The Dirichlet problem of prescribed mean curvature equations is well posed, if the boundery is H-convex. In this article we eliminate the H-convexity condition from a portion $\Gamma$ of the boundary and prove the existence theorem, where the boundary condition is satisfied on $\Gamma$ in the weak sense.
"On the Dirichlet problem of prescribed mean curvature equations without $H$-convexity condition." Nagoya Math. J. 157 177 - 209, 2000.