第8回北東数学解析研究会
アブストラクト
Xiaofeng REN (Utah State Univesity, USA)
Lecture 1. Some mathematical aspects of the Ohta-Kawasaki density functional theory of block copolymers
- Lecture 1. Some mathematical aspects of the Ohta-Kawasaki density functional theory of block copolymers
Abstract. The Ohta-Kawasaki theory of diblock copolymers is a nonlocal variational problem that arises from a density functional theory of diblock copolymer melts. It was created to model the many morphology phases that appear in diblock copolymers, including the lamellar phase, the cylindrical phase, and the spherical phase. The Euler-Lagrange equation of the functional is an integro-differential equation. We will discuss the existence and stability in space of solutions of the lamellar phase. A wide stripe lamellar pattern is found by the Gamma-convergence method and A narrow stripe pattern is found by the Lyapunov-Schmidt reduction method. A stability analysis yields a wriggled lamellar pattern. We also find a multiple droplet pattern as a solution of the Gamma-limit that models the cylindrical phase of a diblock copolymer.
Lecture 2. On a phase field model driven by interface area and interface curvature.
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Abstract. It is known that the Allen-Cahn equation with Neumann boundary condition has no stable nonconstant solution on any convex domain (Casten-Holland, Matano). Many modifications have been proposed that yield richer structures of solutions. Examples include a chiral liquid crystal film problem (Selinger, Wang, Bruinsma and Knobler) and a bending membrane problem (Seul and Andelman). In this talk I will discuss an Allen-Cahn type problem modified by interface curvature, i.e. one adds the interface curvature into the free energy. Every solution of the original Allen-Cahn problem remains a
solution of the new problem. An unstable solution to the old problem becomes stable in the new problem, if the interface curvature part of the free energy is sufficiently large. There also exist solutions to the modified problem that have no counterparts in the original problem. I will show the existence of the so called bubble solutions in this category.
