Variational analysis of a mean curvature flow action functional

TL;DR

This paper analyzes a variational functional related to mean curvature flow, exploring its properties, minimization, and specific solutions like concentric spheres, providing insights into stochastically perturbed geometric evolutions.

Contribution

It introduces a generalized formulation of the reduced Allen-Cahn action functional, proves key mathematical properties, and characterizes minimal solutions for symmetric cases.

Findings

Proved compactness and lower semicontinuity of the functional

Derived Euler-Lagrange equations for stationary points

Explicitly characterized minimal solutions for concentric spheres

Abstract

We consider the reduced Allen-Cahn action functional, which appears 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 sum of the squares of the mean curvature and of the velocity vector, integrated over time and space. For given initial and final conditions we investigate the corresponding action minimization problem. We give a generalized formulation and prove compactness and lower semicontinuity properties of the action functional. Furthermore we characterize the Euler-Lagrange equation for smooth stationary points and investigate conserved quantities. Finally we present an explicit example and consider concentric spheres as initial and final data and characterize in dependence of the given time span the properties of the minimal rotationally symmetric connection.