Capillary problem and nonparametric mean curvature flow

We consider the graphs over a bounded strictly convex domain $\Omega$ in $\mathbb{R}^n$ with prescribed variable contact angle with the boundary cylinder $\partial\Omega\times \mathbb{R}$, which move by nonparametric mean curvature flow. When the contact angle is nearly perpendicular, we show that the solutions converge to ones which move by translation. Subsequently, the existence and uniqueness of smooth solutions to the capillary problem on the strictly convex domain are also discussed. As for the hypersurface evolving along with the mean curvature flow in Riemannian manifold endowed with a Killing vector field, similar results are also obtained. Lastly, if time permits, we introduce a new mean curvature type flow with capillary boundary in the unit ball, which preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases the energy functional. We show that it has longtime existence and converges to the spherical cap.