Conformal metrics on the four-dimensional half sphere with symmetric $Q$ and $T$ curvatures
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913485445660672 |
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| author | Cruz-Blázquez, Sergio DelaTorre, Azahara |
| author_facet | Cruz-Blázquez, Sergio DelaTorre, Azahara |
| contents | In this paper, we address the problem of prescribing non-constant $Q$ and boundary $T$ curvatures on the upper hemisphere $\mathbb{S}^4_+\subset \mathbb{R}^5$, via a conformal change of the background metric. This is equivalent to solve a fourth-order non-linear elliptic boundary value problem with a third-order non-linear equation and homogeneous Neumann conditions at the boundary. The problem admits a Mean-field type variational formulation, similar to the one obtained by Cruz-Blázquez and Ruiz for a related problem in two dimensions, with the associated energy functional being bounded from below but, in general, not coercive. By imposing symmetry conditions, we are able to prove the existence of minimizers, especially when $Q,T\geq 0$. To the best of our knowledge, these are the first existence results obtained for this setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16311 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conformal metrics on the four-dimensional half sphere with symmetric $Q$ and $T$ curvatures Cruz-Blázquez, Sergio DelaTorre, Azahara Analysis of PDEs 35B53, 35B38, 35G30 In this paper, we address the problem of prescribing non-constant $Q$ and boundary $T$ curvatures on the upper hemisphere $\mathbb{S}^4_+\subset \mathbb{R}^5$, via a conformal change of the background metric. This is equivalent to solve a fourth-order non-linear elliptic boundary value problem with a third-order non-linear equation and homogeneous Neumann conditions at the boundary. The problem admits a Mean-field type variational formulation, similar to the one obtained by Cruz-Blázquez and Ruiz for a related problem in two dimensions, with the associated energy functional being bounded from below but, in general, not coercive. By imposing symmetry conditions, we are able to prove the existence of minimizers, especially when $Q,T\geq 0$. To the best of our knowledge, these are the first existence results obtained for this setting. |
| title | Conformal metrics on the four-dimensional half sphere with symmetric $Q$ and $T$ curvatures |
| topic | Analysis of PDEs 35B53, 35B38, 35G30 |
| url | https://arxiv.org/abs/2408.16311 |