On the Strong Maximum Principle and the Large Time Behavior of Generalized Mean Curvature Flow with the Neumann Boundary Condition

A generalized mean curvature flow is considered in a cylindrical domain under the right angle boundary condition. Its large time behavior is studied by analyzing the level set equation of the flow when the initial hypersurface is not necessarily graph-like. For a convex domain it is shown that the solution of the level set equation converges to a function whose level sets are perpendicular to the lateral boundary of the domain as time tends to infinity (when the initial data is constant outside some bounded set). For this purpose a version of the strong maximum principle is established for the level set minimal surface equation although the equation is degenerate