A parabolic free transmission problem: flat free boundaries are smooth
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| author | Kriventsov, Dennis Soria-Carro, María |
| author_facet | Kriventsov, Dennis Soria-Carro, María |
| contents | We study a two-phase parabolic free boundary problem motivated by the jump of conductivity in composite materials that undergo a phase transition. Each phase is governed by a heat equation with distinct thermal conductivity, and a transmission-type condition is imposed on the free interface. We establish strong regularity properties of the free boundary: first, we prove that flat free boundaries are $C^{1,α}$ by means of a linearization technique and compactness arguments. Then we use the Hodograph transform to achieve higher regularity. To this end, we prove a new Harnack-type inequality and develop the Schauder theory for parabolic linear transmission problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16805 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A parabolic free transmission problem: flat free boundaries are smooth Kriventsov, Dennis Soria-Carro, María Analysis of PDEs 35B65, 35R35 (Primary) 35K05 (Secondary) We study a two-phase parabolic free boundary problem motivated by the jump of conductivity in composite materials that undergo a phase transition. Each phase is governed by a heat equation with distinct thermal conductivity, and a transmission-type condition is imposed on the free interface. We establish strong regularity properties of the free boundary: first, we prove that flat free boundaries are $C^{1,α}$ by means of a linearization technique and compactness arguments. Then we use the Hodograph transform to achieve higher regularity. To this end, we prove a new Harnack-type inequality and develop the Schauder theory for parabolic linear transmission problems. |
| title | A parabolic free transmission problem: flat free boundaries are smooth |
| topic | Analysis of PDEs 35B65, 35R35 (Primary) 35K05 (Secondary) |
| url | https://arxiv.org/abs/2402.16805 |