The Impact of the Domain Boundary on an Inhibitory System: Existence and Location of a Stationary Half Disc

Abstract

The nonlocal geometric variational problem derived from the Ohta-Kawasaki diblock copolymer theory is an inhibitory system with self-organizing properties. The free energy, defined on subsets of a prescribed measure in a domain, is a sum of a local perimeter functional and a nonlocal energy given by the Green's function of Poisson's equation on the domain with the Neumann boundary condition. The system has the property of preventing a disc from drifting towards the domain boundary. This raises the question of whether a stationary set may have its interface touch the domain boundary. It is proved that a small, perturbed half disc exists as a stable stationary set, where the circular part of its boundary is inside the domain, as the interface, and the almost flat part of its boundary coincides with part of the domain boundary. The location of the half disc depends on two quantities: the curvature of the domain boundary, and a remnant of the Green's function after one removes the fundamental solution and a reflection of the fundamental solution. This reflection is defined with respect to any sufficiently smooth domain boundary. It is an interesting new concept that generalizes the familiar notions of mirror image and circle inversion. When the nonlocal energy is weighted less against the local energy, the stationary half disc sits near a maximum of the curvature; when the nonlocal energy is weighted more, the half disc appears near a minimum of the remnant function. There is also an intermediate case where the half disc is near a minimum of a combination of the curvature and the remnant function.

The impact of the domain boundary on an inhibitory system: Interior discs and boundary half discs

“…Note that the second term in 1.10 is twice the fundamental solution of −∆. If ξ * ,b ∈ ∂D is the center of the perturbed half disc stationary set found in [9] and the parameters ω and γ are in the same range as in this paper specified in Theorem 1.1, then ξ * ,b is close to a minimum of the function…”

“…Finding a stationary set whose interface meets the domain boundary is a difficult problem. The first non-trivial result came in our work [9]. When ω is sufficiently small and γ is suitably large, there exists a stationary set shaped like a perturbed half disc, stable in some sense, whose boundary inside D (a perturbed half circle) meets ∂D perpendicularly.…”

“…Here we consider an ideal situation where micro-domains are clearly separated from each other by interfaces with zero width [7,10,4]. Mathematically, let D be a bounded domain in R 2 ; D is assumed to be of class C 5 , a condition necessary for many results in [9]. The energy functional is defined for every Lebesgue measurable subset Ω of D whose Lebesgue measure is fixed at ω|D|:…”

“…Here P D (Ω) is the perimeter of Ω in D. In the case that Ω is piecewise C 1 , it is the length of the part of the boundary of Ω that is inside D. For a general Ω, see [11,9] for the definition of P D (Ω). The part of the boundary of Ω inside D is called the interface of Ω.…”

“…When ω is sufficiently small and γ is suitably large, there exists a stationary set shaped like a perturbed half disc, stable in some sense, whose boundary inside D (a perturbed half circle) meets ∂D perpendicularly. A crucial quantity introduced in [9…”