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Abstract: We consider a pattern-forming system in two space dimensions defined by anenergy G e. The functional G e models strong phase separation in AB diblockcopolymer melts, and patterns are represented by {0,1}-valued functions; thevalues 0 and 1 correspond to the A and B phases. The parameter e is the ratiobetween the intrinsic, material length scale and the scale of the domain. Weshow that in the limit as e goes to 0 any sequence u e of patterns withuniformly bounded energy G eu e becomes stripe-like: the pattern becomeslocally one-dimensional and resembles a periodic stripe pattern of periodicityOe. In the limit the stripes become uniform in width and increasinglystraight.Our results are formulated as a convergence theorem, which states that thefunctional G e Gamma-converges to a limit functional G 0. This limit functionalis defined on fields of rank-one projections, which represent the localdirection of the stripe pattern. The functional G 0 is only finite if theprojection field solves a version of the Eikonal equation, and in that case itis the L^2-norm of the divergence of the projection field, or equivalently theL^2-norm of the curvature of the field.At the level of patterns the converging objects are the jump measures|gradu e| combined with the projection fields corresponding to the tangentsto the jump set. The central inequality from Peletier and Roeger, Archive forRational Mechanics and Analysis, to appear, provides the initial estimate andleads to weak measure-function-pair convergence. We obtain strong convergenceby exploiting the non-intersection property of the jump set.



Author: Mark A. Peletier, Marco Veneroni

Source: https://arxiv.org/



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