Minimizing Harmonic Maps on the Unit Ball with Tangential Anchoring
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916574462476288 |
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| author | Bronsard, Lia Colinet, Andrew Stantejsky, Dominik |
| author_facet | Bronsard, Lia Colinet, Andrew Stantejsky, Dominik |
| contents | Since the seminal work of Schoen-Uhlenbeck, many authors have studied properties of harmonic maps satisfying Dirichlet boundary conditions. In this article, we instead investigate regularity and symmetry of $\mathbb{S}^2-$valued minimizing harmonic maps subject to a tangency constraint in the model case of the unit ball in $\mathbb{R}^{3}$. In particular, we obtain a monotonicity formula respecting tangentiality on a curved boundary in order to show optimal regularity up to the boundary. We introduce novel sufficient conditions under which the minimizer must exhibit symmetries. Under a symmetry assumption, we present a delineation of the singularities of minimizers, namely that a mimimizer has exactly two point singularities, located on the boundary at opposite points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11565 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimizing Harmonic Maps on the Unit Ball with Tangential Anchoring Bronsard, Lia Colinet, Andrew Stantejsky, Dominik Analysis of PDEs Classical Analysis and ODEs 49K20, 49Q20, 35B38, 35A16, 35B07, 49S05 Since the seminal work of Schoen-Uhlenbeck, many authors have studied properties of harmonic maps satisfying Dirichlet boundary conditions. In this article, we instead investigate regularity and symmetry of $\mathbb{S}^2-$valued minimizing harmonic maps subject to a tangency constraint in the model case of the unit ball in $\mathbb{R}^{3}$. In particular, we obtain a monotonicity formula respecting tangentiality on a curved boundary in order to show optimal regularity up to the boundary. We introduce novel sufficient conditions under which the minimizer must exhibit symmetries. Under a symmetry assumption, we present a delineation of the singularities of minimizers, namely that a mimimizer has exactly two point singularities, located on the boundary at opposite points. |
| title | Minimizing Harmonic Maps on the Unit Ball with Tangential Anchoring |
| topic | Analysis of PDEs Classical Analysis and ODEs 49K20, 49Q20, 35B38, 35A16, 35B07, 49S05 |
| url | https://arxiv.org/abs/2501.11565 |