Interior Hölder regularity of the linearized Monge-Ampère equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929352285880320 |
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| author | Wang, Ling |
| author_facet | Wang, Ling |
| contents | In this paper, we investigate the interior Hölder regularity of solutions to the linearized Monge-Ampère equation. In particular, we focus on the cases with singular right-hand side, which arise from the study of the semigeostrophic equation and singular Abreu equations. In the two-dimensional case, we give a new proof of the Caffarelli-Gutiérrez Hölder estimate (\textit{Amer. J. Math.} \textbf{119} (1997), no.\,2, 423-465) and the result of Le (\textit{Comm. Math. Phys.} \textbf{360} (2018), no.\,1, 271-305) for the linearized Monge-Ampère equation with singular right-hand side term in divergence form. The main new ingredient in the proof contains the application of the partial Legendre transform to the linearized Monge-Ampère equation. Building on this idea, we also establish a new Moser-Trudinger type inequality in dimension two. In higher dimensions, we derive the interior Hölder estimate under certain integrability assumptions on the coefficients using De Giorgi's iteration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13297 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interior Hölder regularity of the linearized Monge-Ampère equation Wang, Ling Analysis of PDEs In this paper, we investigate the interior Hölder regularity of solutions to the linearized Monge-Ampère equation. In particular, we focus on the cases with singular right-hand side, which arise from the study of the semigeostrophic equation and singular Abreu equations. In the two-dimensional case, we give a new proof of the Caffarelli-Gutiérrez Hölder estimate (\textit{Amer. J. Math.} \textbf{119} (1997), no.\,2, 423-465) and the result of Le (\textit{Comm. Math. Phys.} \textbf{360} (2018), no.\,1, 271-305) for the linearized Monge-Ampère equation with singular right-hand side term in divergence form. The main new ingredient in the proof contains the application of the partial Legendre transform to the linearized Monge-Ampère equation. Building on this idea, we also establish a new Moser-Trudinger type inequality in dimension two. In higher dimensions, we derive the interior Hölder estimate under certain integrability assumptions on the coefficients using De Giorgi's iteration. |
| title | Interior Hölder regularity of the linearized Monge-Ampère equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.13297 |