A toroidal tube solution to a problem involving mean curvature and Newtonian potential

Abstract

The Ohta-Kawasaki theory for block copolymer morphology and the Gierer-Meinhardt theory for morphogenesis in cell development both give rise to a nonlocal geometric problem. One seeks a set in R 3 which satisfies an equation that links the mean curvature of the boundary of the set to the Newtonian potential of the set. An axisymmetric, torus shaped, tube like solution exists in R 3 if the lone parameter of the problem is sufficiently large. A cross section of the torus is small and the distance from the center of the cross section to the axis of symmetry is large. The solution is stable in the class of axisymmetric sets in a particular sense. This work is motivated by the recent discovery of a toroidal morphological phase in a triblock copolymer.

Asymmetric and Symmetric Double Bubbles in a Ternary Inhibitory System

“…All solutions to (1.10) in one dimension are known to be local minimizers of J B [26]. Many solutions in two and three dimensions have been found that match the morphological phases in diblock copolymers [24,30,29,31,32,15,16,33,35,39]. Global minimizers of J B are studied in [2,37,19,5,18,17,11] for various parameter ranges.…”

“…Equation (1.10) or the functional (1.9) may be derived from the OhtaKawasaki theory [23] for diblock copolymers; see [21,26]. The equation can also be derived from the Gierer-Meinhardt system [33]. This binary problem has been studied intensively in recent years.…”

On global minimizers for a variational problem with long-range interactions

“…There are also many interesting questions associated with local minimizers associated with (5.6) (see for example [46,60]). …”

A stationary core-shell assembly in a ternary inhibitory system

“…on ∂Ω ∩ D. Equation (1.17) or functional (1.16) may be derived from the Ohta-Kawasaki theory [17] for diblock copolymers; see [16,19]. The equation can also be derived from the Gierer-Meinhardt system [25]. This binary problem has been studied intensively in recent years.…”

“…There is even a dynamic counterpart of (1.17), and Fife and Hilhorst proved that any time dependent solution converges to one of the local minimizers [8]. Many solutions in two and three dimensions have been found that match the morphological phases in diblock copolymers [18,22,21,23,24,11,12,25,28,31]. Global minimizers of J B are studied in [2,30,14,4,13,10] for various parameter ranges.…”