$\Gamma$-convergence for higher order nonlocal phase transitions

TL;DR

This paper investigates the asymptotic behavior of fractional Allen-Cahn energies for different fractional parameters, establishing Gamma-convergence to classical or fractional perimeters and challenging previous expectations about curvature-dependent limits.

Contribution

It proves Gamma-convergence of fractional Allen-Cahn energies to perimeter functionals for a range of s, revealing new insights into nonlocal phase transitions.

Findings

First variation contribution vanishes as epsilon approaches zero

Gamma-convergence to classical or fractional perimeter depending on s

Contradicts previous expectations of curvature-dependent limits in certain regimes

Abstract

For every $0 < s <3/4$, we study the asymptotic behavior of the $\varepsilon$-rescaled sum of the $s$-fractional Allen-Cahn energy and the squared $L^2$-norm of its first variation. We prove that the contribution of the first variation vanishes as $\varepsilon \to 0$. This implies the Gamma-convergence of the initial sum to either the classical perimeter or to the $2s$-fractional perimeter, depending on whether $s \ge 1/2$ or not. This contradicts the expectation of finding curvature-dependent terms in the limit, as suggested by the regime $3/4 \le s < 1$, and as known to hold in low dimensions in the local case.