The mean curvature type hypersurfaces with prescribed gradient image

In this paper, we consider the existence of mean curvature type hypersurfaces with prescribed gradient image. Let $\mathrm{\Omega }$ and $\tilde{\mathrm{\Omega }}$ be uniformly convex bounded domains in ${\mathbb{R}}^n$ with smooth boundary. We show that there exists unique convex solutions for the second boundary value problem of mean curvature type equations.