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, consisting of minimizing the bulk energy$$I[u] = \int_0^{l_x} \int_0^{l_y} u_x^2 + \sigma |u_{yy}| $$where | u y | = 1 a.e. and u (0,·) = 0. Kohn and Müller [R. V. Kohn and S. Müller, Comm. Pure and Appl. Math. 47 (1994), 405] proved the existence of a minimizer for σ > 0 and obtained bounds on the total energy that suggested 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 sequence$$u^j(x,y) = \theta^{-2j/3} u(\theta^jx, \theta^{2j/3}y)$$(0 < θ < 1) has a strongly converging subsequence in W 1,2 . © 2000 John Wiley & Sons, Inc.