The role of the curvature of a surface in the shape of the solutions to elliptic equations
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866908598630612992 |
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| author | Gladiali, Francesca Grossi, Massimo Provenzano, Luigi |
| author_facet | Gladiali, Francesca Grossi, Massimo Provenzano, Luigi |
| contents | We prove uniqueness and non-degeneracy of the critical point of positive, semi-stable solutions of $-Δu=f(u)$ with Dirichlet boundary conditions for a class of star-shaped domains of the sphere and of the hyperbolic plane satisfying a geometric condition. In the spherical case, this condition is weaker than convexity, while in the hyperbolic case it is weaker than horoconvexity. Finally, we construct examples showing that this geometric condition is indeed optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The role of the curvature of a surface in the shape of the solutions to elliptic equations Gladiali, Francesca Grossi, Massimo Provenzano, Luigi Analysis of PDEs We prove uniqueness and non-degeneracy of the critical point of positive, semi-stable solutions of $-Δu=f(u)$ with Dirichlet boundary conditions for a class of star-shaped domains of the sphere and of the hyperbolic plane satisfying a geometric condition. In the spherical case, this condition is weaker than convexity, while in the hyperbolic case it is weaker than horoconvexity. Finally, we construct examples showing that this geometric condition is indeed optimal. |
| title | The role of the curvature of a surface in the shape of the solutions to elliptic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.15098 |