On the mean curvature type flow for convex capillary hypersurfaces in the ball

In this paper, we study the mean curvature type flow for hypersurfaces in the unit Euclidean ball with capillary boundary, which was introduced by Wang and Xia (Commun Anal Geom, 2019) and Wang and Weng (Calc Var Partial Differ Equ 59(5):149–174, 2020). We show that if the initial hypersurface is strictly convex, then the solution of this flow is strictly convex for t> 0 , exists for all positive time and converges smoothly to a spherical cap. As an application, we prove a family of new Alexandrov–Fenchel inequalities for convex hypersurfaces in the unit Euclidean ball with capillary boundary.