Variational analysis of a mean curvature flow action functional

We consider the reduced Allen-Cahn action functional, which arises as the sharp interface limit of the Allen-Cahn action functional and can be understood as a formal action functional for a stochastically perturbed mean curvature flow. For suitable evolutions of (generalized) hypersurfaces this functional consists of the integral over time and space of the sum of the squares of the mean curvature and of the velocity vectors. Given initial and final conditions, we investigate the associated action minimization problem, for which we propose a weak formulation, and within the latter we prove compactness and lower-semicontinuity properties of a suitably generalized action functional. Furthermore, we derive the Euler-Lagrange equation for smooth stationary trajectories and investigate some related conserved quantities. To conclude, we analyze the explicit case in which the initial and final data are concentric spheres. In this particular situation we characterize the properties of the minimizing rotationally symmetric trajectory in dependence of the given time span.