Stripe patterns in a model for block copolymers

We consider a pattern-forming system in two space dimensions defined by an energy Gε. The functional Gε 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 ε is the ratio between the intrinsic, material length scale and the scale of the domain Ω. We show that in the limit ε → 0 any sequence uε of patterns with uniformly bounded energy Gε(uε) becomes stripe-like: the pattern becomes locally one-dimensional and resembles a periodic stripe pattern of periodicity O(ε). 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 Gε Gamma-converges to a limit functional G0. This limit functional is defined on fields of rank-one projections, which represent the local direction of the stripe pattern. The functional G0 is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L2-norm of the divergence of the projection field, or equivalently the L2-norm of the curvature of the field. At the level of patterns the converging objects are the jump measures |∇uε| 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, 193:475-537, 2009, 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.