Translating Solutions of Non-parametric Mean Curvature Flows with Capillary-Type Boundary Value Problems

In this note, we study the mean curvature flow and the prescribed mean curvature type equation with general capillary-type boundary condition, which is \begin{document}$ u_{\nu} = -\phi(x)(1+|Du|^2)^\frac{1-q}{2} $\end{document} for any parameter \begin{document}$ q>0 $\end{document} . Using the maximum principle, we prove the gradient estimates for the solutions of such a class of boundary value problems. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations. In addition, we study the related additive eigenvalue problem for general boundary value problems and describe the asymptotic behavior of the solution at infinity time. The originality of the paper lies in the range \begin{document}$ 0 , since there are no any related results before. For parabolic case, we generalize the result of Ma-Wang-Wei [ 25 ] to any \begin{document}$ q>0 $\end{document} . And in elliptic case, we generalize the results in [ 32 ] to any \begin{document}$ q\ge 0 $\end{document} and to any bounded smooth domain.