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arXiv · 2401.16718

The pluricomplex Green function of the Monge-Amp\`{e}re equation for $(n-1)$-plurisubharmonic functions and form type $k$-Hessian equations

Shuimu Li
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Abstract

In this paper, we introduce the pluricomplex Green function of the Monge-Amp\`{e}re equation for $(n-1)$-plurisubharmonic functions by solving the Dirichlet problem for the form type Monge-Amp\`{e}re and Hessian equations on a punctured domain. We prove the pluricomplex Green function is $C^{1,\alpha}$ 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.

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Shuimu Li. 2024-01-30. The pluricomplex Green function of the Monge-Amp\`{e}re equation for $(n-1)$-plurisubharmonic functions and form type $k$-Hessian equations. https://arxiv.org/abs/2401.16718

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