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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2504.05655 |
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| _version_ | 1866912408003411968 |
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| author | Fang, Xiang Wei, Juncheng Zheng, Youquan Zhou, Yifu |
| author_facet | Fang, Xiang Wei, Juncheng Zheng, Youquan Zhou, Yifu |
| contents | We consider a Dirichlet problem of the $H$-system \begin{equation*} \begin{cases} Δv = 2v_x\wedge v_y ~& \text{ in }\mathcal{D},\\ v=\varepsilon \tilde g ~& \text{ on }\partial{\mathcal{D}}, \end{cases} \end{equation*} where $\mathcal D\subset \mathbb{R}^2$ is the unit disk, $v:\mathcal D\to \mathbb{R}^3$, and $\tilde g:\partial \mathcal D\to \mathbb{R}^3$ is a given smooth map. As $\varepsilon\to 0^+$, we construct multi-bubble solutions concentrating at distinct points, taking around each point the profile of degree 2 $H$-bubble. This gives a partial answer to a conjecture due to Brezis-Coron and Chanillo-Malchiodi concerning the limiting configuration in the case of higher degrees. This seems to be the first construction in employing higher-degree harmonic maps as the primary configurations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05655 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multi-bubble solutions for the Dirichlet problem of the $H$-system with higher degree Fang, Xiang Wei, Juncheng Zheng, Youquan Zhou, Yifu Analysis of PDEs We consider a Dirichlet problem of the $H$-system \begin{equation*} \begin{cases} Δv = 2v_x\wedge v_y ~& \text{ in }\mathcal{D},\\ v=\varepsilon \tilde g ~& \text{ on }\partial{\mathcal{D}}, \end{cases} \end{equation*} where $\mathcal D\subset \mathbb{R}^2$ is the unit disk, $v:\mathcal D\to \mathbb{R}^3$, and $\tilde g:\partial \mathcal D\to \mathbb{R}^3$ is a given smooth map. As $\varepsilon\to 0^+$, we construct multi-bubble solutions concentrating at distinct points, taking around each point the profile of degree 2 $H$-bubble. This gives a partial answer to a conjecture due to Brezis-Coron and Chanillo-Malchiodi concerning the limiting configuration in the case of higher degrees. This seems to be the first construction in employing higher-degree harmonic maps as the primary configurations. |
| title | Multi-bubble solutions for the Dirichlet problem of the $H$-system with higher degree |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.05655 |