Existence and structure of solutions for general $P$-area minimizing surface

We study existence and structure of solutions to the Dirichlet and Neumann boundary problems associated with minimizers of the functional $I(u)={\int }_{\mathrm{\Omega }}(\phi (x,Du+F)+Hu)dx$, where $\phi (x,\xi )$, among other properties, is convex and homogeneous of degree $1$ with respect to $\xi$. We show that there exists an underlying vector field $N$ that characterizes the existence and structure of all minimizers. We also investigate existence of solutions under the barrier condition on $\mathrm{\partial }\mathrm{\Omega }$. The results in this paper generalize and unify many results in the literature about existence of minimizers of least gradient problems and $P-$area minimizing surfaces.