Evolution of regular bent rectangles by the driven crystalline curvature flow in the plane with a non-uniform forcing term

Abstract

We study the motion of regular bent rectangles driven by singular curvature flow with a driving term. The curvature is being interpreted as a solution to a minimization problem. The evolution equation becomes in a local coordinate a system of Hamilton--Jacobi equations with free boundaries, coupled to a system of ODE's with nonlocal nonlinearities. We establish local-in-time existence of variational solutions to the flow, and uniqueness is proved under additional regularity assumptions on the data.