Branched microstructures: scaling and asymptotic self-similarity
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Submission date: 22. Dec. 1999
published in: Communications on pure and applied mathematics, 53 (2000) 11, p. 1448-1474
DOI number (of the published article): 3.0.CO;2-C - external>10.1002/1097-0312(200011)53:11<1448::AID-CPA6>3.0.CO;2-C
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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
where a.e. and . Kohn and Müller [R. V. Kohn and S. Müller, Comm. Pure and Appl. Math. 47, 405 (1994)] proved the existence of a minimizer for , 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
() has a strongly converging subsequence in .