Hokkaido University Preprint Series in Mathematics
We consider the weighted mean curvature flow with a driving term in the plane. For nisotropy functions this evolution problem degenerates to a first order Hamilton-Jacobi quation with a free boundary. The resulting problem may be written as a Hamilton-Jacobi quation with a spatially non-local and discontinuous Hamiltonian. We prove existence nd uniqueness of solutions. On the way we show a comparison principle and a stability heorem for viscosity solutions.