Giga, if true, would imply the same regularizing effect under the Besov regularity assumption $m\in B^{1/3}_{p,\infty}$ for $p>3$. In this note we establish that regularizing effect in the borderline case $p=6$, above which the Besov regularity assumption implies continuity. If the domain is a disk and $m$ satisfies tangent boundary conditions, we also prove this for $p$ slightly below $6$. |
| title | Another regularizing property of the 2D eikonal equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.20933 |