Abstract
AbstractIn this article we derive an (almost) optimal scaling law for a singular perturbation problem associated with the Tartar square. As in Winter (Eur J Appl Math 8(2):185–207, 1997), Chipot (Numer Math 83(3):325–352, 1999), our upper bound quantifies the well-known construction which is used in the literature to prove the flexibility of the Tartar square in the sense of the flexibility of approximate solutions to the differential inclusion. The main novelty of our article is the derivation of an (up to logarithmic powers matching) ansatz free lower bound which relies on a bootstrap argument in Fourier space and is related to a quantification of the interaction of a nonlinearity and a negative Sobolev space in the form of “a chain rule in a negative Sobolev space”. Both the lower and the upper bound arguments give evidence of the involved “infinite order of lamination”.
Funder
Deutsche Forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
Mechanical Engineering,Mathematics (miscellaneous),Analysis
Reference65 articles.
1. Aumann, R.J., Hart, S.: Bi-convexity and bi-martingales. Israel J. Math. 54(2), 159–180, 1986
2. Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Analysis and Continuum Mechanics, pp. 647–686. Springer, 1989
3. Bella, P., Goldman, M.: Nucleation barriers at corners for a cubic-to-tetragonal phase transformation. Proc. R. Soc. Edinb. Sect. A 145(4), 715–724, 2015
4. Bhattacharya, K.: Microstructure of Martensite: Why it Forms and How It Gives Rise to the Shape-Memory Effect. Oxford University Press, Oxford Series on Materials Modeling, 2003
5. Capella, A., Otto, F.: A rigidity result for a perturbation of the geometrically linear three-well problem. Commun. Pure Appl. Math. 62(12), 1632–1669, 2009
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