On Aviles-Giga limit states with $L^p$ entropy productions
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866914439255556096 |
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| author | Lamy, Xavier Lorent, Andrew Peng, Guanying |
| author_facet | Lamy, Xavier Lorent, Andrew Peng, Guanying |
| contents | The Aviles-Giga energy provides sequences of maps converging to weak solutions $m\colonΩ\subset\mathbb R^2\to\mathbb R^2$ of the eikonal equation \begin{align*} \mathrm{div}\, m=0\text{ in }\mathcal D'(Ω),\quad |m|=1\text{ a.e. in }Ω\,, \end{align*} whose entropy productions $\mathrm{div}\,Φ(m)$ are Radon measures in $Ω$, controlled by the energy. Here, the entropies are all $C^2$ vector fields $Φ\colon\mathbb S^1\to\mathbb R^2$ such that $\mathrm{div}\,Φ(m_*)=0$ for any smooth solution $m_*$. It is conjectured that the entropy production measures are concentrated on the one-dimensional jump set of $m$, as follows from the chain rule if $m$ has bounded variation. In particular, the entropy production measures should vanish if they coincide with $L^p$ functions: this is what we establish in this note if $p$ is not too small and under natural boundary conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01439 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Aviles-Giga limit states with $L^p$ entropy productions Lamy, Xavier Lorent, Andrew Peng, Guanying Analysis of PDEs 49J99 (Primary) 49S99, 35F20 (Secondary) The Aviles-Giga energy provides sequences of maps converging to weak solutions $m\colonΩ\subset\mathbb R^2\to\mathbb R^2$ of the eikonal equation \begin{align*} \mathrm{div}\, m=0\text{ in }\mathcal D'(Ω),\quad |m|=1\text{ a.e. in }Ω\,, \end{align*} whose entropy productions $\mathrm{div}\,Φ(m)$ are Radon measures in $Ω$, controlled by the energy. Here, the entropies are all $C^2$ vector fields $Φ\colon\mathbb S^1\to\mathbb R^2$ such that $\mathrm{div}\,Φ(m_*)=0$ for any smooth solution $m_*$. It is conjectured that the entropy production measures are concentrated on the one-dimensional jump set of $m$, as follows from the chain rule if $m$ has bounded variation. In particular, the entropy production measures should vanish if they coincide with $L^p$ functions: this is what we establish in this note if $p$ is not too small and under natural boundary conditions. |
| title | On Aviles-Giga limit states with $L^p$ entropy productions |
| topic | Analysis of PDEs 49J99 (Primary) 49S99, 35F20 (Secondary) |
| url | https://arxiv.org/abs/2604.01439 |