Uniqueness of the limiting weighted mean-curvature-flow equation

Determine whether the limiting equation for the level-set formulation of weighted mean curvature flow with rough weights has a unique solution.

Background

The paper constructs weak solutions by approximating the rough weighted level-set equation with regularized equations and passing to the limit. The authors explain that the relabeling argument used by Evans and Spruck cannot be applied directly because uniqueness of the limiting equation is not known.

In particular, for a smooth increasing relabeling function, the regularized solutions with relabeled initial data converge along a subsequence to a limit, but the authors cannot identify that limit with the relabeling of the original limiting solution. They therefore establish the Brakke-flow characterization using specially chosen test functions instead of relying on uniqueness.