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 where |uy| = 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 (0 < θ < 1) has a strongly converging subsequence in W1,2.