On the boundary branching set of the one-phase problem
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910547962757120 |
|---|---|
| author | Ferreri, Lorenzo Spolaor, Luca Velichkov, Bozhidar |
| author_facet | Ferreri, Lorenzo Spolaor, Luca Velichkov, Bozhidar |
| contents | We consider minimizers of the one-phase Bernoulli free boundary problem in domains with analytic fixed boundary. In any dimension $d$, we prove that the branching set at the boundary has Hausdorff dimension at most $d-2$. As a consequence, we also obtain an analogous estimate on the branching set for solutions to the two-phase problem under an analytic separation condition. Moreover, as a byproduct of our analysis we obtain strong boundary unique continuation results for quasilinear operators and thin-obstacle variational inequalities. The approach we use is based on the (almost-)monotonicity of a boundary Almgren-type frequency function, obtained via regularity estimates and a Calderón-Zygmund decomposition in the spirit of Almgren-De Lellis-Spadaro. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15230 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the boundary branching set of the one-phase problem Ferreri, Lorenzo Spolaor, Luca Velichkov, Bozhidar Analysis of PDEs 35R35, 35J57 We consider minimizers of the one-phase Bernoulli free boundary problem in domains with analytic fixed boundary. In any dimension $d$, we prove that the branching set at the boundary has Hausdorff dimension at most $d-2$. As a consequence, we also obtain an analogous estimate on the branching set for solutions to the two-phase problem under an analytic separation condition. Moreover, as a byproduct of our analysis we obtain strong boundary unique continuation results for quasilinear operators and thin-obstacle variational inequalities. The approach we use is based on the (almost-)monotonicity of a boundary Almgren-type frequency function, obtained via regularity estimates and a Calderón-Zygmund decomposition in the spirit of Almgren-De Lellis-Spadaro. |
| title | On the boundary branching set of the one-phase problem |
| topic | Analysis of PDEs 35R35, 35J57 |
| url | https://arxiv.org/abs/2407.15230 |