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arXiv · 2409.06215

Asymptotic expansion of a nonlocal phase transition energy

Serena Dipierro·Stefania Patrizi·Enrico Valdinoci·Mary Vaughan
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Abstract

We study the asymptotic behavior of the fractional Allen--Cahn energy functional in bounded domains with prescribed Dirichlet boundary conditions. When the fractional power $s \in (0,\frac12)$, we establish establish the first-order asymptotic development up to the boundary in the sense of $Γ$-convergence. In particular, we prove that the first-order term is the nonlocal minimal surface functional. Also, we show that, in general, the second-order term is not properly defined and intermediate orders may have to be taken into account. For $s \in [\frac12,1)$, we focus on the one-dimensional case and we prove that the first order term is the classical perimeter functional plus a penalization on the boundary. Towards this end, we establish existence of minimizers to a corresponding fractional energy in a half-line, which provides itself a new feature with respect to the existing literature.

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Serena Dipierro, Stefania Patrizi, Enrico Valdinoci, Mary Vaughan. 2024-09-23. Asymptotic expansion of a nonlocal phase transition energy. https://arxiv.org/abs/2409.06215

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