The Eigenvalue Problem for the complex Monge-Ampère operator
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910030431780864 |
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| author | Badiane, Papa Zeriahi, Ahmed |
| author_facet | Badiane, Papa Zeriahi, Ahmed |
| contents | We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge-Ampère operator on a bounded strongly pseudoconvex domain in $\C^n$. We show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in $Ω$ and boundary values $0$. Moreover it is unique up to a positive multiplicative constant.
To this end, we follow the strategy used by P.L. Lions in the real case. However, we have to prove a new theorem on the existence of solutions for some special complex degenerate Monge-Ampère equations. This requires establishing new a priori estimates of the gradient and Laplacian of such solutions using methods and results of L. Caffarelli, J.J. Kohn, L. Nirenberg and J. Spruck \cite{CKNS85} and B. Guan \cite{GuanB98}.
Finally we provide a Pluripotential variational approach to the problem and using our new existence theorem, we prove a Rayleigh quotient type formula for the first eigenvalue of the complex Monge-Ampère operator. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_03285 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Eigenvalue Problem for the complex Monge-Ampère operator Badiane, Papa Zeriahi, Ahmed Complex Variables Analysis of PDEs Differential Geometry 32U05, 32W20, 35J66, 35J96 We prove the existence of the first eigenvalue and an associated eigenfunction with Dirichlet condition for the complex Monge-Ampère operator on a bounded strongly pseudoconvex domain in $\C^n$. We show that the eigenfunction is plurisubharmonic, smooth with bounded Laplacian in $Ω$ and boundary values $0$. Moreover it is unique up to a positive multiplicative constant. To this end, we follow the strategy used by P.L. Lions in the real case. However, we have to prove a new theorem on the existence of solutions for some special complex degenerate Monge-Ampère equations. This requires establishing new a priori estimates of the gradient and Laplacian of such solutions using methods and results of L. Caffarelli, J.J. Kohn, L. Nirenberg and J. Spruck \cite{CKNS85} and B. Guan \cite{GuanB98}. Finally we provide a Pluripotential variational approach to the problem and using our new existence theorem, we prove a Rayleigh quotient type formula for the first eigenvalue of the complex Monge-Ampère operator. |
| title | The Eigenvalue Problem for the complex Monge-Ampère operator |
| topic | Complex Variables Analysis of PDEs Differential Geometry 32U05, 32W20, 35J66, 35J96 |
| url | https://arxiv.org/abs/2306.03285 |