Half Disc Stationary Sets on the Boundary of a Binary Inhibitory System
Open AccessThe Ohta-Kawasaki density functional theory for diblock copolymers gives rise to nonlocal geometric variational problems on inhibitory systems with self-organizing properties. Within a proper range of the block composition parameter, stable stationary patterns of Caccioppoli sets form that roughly take the shape of discs. The energy functional for this type of problem consists of the sum of two terms: a local term that depends on the length of the perimeter of the sets, and a nonlocal term that is derived from the Green's function of Poisson's equation with the Neumann boundary condition.This work studies the existence of stable solutions that touch the domain boundary. Small, half-disc like sets exist as stationary solutions along the domain boundary with the circular part of the disc in the interior of the domain. The location of this stationary set is dependent on two known quantities: the curvature of the domain boundary and a remnant of the Green's function after one removes the fundamental solution and its reflection about the boundary. When the local energy is given more weight, the half disc stationary set appears near the maximum of the curvature of the boundary; when the nonlocal energy is given more weight, the half disc appears near a minimum of the remnant function. In the intermediary case where both receive significant weight, the half disc appears near a minimum of a combination of both the curvature and remnant function. Subsequently the case of a stationary assembly consisting of a combination of interior and boundary discs is studied, and conditions for existence and optimal disc location are presented.
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