Schauder estimates for parabolic equations with degenerate or singular weights
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912001477836800 |
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| author | Audrito, Alessandro Fioravanti, Gabriele Vita, Stefano |
| author_facet | Audrito, Alessandro Fioravanti, Gabriele Vita, Stefano |
| contents | We establish some $C^{0,α}$ and $C^{1,α}$ regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane $Σ$ as a power $a > -1$ of the distance to $Σ$. The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar $C^{1,α}$ estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06038 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schauder estimates for parabolic equations with degenerate or singular weights Audrito, Alessandro Fioravanti, Gabriele Vita, Stefano Analysis of PDEs 35B65 We establish some $C^{0,α}$ and $C^{1,α}$ regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane $Σ$ as a power $a > -1$ of the distance to $Σ$. The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar $C^{1,α}$ estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type. |
| title | Schauder estimates for parabolic equations with degenerate or singular weights |
| topic | Analysis of PDEs 35B65 |
| url | https://arxiv.org/abs/2401.06038 |