The pluricomplex Green function of the Monge-Ampère equation for $(n-1)$-plurisubharmonic functions and form type $k$-Hessian equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910367291015168 |
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| author | Li, Shuimu |
| author_facet | Li, Shuimu |
| contents | In this paper, we introduce the pluricomplex Green function of the Monge-Ampère equation for $(n-1)$-plurisubharmonic functions by solving the Dirichlet problem for the form type Monge-Ampère and Hessian equations on a punctured domain. We prove the pluricomplex Green function is $C^{1,α}$ by constructing approximating solutions and establishing uniform a priori estimates for the gradient and the complex Hessian. The singular solutions turn out to be smooth for the $k$-Hessian equations for $(n-1)$-$k$-admissible functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16718 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The pluricomplex Green function of the Monge-Ampère equation for $(n-1)$-plurisubharmonic functions and form type $k$-Hessian equations Li, Shuimu Analysis of PDEs Complex Variables 35J08, 35J60, 35J96 In this paper, we introduce the pluricomplex Green function of the Monge-Ampère equation for $(n-1)$-plurisubharmonic functions by solving the Dirichlet problem for the form type Monge-Ampère and Hessian equations on a punctured domain. We prove the pluricomplex Green function is $C^{1,α}$ by constructing approximating solutions and establishing uniform a priori estimates for the gradient and the complex Hessian. The singular solutions turn out to be smooth for the $k$-Hessian equations for $(n-1)$-$k$-admissible functions. |
| title | The pluricomplex Green function of the Monge-Ampère equation for $(n-1)$-plurisubharmonic functions and form type $k$-Hessian equations |
| topic | Analysis of PDEs Complex Variables 35J08, 35J60, 35J96 |
| url | https://arxiv.org/abs/2401.16718 |