Existence of a capillary surface with prescribed contact angle in $M\times \mathbb{R}$

We study the prescribed mean curvature equation with a prescribed boundary contact angle condition in $M\times \mathbb{R}$ where $M^n$ is a Riemannian submanifold in ${\mathbb{R}}^{n+1}$. The main purpose is to establish a priori gradient estimates for solutions, from which the long time existence of the solution are derived.