1. Asymptotic expansion of a nonlocal phase transition energy
- Author
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Dipierro, Serena, Patrizi, Stefania, Valdinoci, Enrico, and Vaughan, Mary
- Subjects
Mathematics - Analysis of PDEs ,82B26, 35R11, 49J45, 35A01 - 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 $\Gamma$-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., Comment: 57 pages. Corrected Theorems 1.1 and 1.5. Added Theorem 1.9 and Corollary 1.10
- Published
- 2024