Classification of solutions of an elliptic Hamilton-Jacobi equation
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909817923174400 |
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| author | Porretta, Alessio Souplet, Philippe |
| author_facet | Porretta, Alessio Souplet, Philippe |
| contents | We show that any classical solution of the diffusive Hamilton-Jacobi (DHJ) equation $-Δu= |\nabla u|^p$ in a half-space with zero boundary conditions for $1<p\le 2$ is necessarily one-dimensional. This improves the previously known result, which required an extra assumption of boundedness from above. Combined with the existing analogous result for $p>2$, our result completes the full classification picture of the Dirichlet problem for equation (DHJ) in a half-space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of solutions of an elliptic Hamilton-Jacobi equation Porretta, Alessio Souplet, Philippe Analysis of PDEs We show that any classical solution of the diffusive Hamilton-Jacobi (DHJ) equation $-Δu= |\nabla u|^p$ in a half-space with zero boundary conditions for $1<p\le 2$ is necessarily one-dimensional. This improves the previously known result, which required an extra assumption of boundedness from above. Combined with the existing analogous result for $p>2$, our result completes the full classification picture of the Dirichlet problem for equation (DHJ) in a half-space. |
| title | Classification of solutions of an elliptic Hamilton-Jacobi equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.00146 |