On the regularity of solutions to the Hamilton-Jacobi equations for the N-body problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912501084454912 |
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| author | Berti, Diego Polimeni, Davide Terracini, Susanna |
| author_facet | Berti, Diego Polimeni, Davide Terracini, Susanna |
| contents | We prove that certain suitably renormalized value functions associated with the $d$-dimensional ($d\geq2$) $N$-body problem corresponding to different limiting shapes of expanding solutions, under the assumption that the center of mass is at the origin, are viscosity solutions of the associated Hamilton-Jacobi equation. We analyze their singularities, defined as the initial configurations for which the minimizer of the associated variational problem is not unique. Moreover, we estimate the size of the closure of the singular set by proving its $\mathcal{H}^{d(N-1)-1}$-rectifiability, and we provide an upper bound on the Hausdorff dimension of the set of regular conjugate points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19170 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the regularity of solutions to the Hamilton-Jacobi equations for the N-body problem Berti, Diego Polimeni, Davide Terracini, Susanna Dynamical Systems Analysis of PDEs 70F10, 70H20, 70G75, 49L25 We prove that certain suitably renormalized value functions associated with the $d$-dimensional ($d\geq2$) $N$-body problem corresponding to different limiting shapes of expanding solutions, under the assumption that the center of mass is at the origin, are viscosity solutions of the associated Hamilton-Jacobi equation. We analyze their singularities, defined as the initial configurations for which the minimizer of the associated variational problem is not unique. Moreover, we estimate the size of the closure of the singular set by proving its $\mathcal{H}^{d(N-1)-1}$-rectifiability, and we provide an upper bound on the Hausdorff dimension of the set of regular conjugate points. |
| title | On the regularity of solutions to the Hamilton-Jacobi equations for the N-body problem |
| topic | Dynamical Systems Analysis of PDEs 70F10, 70H20, 70G75, 49L25 |
| url | https://arxiv.org/abs/2507.19170 |