Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909082545291264 |
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| author | Zhu, Siyao Wu, Hongguang Cui, Xiaojun |
| author_facet | Zhu, Siyao Wu, Hongguang Cui, Xiaojun |
| contents | In this paper, we show that any globally hyperbolic space-time admits at least one globally defined distance-like function, which is a viscosity solution to the Lorentzian eikonal equation. According to whether the time orientation is changed, we divide the set of viscosity solutions into some subclasses. We show if the time orientation is consistent, then a viscosity solution has a variational representation locally. As a result, such a viscosity solution is locally semiconcave, as the one in the Riemannian case. Also, if the time orientation of a viscosity solution is non-consistent, we analyse its peculiar properties which make this kind of viscosity solutions are totally different from the ones where the Hamiltonians are convex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_15006 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times Zhu, Siyao Wu, Hongguang Cui, Xiaojun Analysis of PDEs Dynamical Systems In this paper, we show that any globally hyperbolic space-time admits at least one globally defined distance-like function, which is a viscosity solution to the Lorentzian eikonal equation. According to whether the time orientation is changed, we divide the set of viscosity solutions into some subclasses. We show if the time orientation is consistent, then a viscosity solution has a variational representation locally. As a result, such a viscosity solution is locally semiconcave, as the one in the Riemannian case. Also, if the time orientation of a viscosity solution is non-consistent, we analyse its peculiar properties which make this kind of viscosity solutions are totally different from the ones where the Hamiltonians are convex. |
| title | Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2309.15006 |