In this paper, weestablish global 𝐶 2 estimates for a class of mixed Hessian equations with the Neumann boundary condition and obtain the existence theorem of 𝑘admissible solutions for the classical Neumann problem of these mixed Hessian equations.

<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>

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In this article, we study a locally constrained mean curvature flow for star-shaped hypersurfaces with capillary boundary in the half-space. We prove its long-time existence and the global convergence to a spherical cap. Furthermore, the capillary quermassintegrals defined in [29] evolve monotonically along the flow, and hence we establish a class of new Alexandrov–Fenchel inequalities for convex hypersurfaces with capillary boundary in the half-space.