Weak acceleration for interfacial motions

Abstract

We provide two formulations of the acceleration for the motion of the interfaces in the framework of geometric measure theory. The first formulation is considered using varifolds, which are generalised surfaces seen as measures. The second formulation is considered using Caccioppoli sets, which are known as sets of finite perimeter. Both formulations arise from the time derivative of an integral quantity on the hypersurface associated with a weighted momentum. We further prove that the two notions of weak acceleration coincide under suitable assumptions. This result establishes a weak formulation of acceleration relevant to the study of weak solutions of hyperbolic mean curvature flow.