Variational analysis of a mean curvature flow action functional

Abstract

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.

Appendix. Notations and results from differential geometry

Let $M$ be an $n$-dimensional smooth differentiable manifold without boundary and consider a smooth immersion $\phi : M \rightarrow {\mathbb {R}}^{n+1}$. Denoting with $(x^1 ,\ldots , x^n)$ a local coordinate system on $M$ and $(\frac{\partial }{\partial x_1} ,\ldots , \frac{\partial }{\partial x_n}) : = (e_1 ,\ldots , e_n)$ the associated base for the tangent space, the Riemannian metric $g$ induced by $\phi$ on $M$ via pullback reads as follows:

$$\begin{aligned} g_{ij} := \partial _i \phi \cdot \partial _j \phi \,, \end{aligned}$$

(7.1)

where $\cdot$ denotes the standard scalar product in ${\mathbb {R}}^{n+1}$. We will denote with $\nabla$ the covariant derivative associated to the Levi-Civita connection of $g$.

Since $\phi (M)$ has codimension one in ${\mathbb {R}}^{n+1}$, it follows that $M$ is orientable and at each point of $\phi (M)$ there is a well defined smooth inner unit normal vector field which we call $\nu$. We define the second fundamental form of $\phi (M)$ according to

$$\begin{aligned} {\mathrm A}= ({\mathrm h}_{ij})_{ij} := (\nu \cdot \partial ^2_{ij} \phi )_{ij} \,, \end{aligned}$$

(7.2)

which implies that ${\mathrm A}$ is a symmetric 2-tensor on $\phi (M)$.

The mean curvature of the couple $(M,\phi )$ is defined as the trace of the second fundamental form and is denoted by $\vec {{\mathrm H}}$, while the scalar mean curvature is given by ${\mathrm H}:= \varvec{{\mathrm H}} \cdot \nu$.

Within this setting, the Gauss-Weingarten relations read

$$\begin{aligned} \partial ^2_{ij} \phi = \Gamma ^k_{ij} \partial _k \phi + {\mathrm h}_{ij} \nu \quad \mathrm and \quad \partial _i \nu = - {\mathrm h}_{ik} g^{kl} \partial _l \phi \,. \end{aligned}$$

(7.3)

The Bianchi identities for the curvature tensor of the immersed manifold are equivalent to

$$\begin{aligned} \nabla _i {\mathrm h}_{jk} = \nabla _j {\mathrm h}_{ik} \,. \end{aligned}$$

(7.4)