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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.02121 |
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| _version_ | 1866917443640754176 |
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| author | Lamy, Xavier Lorent, Andrew Peng, Guanying |
| author_facet | Lamy, Xavier Lorent, Andrew Peng, Guanying |
| contents | We study differential inclusions $Du\in Π$ in an open set $Ω\subset\mathbb R^2$, where $Π\subset \mathbb R^{2\times 2}$ is a compact connected $C^2$ curve without rank-one connections, but non-elliptic: tangent lines to $Π$ may have rank-one connections, so that classical regularity and rigidity results do not apply. For a wide class of such curves $Π$, we show that $Du$ is locally Lipschitz outside a discrete set, and is rigidly characterized around each singularity. Moreover, in the partially elliptic case where at least one tangent line to $Π$ has no rank-one connections, or under some topological restrictions on the tangent bundle of $Π$, there are no singularities. This goes well beyond previously known particular cases related to Burgers' equation and to the Aviles-Giga functional. The key is the identification and appropriate use of a general underlying structure: an infinite family of conservation laws, called entropy productions in reference to the theory of scalar conservation laws. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02121 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On regularity and rigidity of $2\times 2$ differential inclusions into non-elliptic curves Lamy, Xavier Lorent, Andrew Peng, Guanying Analysis of PDEs We study differential inclusions $Du\in Π$ in an open set $Ω\subset\mathbb R^2$, where $Π\subset \mathbb R^{2\times 2}$ is a compact connected $C^2$ curve without rank-one connections, but non-elliptic: tangent lines to $Π$ may have rank-one connections, so that classical regularity and rigidity results do not apply. For a wide class of such curves $Π$, we show that $Du$ is locally Lipschitz outside a discrete set, and is rigidly characterized around each singularity. Moreover, in the partially elliptic case where at least one tangent line to $Π$ has no rank-one connections, or under some topological restrictions on the tangent bundle of $Π$, there are no singularities. This goes well beyond previously known particular cases related to Burgers' equation and to the Aviles-Giga functional. The key is the identification and appropriate use of a general underlying structure: an infinite family of conservation laws, called entropy productions in reference to the theory of scalar conservation laws. |
| title | On regularity and rigidity of $2\times 2$ differential inclusions into non-elliptic curves |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.02121 |