A convergence result for the derivation of front propagation in nonlocal phase field models

TL;DR

This paper establishes a mathematical link between fractional Laplacians and mean curvature of surfaces, providing a foundation for understanding front propagation in fractional reaction-diffusion models.

Contribution

It proves that mean curvature can be derived as a limit of functions related to fractional Laplacians, connecting nonlocal operators to classical geometric evolution.

Findings

Mean curvature arises as a limit of fractional Laplacian-based functions.

The approach recovers both fractional and classical mean curvature depending on the fractional order.

This result is crucial for deriving front evolution in fractional reaction-diffusion equations.

Abstract

We prove that the mean curvature of a smooth surface in $\mathbb{R}^n$, $n\geq 2$, arises as the limit of a sequence of functions that are intrinsically related to the difference between an $n$- and $1$-dimensional fractional Laplacian of a phase transition. Depending on the order of the fractional Laplace operator, we recover the fractional mean curvature or the classical mean curvature of the surface. Moreover, we show that this is an essential ingredient for deriving the evolution of fronts in fractional reaction-diffusion equations such as those for atomic dislocations in crystals.