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

Abstract

We study existence and structure of solutions to the Dirichlet and Neumann boundary problems associated with minimizers of the functional $I(u)={\int }_{\hat{I}}\text{c◯}(\ddot{I}•(x,Du+F)+Hu)dx$, where $\ddot{I}•(x,\hat{I}¾)$, among other properties, is convex and homogeneous of degree $1$ with respect to $\hat{I}¾$. 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 $\partial \hat{I}\text{c◯}$. 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.

arXiv admin note: text overlap with arXiv:2104.08624