Branched microstructures: Scaling and asymptotic self-similarity

We address some properties of a scalar two-dimensional model that has been proposed to describe microstructure in martensitic phase transformations, c onsisting of minimizing the bulk energy l[u] = integral(0)(lx) integral(0)(ly) u(x)(2) + sigma\u(yy)\ where \u(y)\ = 1 a.e. and u(0,.) = 0. Kohn and Muller [R. V. Kohn and S. Mu ller. Comm. Pure and Appl. Math. 47 (1994), 405] proved the existence of a minimizer for sigma > 0 and obtained bounds on the total energy that sugges ted self-similarity of the minimizer. Building upon their work, we derive a local upper bound on the energy and on the minimizer itself and show that the minimizer u is asymptotically self-similar in the sense that the sequen ce u(j)(x, y) = theta(-2j/3)u(theta(j)x.theta(2j/3)y) (0 < theta < 1) has a strongly converging subsequence in W-1,W-2. (C) 2000 John Wiley & Sons, Inc.