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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2509.16692 |
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| _version_ | 1866914049173749760 |
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| author | Lamy, Xavier Marconi, Elio |
| author_facet | Lamy, Xavier Marconi, Elio |
| contents | Weak solutions $m\colonΩ\subset\mathbb{R}^2\to\mathbb{R}^2$ of the eikonal equation \begin{align*} |m|=1\text{ a.e. and }\mathrm{div}\: m =0\,, \end{align*} arise naturally as sharp interface limits of bounded energy configurations in various physically motivated models, including the Aviles-Giga energy. The distributions $μ_Φ=\mathrm{div}\,Φ(m)$, defined for a class of smooth vector fields $Φ$ called entropies, carry information about singularities and energy cost. If these entropy productions are Radon measures, a long-standing conjecture predicts that they must be concentrated on the 1-rectifiable jump set of $m$, as they do if $m$ has bounded variation (BV) thanks to the chain rule. We establish this concentration property, for a large class of entropies, under the Besov regularity assumption \begin{align*} m\in B^{1/p}_{p,\infty} \quad \Leftrightarrow \quad \sup_{h\in \mathbb R^2\setminus\lbrace 0\rbrace} \frac{\|m(\cdot +h)-m\|_{L^p }}{|h|^{1/p}} <\infty\,, \end{align*} for any $1\leq p<3$, thus going well beyond the BV setting ($p=1$) and leaving only the borderline case $p=3$ open. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16692 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rectifiability of entropy productions for weak solutions of the 2D eikonal equation with supercritical regularity Lamy, Xavier Marconi, Elio Analysis of PDEs 35L60 Weak solutions $m\colonΩ\subset\mathbb{R}^2\to\mathbb{R}^2$ of the eikonal equation \begin{align*} |m|=1\text{ a.e. and }\mathrm{div}\: m =0\,, \end{align*} arise naturally as sharp interface limits of bounded energy configurations in various physically motivated models, including the Aviles-Giga energy. The distributions $μ_Φ=\mathrm{div}\,Φ(m)$, defined for a class of smooth vector fields $Φ$ called entropies, carry information about singularities and energy cost. If these entropy productions are Radon measures, a long-standing conjecture predicts that they must be concentrated on the 1-rectifiable jump set of $m$, as they do if $m$ has bounded variation (BV) thanks to the chain rule. We establish this concentration property, for a large class of entropies, under the Besov regularity assumption \begin{align*} m\in B^{1/p}_{p,\infty} \quad \Leftrightarrow \quad \sup_{h\in \mathbb R^2\setminus\lbrace 0\rbrace} \frac{\|m(\cdot +h)-m\|_{L^p }}{|h|^{1/p}} <\infty\,, \end{align*} for any $1\leq p<3$, thus going well beyond the BV setting ($p=1$) and leaving only the borderline case $p=3$ open. |
| title | Rectifiability of entropy productions for weak solutions of the 2D eikonal equation with supercritical regularity |
| topic | Analysis of PDEs 35L60 |
| url | https://arxiv.org/abs/2509.16692 |