The Neumann problem for a class of mixed complex Hessian equations

Abstract

<p style='text-indent:20px;'>In this paper, we consider the Neumann problem of a class of mixed complex Hessian equations <inline-formula><tex-math id="M1">\begin{document}$ \sigma_k(\partial \bar{\partial} u) = \sum\limits _{l = 0}^{k-1} \alpha_l(z) \sigma_l (\partial \bar{\partial} u) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ 2 \leq k \leq n $\end{document}</tex-math></inline-formula>, and establish the global <inline-formula><tex-math id="M3">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> estimates and reduce the global second derivative estimate to the estimate of double normal second derivatives on the boundary. In particular, we can prove the global <inline-formula><tex-math id="M4">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> estimates and the existence theorems when <inline-formula><tex-math id="M5">\begin{document}$ k = n $\end{document}</tex-math></inline-formula>.</p>