Existence and structure of solutions for general $ P $-area minimizing surfaces
Amir Moradifam, Alexander Rowell · Communications on Pure & Applied Analysis · 2025
We study the existence and structure of solutions to the Dirichlet and Neumann boundary problems associated with minimizers of the functional $ I(u) = \int_{\Omega} ({{\varphi}}(x, D u + F)+Hu) \, dx $, where $ {{\varphi}} (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 the existence of solutions under the barrier condition on $ \partial \Omega $. The results in this paper generalize and unify many results in the literature about the existence of minimizers of least gradient problems and $ P- $area minimizing surfaces.