On free boundary problems shaped by varying singularities
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918194624593920 |
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| author | Araújo, Damião Sobral, Aelson Teixeira, Eduardo V. Urbano, José Miguel |
| author_facet | Araújo, Damião Sobral, Aelson Teixeira, Eduardo V. Urbano, José Miguel |
| contents | We start the investigation of free boundary variational models featuring varying singularities. The theory depends strongly on the nature of the singular power $γ(x)$ and how it changes. Under a mild continuity assumption on $γ(x)$, we prove the optimal regularity of minimizers. Such estimates vary point-by-point, leading to a continuum of free boundary geometries. We also conduct an extensive analysis of the free boundary shaped by the singularities. Utilizing a new monotonicity formula, we show that if the singular power $γ(x)$ varies in a $W^{1,n^{+}}$ fashion, then the free boundary is locally a $C^{1,δ}$ surface, up to a negligible singular set of Hausdorff co-dimension at least $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08071 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On free boundary problems shaped by varying singularities Araújo, Damião Sobral, Aelson Teixeira, Eduardo V. Urbano, José Miguel Analysis of PDEs 35R35 (Primary), 35J75 (Secondary) We start the investigation of free boundary variational models featuring varying singularities. The theory depends strongly on the nature of the singular power $γ(x)$ and how it changes. Under a mild continuity assumption on $γ(x)$, we prove the optimal regularity of minimizers. Such estimates vary point-by-point, leading to a continuum of free boundary geometries. We also conduct an extensive analysis of the free boundary shaped by the singularities. Utilizing a new monotonicity formula, we show that if the singular power $γ(x)$ varies in a $W^{1,n^{+}}$ fashion, then the free boundary is locally a $C^{1,δ}$ surface, up to a negligible singular set of Hausdorff co-dimension at least $2$. |
| title | On free boundary problems shaped by varying singularities |
| topic | Analysis of PDEs 35R35 (Primary), 35J75 (Secondary) |
| url | https://arxiv.org/abs/2401.08071 |