2023/08/14 by Tobias Ried, Ried, Tobias, Carlos Román +1
Computer Science · Materials Science · Physics and Astronomy · #49N60 (82D40 74N15) #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Solidification and crystal growth phenomena #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2308.07239
openalex publication_date 2023/08/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the occurrence of domain branching in a class of (d+1)-dimensional sharp interface models featuring the competition between an interfacial energy and a non-local field energy. Our motivation comes from branching in uniaxial ferromagnets corresponding to d=2, but our result also covers twinning in shape-memory alloys near an austenite-twinned-martensite interface (corresponding to d=1, thereby recovering a result of Conti [Comm. Pure Appl. Math. 53 (2000), 1448-1474. https://doi.org/cgkvm4 ]). We prove that the energy density of a minimising configuration in a large cuboid domain QL,T=[-L,L]d× [0,T] scales like T-(2)/(3) (irrespective of the dimension d) if L≫ T(2)/(3). While this already provides a lot of insight into the nature of minimisers, it does not characterise their behaviour close to the top and bottom boundaries of the sample, i.e. in the region where the branching is concentrated. More significantly, we show that minimisers exhibit a self-similar behaviour near the top and bottom boundaries in a statistical sense through local energy bounds: for any minimiser in QL,T, the energy density in a small cuboid Qℓ,t centred at the top or bottom boundaries of the sample, with side lengths ℓ ≫ t(2)/(3), satisfies the same scaling law, that is, it is of order t-(2)/(3).