arXiv subjects

Alexander Rowell

Publications and source records attributed to Alexander Rowell.

2 recordsLinked to original sources

arXiv2022

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 $I(u)=\int_{\Omega} (\phi(x, D u + F)+Hu) \, dx$, 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 $\partial \Omega$. 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.

math.AP↗

arXiv2021

Existence and structure of P-area minimizing surfaces in the Heisenberg group

We study existence and structure of $P-$area minimizing surfaces in the Heisenberg group under Dirichlet and Neumann boundary conditions. We show that there exists an underlying vector field $N$ that characterized existence and structure of $P$-area minimizing surfaces. This vector field exists even if there is no $P$-area minimizing surface satisfying the prescribed boundary conditions. We prove that if $\partial \Omega$ satisfies a so called Barrier condition, it is sufficient to guarantee existence of such surfaces. Our approach is completely different from previous methods in the literature and makes major progress in understanding existence of $P$-area minimizing surfaces.

math.DG↗