Bent rectangles as viscosity solutions over a circle

We study the motion of the so-called bent rectangles by the singular weighted mean curvature. We are interested in the curves which can be rendered as graphs over a smooth one-dimensional reference manifold. We establish a sufficient condition for that. Once we deal with graphs we can have the tools of the viscosity theory available, like the Comparison Principle. With its help we establish uniqueness of variational solutions constructed by the authors (Giga et al., 2013). In addition, we establish a criterion for the mobility coefficient guaranteeing vertex preservation.