Branched microstructures: Scaling and asymptotic self-similarity

✍ Scribed by Sergio Conti

• Publisher: John Wiley and Sons
• Year: 2000
• Tongue: English
• Weight: 234 KB
• Volume: 53
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/1097-0312(200011)53:11<1448::aid-cpa6>3.0.co;2-c

✦ Synopsis

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 |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 selfsimilarity 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) = θ -2 j/3 u(θ j x, θ 2 j/3 y) (0 < θ < 1) has a strongly converging subsequence in W 1,2 .