A mean field problem approach for the double curvature prescription problem
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866916430041055232 |
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| author | Battaglia, Luca López-Soriano, Rafael |
| author_facet | Battaglia, Luca López-Soriano, Rafael |
| contents | In this paper we establish a new mean field-type formulation to study the problem of prescribing Gaussian and geodesic curvatures on compact surfaces with boundary, which is equivalent to the following Liouville-type PDE with nonlinear Neumann conditions: $$\left\{\begin{array}{ll} -Δu+2K_g=2Ke^u&\text{in }Σ\\ \partial_νu+2h_g=2he^\frac u2&\text{on }\partialΣ. \end{array}\right.$$ We provide three different existence results in the cases of positive, zero and negative Euler characteristics by means of variational techniques. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07735 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A mean field problem approach for the double curvature prescription problem Battaglia, Luca López-Soriano, Rafael Analysis of PDEs Differential Geometry 35J20, 58J32 In this paper we establish a new mean field-type formulation to study the problem of prescribing Gaussian and geodesic curvatures on compact surfaces with boundary, which is equivalent to the following Liouville-type PDE with nonlinear Neumann conditions: $$\left\{\begin{array}{ll} -Δu+2K_g=2Ke^u&\text{in }Σ\\ \partial_νu+2h_g=2he^\frac u2&\text{on }\partialΣ. \end{array}\right.$$ We provide three different existence results in the cases of positive, zero and negative Euler characteristics by means of variational techniques. |
| title | A mean field problem approach for the double curvature prescription problem |
| topic | Analysis of PDEs Differential Geometry 35J20, 58J32 |
| url | https://arxiv.org/abs/2309.07735 |