On the Grad-Mercier equation and Semilinear Free Boundary Problems
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909582227406848 |
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| author | Caffarelli, Luis Farah, Antonio Restrepo, Daniel |
| author_facet | Caffarelli, Luis Farah, Antonio Restrepo, Daniel |
| contents | In this paper, we establish regularity and uniqueness results for Grad-Mercier type equations that arise in the context of plasma physics. We show that solutions of this problem naturally develop a dead core, which corresponds to the set where the solutions become identically equal to their maximum. We prove uniqueness, sharp regularity, and non-degeneracy bounds for solutions under suitable assumptions on the reaction term. Of independent interest, our methods allow us to prove that the free boundaries of a broad class of semilinear equations have locally finite $H^{n-1}$ measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12548 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Grad-Mercier equation and Semilinear Free Boundary Problems Caffarelli, Luis Farah, Antonio Restrepo, Daniel Analysis of PDEs In this paper, we establish regularity and uniqueness results for Grad-Mercier type equations that arise in the context of plasma physics. We show that solutions of this problem naturally develop a dead core, which corresponds to the set where the solutions become identically equal to their maximum. We prove uniqueness, sharp regularity, and non-degeneracy bounds for solutions under suitable assumptions on the reaction term. Of independent interest, our methods allow us to prove that the free boundaries of a broad class of semilinear equations have locally finite $H^{n-1}$ measure. |
| title | On the Grad-Mercier equation and Semilinear Free Boundary Problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.12548 |