Complex Alexandrov-Bakelman-Pucci estimate and its applications
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929529492078592 |
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| author | Liu, Junbang |
| author_facet | Liu, Junbang |
| contents | We prove an Alexandrov-Bakelman-Pucci type estimate, which involves the integral of the determinant of the complex Hessian over a certain subset. It improves the classical ABP estimate adapted (by inequality $2^{2n}|\det(u_{i\bar{j}})|^2\geq |\det(\nabla^2u)|$) to complex setting. We give an application of it to derive sharp gradient estimates for complex Monge-Ampère equations. The approach is based on the De Giorgi iteration method developed by Guo-Phong-Tong for equations of complex Monge-Ampère type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04395 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complex Alexandrov-Bakelman-Pucci estimate and its applications Liu, Junbang Differential Geometry Analysis of PDEs We prove an Alexandrov-Bakelman-Pucci type estimate, which involves the integral of the determinant of the complex Hessian over a certain subset. It improves the classical ABP estimate adapted (by inequality $2^{2n}|\det(u_{i\bar{j}})|^2\geq |\det(\nabla^2u)|$) to complex setting. We give an application of it to derive sharp gradient estimates for complex Monge-Ampère equations. The approach is based on the De Giorgi iteration method developed by Guo-Phong-Tong for equations of complex Monge-Ampère type. |
| title | Complex Alexandrov-Bakelman-Pucci estimate and its applications |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2410.04395 |