$\hat{I}“$-convergence for higher order nonlocal phase transitions

Abstract

For every $0<s<3/4$, we study the asymptotic behavior of the $\epsilon$-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 $\epsilon \to 0$. 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.