Bifurcation for a sharp interface generation problem

As opposed to the widely studied bifurcation phenomena for maps or PDE problems, we are concerned with bifurcation for stationary points of a non-local 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 $\Gamma$-limit of the FitzHugh-Nagumo energy functional in a (flat) square torus in $R^2$ 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.