As opposed to the widely studied bifurcation phenomena for maps or PDE problems, we are concerned with bifurcation for stationary points of a nonlocal variational functional defined not on functions but on sets of finite perimeter, and involving a nonlocal term. This sharp interface model (1.2), arised as the Γ-limit of the FitzHugh–Nagumo energy functional in a (flat) square torus in R2 of size T, possesses lamellar stationary points of various widths with well-understood stability ranges and exhibits many interesting phenomena of pattern formation as well as wave propagation. We prove that when the lamella loses its stability, bifurcation occurs, leading to a two-dimensional branch of nonplanar stationary points. Thinner nonplanar structures, achieved through a smaller T, or multiple layered lamellae in the same-sized torus, are more stable. To the best of our knowledge, bifurcation for nonlocal problems in a geometric measure theoretic setting is an entirely new result.
1
University of Parma, Italy
2
National Tsing Hua University, Hsinchu, Taiwan
3
University of Connecticut, Storrs, USA
Emilio D. Acerbi; Chao-Nien Chen; Yung Sze Choi. Bifurcation for a sharp interface generation problem. Interfaces and free boundaries, Tome 27 (2025) no. 3, pp. 403-457. doi: 10.4171/ifb/538
@article{10_4171_ifb_538,
author = {Emilio D. Acerbi and Chao-Nien Chen and Yung Sze Choi},
title = {Bifurcation for a sharp interface generation problem},
journal = {Interfaces and free boundaries},
pages = {403--457},
year = {2025},
volume = {27},
number = {3},
doi = {10.4171/ifb/538},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/538/}
}
TY - JOUR
AU - Emilio D. Acerbi
AU - Chao-Nien Chen
AU - Yung Sze Choi
TI - Bifurcation for a sharp interface generation problem
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PY - 2025
SP - 403
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VL - 27
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