The Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$, with possible infinite boundary data

Abstract

In this paper, we study the Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$ with possible infinite boundary data, where $\mathrm{Sol}_{3}$ is the non-Abelian solvable $3$-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometries. Our main result is a Jenkins–Serrin type theorem which establishes necessary and sufficient conditions for the existence and uniqueness of certain minimal Killing graphs with a non-unitary Killing vector field in $\mathrm{Sol}_{3}$.