Evolution by Non-Convex Functionals

TL;DR: In this article, a semi-group solution concept for flows that are generated by generalized minimizers of non-convex energy functionals is established. But the main part of this work consists in exemplary validation of the solution concept.

Abstract: We establish a semi-group solution concept for flows that are generated by generalized minimizers of non-convex energy functionals. We use relaxation and convexification to define these generalized minimizers. The main part of this work consists in exemplary validation of the solution concept for a non-convex energy functional. For rotationally invariant initial data it is compared with the solution of the mean curvature flow equation. The basic example relates the mean curvature flow equation with a sequence of iterative minimizers of a family of non-convex energy functionals. Together with the numerical evidence this corroborates the claim that the non-convex semi-group solution concept defines, in general, a solution of the mean curvature equation.