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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.01607 |
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| _version_ | 1866912410742292480 |
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| author | De Silva, Daniela Jeon, Seongmin Shahgholian, Henrik |
| author_facet | De Silva, Daniela Jeon, Seongmin Shahgholian, Henrik |
| contents | In this paper, we study superlinear systems that give rise to free boundaries. Such systems appear for example from the minimization of the energy functional $$ \int_Ω\left(|\nabla\mathbf{u}|^2+\frac2p|\mathbf{u}|^p\right),\quad 0<p<1, $$ but solutions can be also understood in an ad hoc viscosity way. First, we prove the optimal regularity of minimizers using a variational approach. Then, we apply a linearization technique to establish the $C^{1,α}$-regularity of the ``flat'' part of the free boundary via a viscosity method. Finally, for minimizing free boundaries, we extend this result to analyticity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01607 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The free boundary for a superlinear system De Silva, Daniela Jeon, Seongmin Shahgholian, Henrik Analysis of PDEs 35R35, 35J60 In this paper, we study superlinear systems that give rise to free boundaries. Such systems appear for example from the minimization of the energy functional $$ \int_Ω\left(|\nabla\mathbf{u}|^2+\frac2p|\mathbf{u}|^p\right),\quad 0<p<1, $$ but solutions can be also understood in an ad hoc viscosity way. First, we prove the optimal regularity of minimizers using a variational approach. Then, we apply a linearization technique to establish the $C^{1,α}$-regularity of the ``flat'' part of the free boundary via a viscosity method. Finally, for minimizing free boundaries, we extend this result to analyticity. |
| title | The free boundary for a superlinear system |
| topic | Analysis of PDEs 35R35, 35J60 |
| url | https://arxiv.org/abs/2506.01607 |