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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.01811 |
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| _version_ | 1866913973138358272 |
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| author | Fu, Haotong Wang, Huaijie Wang, Wei |
| author_facet | Fu, Haotong Wang, Huaijie Wang, Wei |
| contents | For the Landau-de Gennes functional modeling nematic liquid crystals in dimension three, we prove that, if the energy is bounded by $C(\log\frac{1}{\varepsilon}+1)$, then the sequence of minimizers $\{\mathbf{Q}_{\varepsilon}\}_{\varepsilon\in (0,1)}$ is relatively compact in $W_{\operatorname{loc}}^{1,p}$ for every $1<p<2$. This extends the classical compactness theorem of Bourgain-Brézis-Mironescu [Publ. Math., IHÉS, 99:1-115, 2004] for complex Ginzburg-Landau minimizers to the $\mathbb R\mathbf P^2$-valued Landau-de Gennes setting. Moreover, We obtain local bounds on the integral of the bulk energy potential that are uniform in $ \varepsilon $, improving the estimate that follows directly from the assumption. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01811 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform estimates of Landau-de Gennes minimizers in the vanishing elasticity limit with line defects Fu, Haotong Wang, Huaijie Wang, Wei Analysis of PDEs Mathematical Physics For the Landau-de Gennes functional modeling nematic liquid crystals in dimension three, we prove that, if the energy is bounded by $C(\log\frac{1}{\varepsilon}+1)$, then the sequence of minimizers $\{\mathbf{Q}_{\varepsilon}\}_{\varepsilon\in (0,1)}$ is relatively compact in $W_{\operatorname{loc}}^{1,p}$ for every $1<p<2$. This extends the classical compactness theorem of Bourgain-Brézis-Mironescu [Publ. Math., IHÉS, 99:1-115, 2004] for complex Ginzburg-Landau minimizers to the $\mathbb R\mathbf P^2$-valued Landau-de Gennes setting. Moreover, We obtain local bounds on the integral of the bulk energy potential that are uniform in $ \varepsilon $, improving the estimate that follows directly from the assumption. |
| title | Uniform estimates of Landau-de Gennes minimizers in the vanishing elasticity limit with line defects |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2508.01811 |