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MiS Preprint
73/1999
Branched microstructures: scaling and asymptotic self-similarity
Sergio Conti
Abstract
We address some properties of a scalar 2D model which has been proposed to describe microstructure in martensitic phase transformations, consisting in minimizing the bulk energy $$I[u] = \int^1_0 \int^h_0 u^2_x + \sigma|u_{yy}|$$ where $|u_y|=1$ a.e. and $u(0,\cdot )=0$. Kohn and Müller [R. V. Kohn and S. Müller, Comm. Pure and Appl. Math. 47, 405 (1994)] proved the existence of a minimizer for $\sigma > 0$, and obtained bounds on the total energy which 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 $$u^j (x,y)=\theta^{-2j/3} u(\theta^j x, \theta^{2j/3}y)$$ ($0<\theta<1$) has a strongly converging subsequence in $W^{1,2}$.
