Weak solutions to mean curvature flow respecting obstacles

Weak Solutions to Mean Curvature Flow Respecting Obstacles I: the Graphical Case

We consider the problem of evolving hypersurfaces by mean curvature flow in the presence of obstacles, that is domains which the flow is not allowed to enter. In this paper, we treat the case of complete graphs and explain how the approach of M. Sáez and the second author [13] yields a global weak solution to the original problem for general initial data and onesided obstacles.

Mean curvature flow with obstacles

We consider the evolution of fronts by mean curvature in the presence of obstacles. We construct a weak solution to the flow by means of a variational method, corresponding to an implicit time-discretization scheme. Assuming the regularity of the obstacles, in the two-dimensional case we show existence and uniqueness of a regular solution before the onset of singularities. Finally, we discuss a...

Weak Solutions for the Mean Curvature Flow of Planar Graphs

Translating Solutions to Lagrangian Mean Curvature Flow

We prove some non-existence theorems for translating solutions to Lagrangian mean curvature flow. More precisely, we show that translating solutions with an L bound on the mean curvature are planes and that almost-calibrated translating solutions which are static are also planes. Recent work of D. Joyce, Y.-I. Lee, and M.-P. Tsui, shows that these conditions are optimal.

Mean curvature flow with obstacles: existence, uniqueness and regularity of solutions

We show short time existence and uniqueness of C solutions to the mean curvature flow with obstacles, when the obstacles are of class C. If the initial interface is a periodic graph we show long time existence of the evolution and convergence to a minimal constrained hypersurface.