2024/10/30 by Dipierro, Serena, Valdinoci, Enrico, Vaughan, Mary · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.22873
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 ∈ (0,1) and a first-order asymptotic development in the Γ-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 μ> 2-2s. In view of this range value for μ, it would be interesting to develop a new asymptotic development for the Γ-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.