Title: On Rigidity, Flexibility and Scaling Laws
Abstract: Matrix-valued differential inclusions associated with highly non-convex sets arise in numerous physical applications. One such example is the modelling of shape-memory alloys. In these settings, often the exact differential inclusions display a striking dichotomy between rigidity and flexibility in that
* solutions of sufficiently high regularity obey the "characteristic equations'' determined by the differential inclusion, the solutions are rigid in this sense,
* low regularity solutions are highly non-unique and hence extremely flexible.
In this talk, I discuss quantitative versions of this dichotomy and explore the role of scaling in associated singular perturbation problems in the limit of vanishing strength of the regularization parameter in various set-ups, including the Tartar square and nucleation results. We highlight the connection between the complexity of the resulting microstructure and the order of lamination of the boundary/nucleation data.
This talk is based on joint work with Jamie Taylor, Antonio Tribuzio, Christian Zillinger and Barbara Zwicknagl.