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We study existence and structure of solutions to the Dirichlet and Neumannboundary problems associated with minimizers of the functional$I(u)=\int_{\Omega} (\phi(x, D u + F)+Hu) \, dx$, where $\phi (x, \xi)$, amongother properties, is convex and homogeneous of degree $1$ with respect to$\xi$. We show that there exists an underlying vector field $N$ thatcharacterizes the existence and structure of all minimizers. In particular weprove that if $\Omega$ satisfies a so called barrier condition, then $ I(u)$has a minimizer in $BV_f(\Omega)=\{u\in BV(\Omega): u|_{\partial \Omega}=f\}$.The results in this paper generalize and unify many results in the literatureabout existence of minimizers of least gradient problems and $P-$areaminimizing surfaces.

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