Calculus of Variations and Geometric Measure Theory
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A. Ruland - A. Tribuzio

On Scaling Laws for Multi-Well Nucleation Problems without Gauge Invariances

created by tribuzio on 22 Jun 2022

[BibTeX]

preprint

Inserted: 22 jun 2022

Year: 2022

ArXiv: 2206.05164 PDF

Abstract:

In this article we study scaling laws for simplified multi-well nucleation problems without gauge invariances which are motivated by models for shape-memory alloys. Seeking to explore the role of the order of lamination on the energy scaling for nucleation processes, we provide scaling laws for various model problems in two and three dimensions. In particular, we discuss (optimal) scaling results in the volume and the singular perturbation parameter for settings in which the surrounding parent phase is in the first, the second and the third order lamination convex hull of the wells of the nucleating phase. Furthermore, we provide a corresponding result for the setting of an infinite order laminate which arises in the context of the Tartar square. In particular, our results provide isoperimetric estimates in situations in which strong nonlocal anisotropies are present.

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