2015/08/07 by Barbara Nelli, Nelli, Barbara, Ricardo Sá Earp +3
Computer Science · Mathematics · #35J25 #53A10 #53C42 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1508.01724
openalex publication_date 2015/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the minimal surface equation in the Heisenberg space, Nil3. A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved (our proof holds in the Euclidean space as well). We solve the Dirichlet problem for the minimal surface equation over bounded and unbounded convex domains, taking bounded, piecewise continuous boundary value. We are able to construct a Scherk type minimal surface and we use it as a barrier to construct non trivial minimal graphs over a wedge of angle between π/2 ,and πtaking non negative continuous boundary data, having at least quadratic growth. In the case of an half- plane, we are also able to give solutions (with either linear or quadratic growth), provided some geometric hypothesis on the boundary data. Finally, some open problem arising from our work, are posed.