A threshold for higher-order asymptotic development of genuinely nonlocal phase transition energies

TL;DR

This paper investigates the asymptotic behavior of nonlocal phase transition energies with fractional order, revealing limitations on higher-order expansions and suggesting the need for new fractional order-based asymptotic frameworks.

Contribution

It establishes the non-existence of meaningful second-order and higher fractional order asymptotic expansions for nonlocal phase transition energies.

Findings

No second-order asymptotic expansion exists.

Higher fractional order expansions beyond 2-2s are not meaningful.

Results hold in all space dimensions with mild exterior data assumptions.

Abstract

We study the higher-order asymptotic development of a nonlocal phase transition energy in bounded domains and with prescribed external boundary conditions. The energy under consideration has fractional order $2s \in (0,1)$ and a first-order asymptotic development in the $\Gamma$-sense as described by the fractional perimeter functional. We prove that there is no meaningful second-order asymptotic expansion and, in fact, no asymptotic expansion of fractional order $\mu > 2-2s$. In view of this range value for $\mu$, it would be interesting to develop a new asymptotic development for the $\Gamma$-convergence of our energy functional which takes into account fractional orders. The results obtained here are also valid in every space dimension and with mild assumptions on the exterior data.