嵌段共聚物模型中的条纹图案
Stripe patterns in a model for block copolymers
摘要：
我们考虑由能量G_e定义的二维空间中的图案形成系统。泛函G_e模型描述了AB型两嵌段共聚物熔体中的强相分离，其模式由{0，1}值函数表示，其中0和1分别对应于A相和B相。参数e是本征材料长度尺度与畴尺度之间的比率。我们证明了在极限下（当e趋于0时），任何具有一致有界能量G_e（u_e）的图案序列u_e都变成了条状：图案变成局部一维的，类似于周期为O（e）的周期性条纹图案。在极限情况下，条纹的宽度变得均匀，并且越来越直。我们的结果被表述为一个收敛定理，即泛函G_e γ-收敛于极限泛函G_0。该极限泛函定义在表示条纹图案的局部方向的秩一投影的域上。泛函G_0只有在投影场解了程函方程的一个版本时才是有限的，在这种情况下，它是投影场的散度的L^2范数，或者等价地是场的曲率的L^2范数。在模式层次上，收敛对象是跳测度|梯度（u_e）|与对应于跳跃集合的切线的投影场相结合。Peletier & Roeger的中心不等式（Archive for Rational Mechanics and Analysis）提供了初始估计，并导致弱测度函数对收敛。利用跳集的不交性，我们得到了强收敛性。
英文摘要：
We consider a pattern-forming system in two space dimensions defined by an energy G_e. The functional G_e models strong phase separation in AB diblock copolymer melts, and patterns are represented by {0,1}-valued functions; the values 0 and 1 correspond to the A and B phases. The parameter e is the ratio between the intrinsic, material length scale and the scale of the domain. We show that in the limit (as e goes to 0) any sequence u_e of patterns with uniformly bounded energy G_e(u_e) becomes stripe-like: the pattern becomes locally one-dimensional and resembles a periodic stripe pattern of periodicity O(e). In the limit the stripes become uniform in width and increasingly straight. Our results are formulated as a convergence theorem, which states that the functional G_e Gamma-converges to a limit functional G_0. This limit functional is defined on fields of rank-one projections, which represent the local direction of the stripe pattern. The functional G_0 is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L^2-norm of the divergence of the projection field, or equivalently the L^2-norm of the curvature of the field. At the level of patterns the converging objects are the jump measures |grad(u_e)| combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier & Roeger, (Archive for Rational Mechanics and Analysis, to appear), provides the initial estimate and leads to weak measure-function-pair convergence. We obtain strong convergence by exploiting the non-intersection property of the jump set.
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