Shuimu Li | Searcharxiv

arXiv2024

The pluricomplex Green function of the Monge-Ampère equation for $(n-1)$-plurisubharmonic functions and form type $k$-Hessian equations

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.

math.AP↗

arXiv2022

$L^p$ estimate for positive harmonic functions near singularities and Bôcher type theorems

In this paper, positive solutions to the Laplace equation with 1-dimensional circular singularities are investigated. First, we establish $L^p$ integrability estimates for such solutions $u$ near the singularities, in comparison with classical $L^1$ estimates. Then we characterize $-Δu$ around the circle. The method we developed may also be adapted to higher dimensional as well as more generalized cases.

math.AP↗