Special cases of the planar least gradient problem
  We study two special cases of the planar least gradient problem. In the first one, the boundary conditions are imposed on a part of the strictly convex domain. In the second case, we impose the Dirichlet data on the boundary of a rectangle, an example of convex but not strictly convex domain. We show the existence of solutions and study their properties for particular cases of data. 
Wojciech G\'orny, Piotr Rybka, Ahmad Sabra
math.AP