Luca Mugnai

The Allen–Cahn action functional in higher dimensions

The Allen‐Cahn action functional is related to the probability of rare events in the stochastically perturbed Allen‐Cahn equation. Formal calculations suggest a reduced action functionalin the sharp interface limit. We prove the corresponding lower bound in two and three space dimensions. One difficulty is that diffuse interfaces may collapse in the limit. We therefore consider the limit of diffuse surface area measures and introduce a generalized velocity and generalized reduced action functional in a class of evolving measures.

Approximation of Helfrich's Functional via Diffuse Interfaces

We give a rigorous proof of the approximability of the so-called Helfrichs functional via diffuse interfaces under a constraint on the ratio between the bending rigidity and the Gaussian rigidity.

Confined Elastic Curves

We consider the problem of minimizing Eulers elastica energy for simple closed curves confined to the unit disk. We approximate a simple closed curve by the zero level set of a function with values

On the coarsening rates for attachment-limited kinetics

+1

A Phase Field Model for the Optimization of the Willmore Energy in the Class of Connected Surfaces

on the inside and

Some aspects of the variational nature of mean curvature flow

-1

On the approximation of the elastica functional in radial symmetry

on the outside of the curve. The outer container now becomes just the domain of the phase field. Diffuse approximations of the elastica energy and the curve length are well known; implementing the topological constraint thus becomes the main difficulty here. We propose a solution based on a diffuse approximation of the winding number, present a proof that one can approximate a given sharp interface using a sequence of phase fields, and show some numerical results using finite elements based on subdivision surfaces.

Convergence of Perturbed Allen Cahn Equations to Forced Mean Curvature Flow

We study the coarsening rates for attachment-limited kinetics which is modeled by volume-preserving mean-curvature flow. Attachment-limited kinetics is observed during solidification processes, in which the system is divided into two domains of the two pure phases, more precisely, islands of a solid phase surrounded by an undercooled liquid phase, and the relaxation process is due to material redistribution form high to low interfacial curvature regions. The interfacial area between the phases decreases in time while the volume of each phase is preserved. Consequently, the domain morphology coarsens. Experiments, heuristics, and numerics suggest that the typical domain size

On a mesoscopic many-body Hamiltonian describing elastic shears and dislocations

\ell

Global solutions to the volume-preserving mean-curvature flow

of the solid islands grows according to the power law